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Timofeev Chapter 9 Integration Problems

fricas

-- Problems 1 - 6 (p. 391-392)
integrate(acos(x)^2/x^5, x)

$\label{eq1}{\left( \begin{array}{@{}l} \displaystyle {{\left({4 \ {{x}^{3}}}+{2 \ x}\right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{4 \ {{x}^{4}}\ {\log \left({x}\right)}}- \ \ \displaystyle {3 \ {{\arccos \left({x}\right)}^{2}}}-{{x}^{2}}$

(1)

Type: Union(Expression(Integer),...)

fricas

integrate(x^2*asin(x)^2, x)

$\label{eq2}{\left( \begin{array}{@{}l} \displaystyle {{\left({6 \ {{x}^{2}}}+{12}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{9 \ {{x}^{3}}\ {{\arcsin \left({x}\right)}^{2}}}- \ \ \displaystyle {2 \ {{x}^{3}}}-{{12}\ x}$

(2)

Type: Union(Expression(Integer),...)

fricas

integrate(x^3*atan(x)^2, x)

$\label{eq3}{\left( \begin{array}{@{}l} \displaystyle -{4 \ {\log \left({{{x}^{2}}+ 1}\right)}}+{{\left({3 \ {{x}^{4}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+ \ \ \displaystyle {{\left(-{2 \ {{x}^{3}}}+{6 \ x}\right)}\ {\arctan \left({x}\right)}}+{{x}^{2}}$

(3)

Type: Union(Expression(Integer),...)

fricas

integrate(atan(x)^2/x^5, x)

$\label{eq4}{\left( \begin{array}{@{}l} \displaystyle {4 \ {{x}^{4}}\ {\log \left({{{x}^{2}}+ 1}\right)}}-{8 \ {{x}^{4}}\ {\log \left({x}\right)}}+ \ \ \displaystyle {{\left({3 \ {{x}^{4}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+{{\left({6 \ {{x}^{3}}}-{2 \ x}\right)}\ {\arctan \left({x}\right)}}-{{x}^{2}}$

(4)

Type: Union(Expression(Integer),...)

fricas

integrate(x^3*acsc(x)^2, x)

$\label{eq5}{\left( \begin{array}{@{}l} \displaystyle {{\left({2 \ {{x}^{2}}}+ 4 \right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{4 \ {\log \left({x}\right)}}+ \ \ \displaystyle {3 \ {{x}^{4}}\ {{acsc \left({x}\right)}^{2}}}+{{x}^{2}}$

(5)

Type: Union(Expression(Integer),...)

fricas

integrate(asec(x)^4/x^5, x)

$\label{eq6}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({{48}\ {{x}^{2}}}+{32}\right)}\ {{asec \left({x}\right)}^{3}}}+ \ \ \displaystyle {{\left(-{{90}\ {{x}^{2}}}-{12}\right)}\ {asec \left({x}\right)}}$

(6)

Type: Union(Expression(Integer),...)

fricas

-- Problems 7 - 18 (p. 397-398)
integrate(asin(x)*sqrt(1-x^2), x)

$\label{eq7}{{2 \ x \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\arcsin \left({x}\right)}^{2}}-{{x}^{2}}}\over 4$

(7)

Type: Union(Expression(Integer),...)

fricas

integrate(acos(x)*sqrt(1-x^2), x)

$\label{eq8}{{2 \ x \ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}-{{\arccos \left({x}\right)}^{2}}+{{x}^{2}}}\over 4$

(8)

Type: Union(Expression(Integer),...)

fricas

integrate(x*acos(x)*sqrt(1-x^2), x)

$\label{eq9}{{{\left({3 \ {{x}^{2}}}- 3 \right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{x}^{3}}-{3 \ x}}\over 9$

(9)

Type: Union(Expression(Integer),...)

fricas

integrate((1-x^2)^(3/2)*asin(x), x)

$\label{eq10}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{4 \ {{x}^{3}}}+{{10}\ x}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ \ \ \displaystyle {3 \ {{\arcsin \left({x}\right)}^{2}}}+{{x}^{4}}-{5 \ {{x}^{2}}}$

(10)

Type: Union(Expression(Integer),...)

fricas

integrate(x*(1-x^2)^(3/2)*asin(x), x)

