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Timofeev Chapter 9 Integration Problems

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

\label{eq1}\frac{{{\left({4 \ {{x}^{3}}}+{2 \  x}\right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{4 \ {{x}^{4}}\ {\log \left({x}\right)}}-{3 \ {{\arccos \left({x}\right)}^{2}}}-{{x}^{2}}}{{12}\ {{x}^{4}}}(1)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*asin(x)^2, x)

\label{eq2}\frac{{{\left({6 \ {{x}^{2}}}+{12}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{9 \ {{x}^{3}}\ {{\arcsin \left({x}\right)}^{2}}}-{2 \ {{x}^{3}}}-{{12}\  x}}{27}(2)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*atan(x)^2, x)

\label{eq3}\frac{-{4 \ {\log \left({{{x}^{2}}+ 1}\right)}}+{{\left({3 \ {{x}^{4}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+{{\left(-{2 \ {{x}^{3}}}+{6 \  x}\right)}\ {\arctan \left({x}\right)}}+{{x}^{2}}}{12}(3)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(x)^2/x^5, x)

\label{eq4}\frac{{4 \ {{x}^{4}}\ {\log \left({{{x}^{2}}+ 1}\right)}}-{8 \ {{x}^{4}}\ {\log \left({x}\right)}}+{{\left({3 \ {{x}^{4}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+{{\left({6 \ {{x}^{3}}}-{2 \  x}\right)}\ {\arctan \left({x}\right)}}-{{x}^{2}}}{{1
2}\ {{x}^{4}}}(4)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*acsc(x)^2, x)

\label{eq5}\frac{{{\left({2 \ {{x}^{2}}}+ 4 \right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{4 \ {\log \left({x}\right)}}+{3 \ {{x}^{4}}\ {{acsc \left({x}\right)}^{2}}}+{{x}^{2}}}{12}(5)
Type: Union(Expression(Integer),...)
fricas
integrate(asec(x)^4/x^5, x)

\label{eq6}\frac{{{\left({{\left({{48}\ {{x}^{2}}}+{32}\right)}\ {{asec \left({x}\right)}^{3}}}+{{\left(-{{90}\ {{x}^{2}}}-{12}\right)}\ {asec \left({x}\right)}}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{{\left({{1
2}\ {{x}^{4}}}-{32}\right)}\ {{asec \left({x}\right)}^{4}}}+{{\left(-{{45}\ {{x}^{4}}}+{{72}\ {{x}^{2}}}+{24}\right)}\ {{asec \left({x}\right)}^{2}}}-{{45}\ {{x}^{2}}}- 3}{{128}\ {{x}^{4}}}(6)
Type: Union(Expression(Integer),...)
fricas
-- Problems 7 - 18 (p. 397