$\label{eq11}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{{15}\ {{x}^{4}}}+{{30}\ {{x}^{2}}}-{15}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{3 \ {{x}^{5}}}- \ \ \displaystyle {{10}\ {{x}^{3}}}+{{15}\ x}$

(11)

Type: Union(Expression(Integer),...)

fricas

integrate(x^3*(1-x^2)^(3/2)*acos(x), x)

$\label{eq12}{\left( \begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle {\left(-{{525}\ {{x}^{6}}}+{{840}\ {{x}^{4}}}-{{105}\ {{x}^{2}}}-{210}\right)}\ {\arccos \left({x}\right)}\ \cdot \ \ \displaystyle {\sqrt{-{{x}^{2}}+ 1}}$

(12)

Type: Union(Expression(Integer),...)

fricas

integrate((1-x^2)^(3/2)*acos(x)/x, x)

$\label{eq13}\int^{ \displaystyle x}{{{{\left(-{{\%A}^{2}}+ 1 \right)}\ {\arccos \left({\%A}\right)}\ {\sqrt{-{{\%A}^{2}}+ 1}}}\over \%A}\ {d \%A}}$

(13)

Type: Union(Expression(Integer),...)

fricas

integrate((1-x^2)^(3/2)*asin(x)/x^6, x)

$\label{eq14}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{4 \ {{x}^{4}}}+{8 \ {{x}^{2}}}- 4 \right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ \ \ \displaystyle {4 \ {{x}^{5}}\ {\log \left({x}\right)}}+{4 \ {{x}^{3}}}- x$

(14)

Type: Union(Expression(Integer),...)

fricas

integrate(x^2*asin(x)/sqrt(1-x^2), x)

$\label{eq15}{-{2 \ x \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\arcsin \left({x}\right)}^{2}}+{{x}^{2}}}\over 4$

(15)

Type: Union(Expression(Integer),...)

fricas

integrate(x^4*asin(x)/sqrt(1-x^2), x)

$\label{eq16}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{4 \ {{x}^{3}}}-{6 \ x}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{3 \ {{\arcsin \left({x}\right)}^{2}}}+ \ \ \displaystyle {{x}^{4}}+{3 \ {{x}^{2}}}$

(16)

Type: Union(Expression(Integer),...)

fricas

integrate(x*asin(x)/(1-x^2)^(3/2), x)

$\label{eq17}{\left( \begin{array}{@{}l} \displaystyle -{2 \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\left(-{{x}^{2}}+ 1 \right)}\ {\log \left({x + 1}\right)}}+ \ \ \displaystyle {{\left({{x}^{2}}- 1 \right)}\ {\log \left({x - 1}\right)}}$

(17)

Type: Union(Expression(Integer),...)

fricas

integrate(x*acos(x)/(1-x^2)^(3/2), x)

$\label{eq18}{\left( \begin{array}{@{}l} \displaystyle -{2 \ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\left({{x}^{2}}- 1 \right)}\ {\log \left({x + 1}\right)}}+ \ \ \displaystyle {{\left(-{{x}^{2}}+ 1 \right)}\ {\log \left({x - 1}\right)}}$

(18)

Type: Union(Expression(Integer),...)

fricas

integrate(asin(x)/(1-x^2)^(5/2), x)

$\label{eq19}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{4 \ {{x}^{3}}}+{6 \ x}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ \ \ \displaystyle {{\left({2 \ {{x}^{4}}}-{4 \ {{x}^{2}}}+ 2 \right)}\ {\log \left({{{x}^{2}}- 1}\right)}}+{{x}^{2}}- 1$

(19)

Type: Union(Expression(Integer),...)

fricas

integrate(x^3*asin(x)/(1-x^2)^(3/2), x)

$\label{eq20}{\left( \begin{array}{@{}l} \displaystyle {{\left({2 \ {{x}^{2}}}- 4 \right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ \ \ \displaystyle {{\left(-{{x}^{2}}+ 1 \right)}\ {\log \left({x + 1}\right)}}+{{\left({{x}^{2}}- 1 \right)}\ {\log \left({x - 1}\right)}}-{2 \ {{x}^{3}}}+ \ \ \displaystyle {2 \ x}$

(20)

Type: Union(Expression(Integer),...)

fricas

-- Problems 19 - 22 (p. 401)
integrate(asin(x)/(x*(1-x^2)^(3/2)), x)

$\label{eq21}\int^{ \displaystyle x}-{{{\arcsin \left({\%A}\right)}\over{{\left({{\%A}^{3}}- \%A \right)}\ {\sqrt{-{{\%A}^{2}}+ 1}}}}\ {d \%A}}$

(21)

Type: Union(Expression(Integer),...)

fricas

integrate(acos(x)/(x^4*sqrt(1-x^2)), x)

$\label{eq22}{{{\left(-{4 \ {{x}^{2}}}- 2 \right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}-{4 \ {{x}^{3}}\ {\log \left({x}\right)}}+ x}\over{6 \ {{x}^{3}}}$

(22)

Type: Union(Expression(Integer),...)

fricas

integrate(x*acos(x)^2*sqrt(1-x^2), x)

$\label{eq23}{\left( \begin{array}{@{}l} \displaystyle {{\left({{\left({9 \ {{x}^{2}}}- 9 \right)}\ {{\arccos \left({x}\right)}^{2}}}-{2 \ {{x}^{2}}}+{14}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ \ \ \displaystyle {{\left({6 \ {{x}^{3}}}-{{18}\ x}\right)}\ {\arccos \left({x}\right)}}$

(23)

Type: Union(Expression(Integer),...)

fricas

integrate(x^2*asin(x)^3/sqrt(1-x^2), x)

$\label{eq24}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{4 \ x \ {{\arcsin \left({x}\right)}^{3}}}+{6 \ x \ {\arcsin \left({x}\right)}}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+ \ \ \displaystyle {{\arcsin \left({x}\right)}^{4}}+{{\left({6 \ {{x}^{2}}}- 3 \right)}\ {{\arcsin \left({x}\right)}^{2}}}-{3 \ {{x}^{2}}}$

(24)

Type: Union(Expression(Integer),...)

fricas

-- Problems 23 - 26 (p. 404-405)
integrate(x*atan(x)/(1+x^2)^2, x)

$\label{eq25}{{{\left({{x}^{2}}- 1 \right)}\ {\arctan \left({x}\right)}}+ x}\over{{4 \ {{x}^{2}}}+ 4}$

(25)

Type: Union(Expression(Integer),...)

fricas

integrate(x*atan(x)/(1+x^2)^3, x)

$\label{eq26}{{{\left({3 \ {{x}^{4}}}+{6 \ {{x}^{2}}}- 5 \right)}\ {\arctan \left({x}\right)}}+{3 \ {{x}^{3}}}+{5 \ x}}\over{{{32}\ {{x}^{4}}}+{{64}\ {{x}^{2}}}+{32}}$

(26)

Type: Union(Expression(Integer),...)

fricas

integrate(x^2*atan(x)/(1+x^2), x)

$\label{eq27}{-{\log \left({{{x}^{2}}+ 1}\right)}-{{\arctan \left({x}\right)}^{2}}+{2 \ x \ {\arctan \left({x}\right)}}}\over 2$

(27)

Type: Union(Expression(Integer),...)

fricas

integrate(x^3*atan(x)/(1+x^2), x)

$\label{eq28}\int^{ \displaystyle x}{{{{{\%A}^{3}}\ {\arctan \left({\%A}\right)}}\over{{{\%A}^{2}}+ 1}}\ {d \%A}}$

(28)

Type: Union(Expression(Integer),...)

fricas

-- Problems 27 - 32 (p. 407-408)
integrate(x^2*atan(x)/(1+x^2)^2, x)

$\label{eq29}{{{\left({{x}^{2}}+ 1 \right)}\ {{\arctan \left({x}\right)}^{2}}}-{2 \ x \ {\arctan \left({x}\right)}}- 1}\over{{4 \ {{x}^{2}}}+ 4}$

(29)

Type: Union(Expression(Integer),...)

fricas

integrate(x^3*atan(x)/(1+x^2)^2, x)

$\label{eq30}\int^{ \displaystyle x}{{{{{\%A}^{3}}\ {\arctan \left({\%A}\right)}}\over{{{\%A}^{4}}+{2 \ {{\%A}^{2}}}+ 1}}\ {d \%A}}$

(30)

Type: Union(Expression(Integer),...)

fricas

integrate(x^5*atan(x)/(1+x^2)^2, x)

$\label{eq31}\int^{ \displaystyle x}{{{{{\%A}^{5}}\ {\arctan \left({\%A}\right)}}\over{{{\%A}^{4}}+{2 \ {{\%A}^{2}}}+ 1}}\ {d \%A}}$

(31)

Type: Union(Expression(Integer),...)

fricas

integrate((1+x^2)*atan(x)/x^2, x)

$\label{eq32}{-{x \ {\log \left({{{x}^{2}}+ 1}\right)}}+{x \ {\log \left({x}\right)}}+{{\left({{x}^{2}}- 1 \right)}\ {\arctan \left({x}\right)}}}\over x$

(32)

Type: Union(Expression(Integer),...)

fricas

integrate((1+x^2)*atan(x)/x^5, x)

$\label{eq33}{{{\left(-{3 \ {{x}^{4}}}-{6 \ {{x}^{2}}}- 3 \right)}\ {\arctan \left({x}\right)}}-{3 \ {{x}^{3}}}- x}\over{{12}\ {{x}^{4}}}$

(33)

Type: Union(Expression(Integer),...)

fricas

integrate((1+x^2)^2*atan(x)/x^5, x)

$\label{eq34}\int^{ \displaystyle x}{{{{\left({{\%A}^{4}}+{2 \ {{\%A}^{2}}}+ 1 \right)}\ {\arctan \left({\%A}\right)}}\over{{\%A}^{5}}}\ {d \%A}}$

(34)

Type: Union(Expression(Integer),...)

fricas

-- Problems 33 - 36 (p. 409)
integrate(atan(x)/(x^2*(1+x^2)), x)

$\label{eq35}{-{x \ {\log \left({{{x}^{2}}+ 1}\right)}}+{2 \ x \ {\log \left({x}\right)}}-{x \ {{\arctan \left({x}\right)}^{2}}}-{2 \ {\arctan \left({x}\right)}}}\over{2 \ x}$

(35)

Type: Union(Expression(Integer),...)

fricas

integrate(atan(x)^2/x^3, x)

$\label{eq36}{\left( \begin{array}{@{}l} \displaystyle -{{{x}^{2}}\ {\log \left({{{x}^{2}}+ 1}\right)}}+{2 \ {{x}^{2}}\ {\log \left({x}\right)}}+ \ \ \displaystyle {{\left(-{{x}^{2}}- 1 \right)}\ {{\arctan \left({x}\right)}^{2}}}-{2 \ x \ {\arctan \left({x}\right)}}$

(36)

Type: Union(Expression(Integer),...)

fricas

integrate((1+x^2)*atan(x)^2/x^5, x)

$\label{eq37}{\left( \begin{array}{@{}l} \displaystyle -{2 \ {{x}^{4}}\ {\log \left({{{x}^{2}}+ 1}\right)}}+{4 \ {{x}^{4}}\ {\log \left({x}\right)}}+ \ \ \displaystyle {{\left(-{3 \ {{x}^{4}}}-{6 \ {{x}^{2}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+ \ \ \displaystyle {{\left(-{6 \ {{x}^{3}}}-{2 \ x}\right)}\ {\arctan \left({x}\right)}}-{{x}^{2}}$

(37)

Type: Union(Expression(Integer),...)

fricas

integrate(x^3*atan(x)^2/(1+x^2)^3, x)

$\label{eq38}{\left( \begin{array}{@{}l} \displaystyle {{\left({5 \ {{x}^{4}}}-{6 \ {{x}^{2}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+ \ \ \displaystyle {{\left({{10}\ {{x}^{3}}}+{6 \ x}\right)}\ {\arctan \left({x}\right)}}+{5 \ {{x}^{2}}}+ 4$

(38)

Type: Union(Expression(Integer),...)

fricas

-- Problems 37 - 43 (p. 412-414)
integrate(asec(x)*sqrt(-1+x^2)/x^2, x)

$\label{eq39}\int^{ \displaystyle x}{{{{asec \left({\%A}\right)}\ {\sqrt{{{\%A}^{2}}- 1}}}\over{{\%A}^{2}}}\ {d \%A}}$

(39)

Type: Union(Expression(Integer),...)

fricas

integrate((-1+x^2)^(5/2)*acsc(x)/x^3, x)

$\label{eq40}{\left( \begin{array}{@{}l} \displaystyle {{\left({4 \ {{x}^{4}}}-{{28}\ {{x}^{2}}}- 6 \right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}-{{28}\ {{x}^{2}}\ {\log \left({x}\right)}}- \ \ \displaystyle {{15}\ {{x}^{2}}\ {{acsc \left({x}\right)}^{2}}}+{2 \ {{x}^{4}}}+ 3$

(40)

Type: Union(Expression(Integer),...)

fricas

integrate(asec(x)*sqrt(-1+x^2)/x^4, x)

$\label{eq41}{{{\left({3 \ {{x}^{2}}}- 3 \right)}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{3 \ {{x}^{2}}}- 1}\over{9 \ {{x}^{3}}}$

(41)

Type: Union(Expression(Integer),...)

fricas

integrate(asec(x)/(-1+x^2)^(5/2), x)

$\label{eq42}{\left( \begin{array}{@{}l} \displaystyle {{\left({8 \ {{x}^{3}}}-{{12}\ x}\right)}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ \ \ \displaystyle {{\left({5 \ {{x}^{4}}}-{{10}\ {{x}^{2}}}+ 5 \right)}\ {\log \left({x + 1}\right)}}+ \ \ \displaystyle {{\left(-{5 \ {{x}^{4}}}+{{10}\ {{x}^{2}}}- 5 \right)}\ {\log \left({x - 1}\right)}}-{2 \ {{x}^{3}}}+{2 \ x}$

(42)

Type: Union(Expression(Integer),...)

fricas

integrate(x^2*asec(x)/(-1+x^2)^(5/2), x)

$\label{eq43}{\left( \begin{array}{@{}l} \displaystyle -{4 \ {{x}^{3}}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ \ \ \displaystyle {{\left(-{{x}^{4}}+{2 \ {{x}^{2}}}- 1 \right)}\ {\log \left({x + 1}\right)}}+ \ \ \displaystyle {{\left({{x}^{4}}-{2 \ {{x}^{2}}}+ 1 \right)}\ {\log \left({x - 1}\right)}}-{2 \ {{x}^{3}}}+{2 \ x}$

(43)

Type: Union(Expression(Integer),...)

fricas

integrate(x^3*asec(x)/(-1+x^2)^(5/2), x)

$\label{eq44}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{6 \ {{x}^{2}}}+ 4 \right)}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ \ \ \displaystyle {{\left({2 \ {{x}^{4}}}-{4 \ {{x}^{2}}}+ 2 \right)}\ {\log \left({{{x}^{2}}- 1}\right)}}+ \ \ \displaystyle {{\left(-{4 \ {{x}^{4}}}+{8 \ {{x}^{2}}}- 4 \right)}\ {\log \left({x}\right)}}-{{x}^{2}}+ 1$

(44)

Type: Union(Expression(Integer),...)

fricas

integrate(x^6*asec(x)/(-1+x^2)^(5/2), x)

$\label{eq45}\int^{ \displaystyle x}{{{{{\%A}^{6}}\ {asec \left({\%A}\right)}}\over{{\left({{\%A}^{4}}-{2 \ {{\%A}^{2}}}+ 1 \right)}\ {\sqrt{{{\%A}^{2}}- 1}}}}\ {d \%A}}$

(45)

Type: Union(Expression(Integer),...)

fricas

-- Problems 44 - 48 (p. 416-417)
integrate(asec(x)/(x^2*sqrt(-1+x^2)), x)

$\label{eq46}{{{asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ 1}\over x$

(46)

Type: Union(Expression(Integer),...)

fricas

integrate(acsc(x)/(x^2*(-1+x^2)^(5/2)), x)

$\label{eq47}{\left( \begin{array}{@{}l} \displaystyle {{\left({{32}\ {{x}^{4}}}-{{48}\ {{x}^{2}}}+{12}\right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ \ \ \displaystyle {{\left(-{{11}\ {{x}^{5}}}+{{22}\ {{x}^{3}}}-{{11}\ x}\right)}\ {\log \left({x + 1}\right)}}+ \ \ \displaystyle {{\left({{11}\ {{x}^{5}}}-{{22}\ {{x}^{3}}}+{{11}\ x}\right)}\ {\log \left({x - 1}\right)}}-{{10}\ {{x}^{4}}}+{{22}\ {{x}^{2}}}- \ \ \displaystyle {12}$

(47)

Type: Union(Expression(Integer),...)

fricas

integrate(acsc(x)^4/(x^2*sqrt(-1+x^2)), x)

$\label{eq48}{\left( \begin{array}{@{}l} \displaystyle {{\left({{acsc \left({x}\right)}^{4}}-{{12}\ {{acsc \left({x}\right)}^{2}}}+{24}\right)}\ {\sqrt{{{x}^{2}}- 1}}}-{4 \ {{acsc \left({x}\right)}^{3}}}+ \ \ \displaystyle {{24}\ {acsc \left({x}\right)}}$

(48)

Type: Union(Expression(Integer),...)

fricas

integrate((-1+x^2)^(3/2)*asec(x)^2/x^5, x)

$\label{eq49}{\left( \begin{array}{@{}l} \displaystyle {{\left({{\left(-{{40}\ {{x}^{2}}}+{16}\right)}\ {{asec \left({x}\right)}^{2}}}+{{17}\ {{x}^{2}}}- 2 \right)}\ {\sqrt{{{x}^{2}}- 1}}}+ \ \ \displaystyle {8 \ {{x}^{4}}\ {{asec \left({x}\right)}^{3}}}+{{\left({{17}\ {{x}^{4}}}-{{40}\ {{x}^{2}}}+ 8 \right)}\ {asec \left({x}\right)}}$

(49)

Type: Union(Expression(Integer),...)

fricas

integrate(asec(x)^3*sqrt(-1+x^2)/x^4, x)

$\label{eq50}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({9 \ {{x}^{2}}}- 9 \right)}\ {{asec \left({x}\right)}^{3}}}+ \ \ \displaystyle {{\left(-{{42}\ {{x}^{2}}}+ 6 \right)}\ {asec \left({x}\right)}}$

(50)

Type: Union(Expression(Integer),...)

fricas

-- Problems 49 - 56 (p. 422)
integrate(asin(sqrt((-a+x)/(a+x))), x)

$\label{eq51}{{\left(x + a \right)}\ {\arcsin \left({\sqrt{{x - a}\over{x + a}}}\right)}}+{{\left(- x - a \right)}\ {\sqrt{{2 \ a}\over{x + a}}}\ {\sqrt{{x - a}\over{x + a}}}}$

(51)

Type: Union(Expression(Integer),...)

fricas

integrate(atan(sqrt((-a+x)/(a+x))), x)

$\label{eq52}{\left( \begin{array}{@{}l} \displaystyle -{a \ {\log \left({{\sqrt{{x - a}\over{x + a}}}+ 1}\right)}}+{a \ {\log \left({{\sqrt{{x - a}\over{x + a}}}- 1}\right)}}+ \ \ \displaystyle {2 \ x \ {\arctan \left({\sqrt{{x - a}\over{x + a}}}\right)}}$

(52)

Type: Union(Expression(Integer),...)

fricas

integrate(atan(x)/(1+x)^3, x)

$\label{eq53}{\left( \begin{array}{@{}l} \displaystyle {{\left(-{{x}^{2}}-{2 \ x}- 1 \right)}\ {\log \left({{{x}^{2}}+ 1}\right)}}+ \ \ \displaystyle {{\left({2 \ {{x}^{2}}}+{4 \ x}+ 2 \right)}\ {\log \left({x + 1}\right)}}-{4 \ {\arctan \left({x}\right)}}-{2 \ x}- 2$

(53)

Type: Union(Expression(Integer),...)

fricas

integrate(atan(-a+x)/(a+x), x)

$\label{eq54}\int^{ \displaystyle x}-{{{\arctan \left({a - \%A}\right)}\over{a + \%A}}\ {d \%A}}$

(54)

Type: Union(Expression(Integer),...)

fricas

integrate(asin(sqrt(1-x^2))/sqrt(1-x^2), x)

$\label{eq55}-{{{\arcsin \left({\sqrt{-{{x}^{2}}+ 1}}\right)}^{2}}\over 2}$

(55)

Type: Union(Expression(Integer),...)

fricas

integrate(x*atan(sqrt(1+x^2))/sqrt(1+x^2), x)

$\label{eq56}{{2 \ {\sqrt{{{x}^{2}}+ 1}}\ {\arctan \left({\sqrt{{{x}^{2}}+ 1}}\right)}}-{\log \left({{{x}^{2}}+ 2}\right)}}\over 2$

(56)

Type: Union(Expression(Integer),...)

fricas

integrate(asin(x)/(1-x)^(5/2), x)

$\label{eq57}{\left( \begin{array}{@{}l} \displaystyle {{\left({{x}^{2}}-{2 \ x}+ 1 \right)}\ {\log{\left({{-{4 \ {\sqrt{-{{x}^{2}}+ 1}}\ {\sqrt{- x + 1}}}+{{\left(-{{x}^{2}}-{2 \ x}+ 3 \right)}\ {\sqrt{2}}}}\over{{{x}^{2}}-{2 \ x}+ 1}}\right)}}}+ \ \ \displaystyle {{\left(-{2 \ {\sqrt{2}}\ {\sqrt{-{{x}^{2}}+ 1}}}+{4 \ {\sqrt{2}}\ {\arcsin \left({x}\right)}}\right)}\ {\sqrt{- x + 1}}}$

(57)

Type: Union(Expression(Integer),...)

fricas

integrate((-1+x)^(5/2)*acsc(x), x)

$\label{eq58}{\left( \begin{array}{@{}l} \displaystyle {{30}\ {\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}\ {\log \left({{{{\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}}+{{x}^{2}}- 1}\over{{{x}^{2}}- 1}}\right)}}- \ \ \displaystyle {{30}\ {\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}\ {\log \left({{{{\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}}-{{x}^{2}}+ 1}\over{{{x}^{2}}- 1}}\right)}}+ \ \ \displaystyle {{\left({{30}\ {{x}^{4}}}-{{120}\ {{x}^{3}}}+{{180}\ {{x}^{2}}}-{{120}\ x}+{30}\right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ \ \ \displaystyle {{12}\ {{x}^{4}}}-{{76}\ {{x}^{3}}}+{{320}\ {{x}^{2}}}+{{76}\ x}-{332}$

(58)

Type: Union(Expression(Integer),...)

fricas

-- Problems 57 - 59 (p. 427)

fricas

)set output tex off

fricas

)set output algebra on

integrate(asin(sinh(x))*sech(x)^4, x)

   (59)
                     6                   5               2             4
           - 4sinh(x)  - 24cosh(x)sinh(x)  + (- 60cosh(x)  - 12)sinh(x)
         + 
                       3                    3
           (- 80cosh(x)  - 48cosh(x))sinh(x)
         + 
                       4            2             2
           (- 60cosh(x)  - 72cosh(x)  - 12)sinh(x)
         + 
                       5            3                               6
           (- 24cosh(x)  - 48cosh(x)  - 24cosh(x))sinh(x) - 4cosh(x)
         + 
                      4            2
           - 12cosh(x)  - 12cosh(x)  - 4
      *
                 +-------------------------------------+
                 |               2           2
                 |     - 2sinh(x)  - 2cosh(x)  + 6
                 |-------------------------------------
                 |       2                            2
                \|sinh(x)  - 2cosh(x)sinh(x) + cosh(x)
         atan(-------------------------------------------)
                      2                             2
              3sinh(x)  + 6cosh(x)sinh(x) + 3cosh(x)  - 1
     + 
                     2                               2
         (- 24sinh(x)  - 48cosh(x)sinh(x) - 24cosh(x)  - 8)
      *
                     2                            2
              sinh(x)  + 2cosh(x)sinh(x) + cosh(x)  - 1
         atan(-----------------------------------------)
                +-------------------------------------+
                |               2           2
                |     - 2sinh(x)  - 2cosh(x)  + 6
                |-------------------------------------
                |       2                            2
               \|sinh(x)  - 2cosh(x)sinh(x) + cosh(x)
     + 
                  4                  3            2            2
           sinh(x)  + 4cosh(x)sinh(x)  + (6cosh(x)  + 2)sinh(x)
         + 
                    3                             4           2
           (4cosh(x)  + 4cosh(x))sinh(x) + cosh(x)  + 2cosh(x)  + 1
      *
          +-------------------------------------+
          |               2           2
          |     - 2sinh(x)  - 2cosh(x)  + 6
          |-------------------------------------
          |       2                            2
         \|sinh(x)  - 2cosh(x)sinh(x) + cosh(x)
  /
               6                   5             2             4
       6sinh(x)  + 36cosh(x)sinh(x)  + (90cosh(x)  + 18)sinh(x)
     + 
                  3                    3
       (120cosh(x)  + 72cosh(x))sinh(x)
     + 
                 4             2             2
       (90cosh(x)  + 108cosh(x)  + 18)sinh(x)
     + 
                 5            3                               6            4
       (36cosh(x)  + 72cosh(x)  + 36cosh(x))sinh(x) + 6cosh(x)  + 18cosh(x)
     + 
                2
       18cosh(x)  + 6

Type: Union(Expression(Integer),...)

fricas

integrate(acot(cosh(x))*cosh(x)/sinh(x)^4, x)

   (60)
                  6                  5             2            4
           sinh(x)  + 6cosh(x)sinh(x)  + (15cosh(x)  - 3)sinh(x)
         + 
                     3                    3
           (20cosh(x)  - 12cosh(x))sinh(x)
         + 
                     4            2            2
           (15cosh(x)  - 18cosh(x)  + 3)sinh(x)
         + 
                    5            3                             6           4
           (6cosh(x)  - 12cosh(x)  + 6cosh(x))sinh(x) + cosh(x)  - 3cosh(x)
         + 
                   2
           3cosh(x)  - 1
      *
         log
                   +-+             2         +-+
                (9\|2  - 12)sinh(x)  + (- 16\|2  + 24)cosh(x)sinh(x)
              + 
                   +-+             2     +-+
                (9\|2  - 12)cosh(x)  + 3\|2  - 4
           /
                     2          2
              sinh(x)  + cosh(x)  + 3
     + 
                +-+       3      +-+              2      +-+       2
           - 32\|2 sinh(x)  - 96\|2 cosh(x)sinh(x)  - 96\|2 cosh(x) sinh(x)
         + 
                +-+       3
           - 32\|2 cosh(x)
      *
                         2sinh(x) + 2cosh(x)
         atan(-----------------------------------------)
                     2                            2
              sinh(x)  + 2cosh(x)sinh(x) + cosh(x)  + 1
     + 
         +-+       4      +-+              3       +-+       2     +-+        2
       4\|2 sinh(x)  + 16\|2 cosh(x)sinh(x)  + (24\|2 cosh(x)  - 8\|2 )sinh(x)
     + 
           +-+       3      +-+                    +-+       4     +-+       2
       (16\|2 cosh(x)  - 16\|2 cosh(x))sinh(x) + 4\|2 cosh(x)  - 8\|2 cosh(x)
     + 
         +-+
       4\|2
  /
          +-+       6      +-+              5
       12\|2 sinh(x)  + 72\|2 cosh(x)sinh(x)
     + 
            +-+       2      +-+        4
       (180\|2 cosh(x)  - 36\|2 )sinh(x)
     + 
            +-+       3       +-+               3
       (240\|2 cosh(x)  - 144\|2 cosh(x))sinh(x)
     + 
            +-+       4       +-+       2      +-+        2
       (180\|2 cosh(x)  - 216\|2 cosh(x)  + 36\|2 )sinh(x)
     + 
           +-+       5       +-+       3      +-+
       (72\|2 cosh(x)  - 144\|2 cosh(x)  + 72\|2 cosh(x))sinh(x)
     + 
          +-+       6      +-+       4      +-+       2      +-+
       12\|2 cosh(x)  - 36\|2 cosh(x)  + 36\|2 cosh(x)  - 12\|2

Type: Union(Expression(Integer),...)

fricas

)set output algebra off

fricas

)set output tex on

integrate(%e^x*asin(tanh(x)), x)

$\label{eq59}\begin{array}{@{}l} \displaystyle -{\log \left({-{{2 \ {\cosh \left({x}\right)}}\over{{\sinh \left({x}\right)}-{\cosh \left({x}\right)}}}}\right)}+ \ \ \displaystyle {{\left({\sinh \left({x}\right)}+{\cosh \left({x}\right)}\right)}\ {\arctan \left({\sinh \left({x}\right)}\right)}}$

(59)

Type: Union(Expression(Integer),...)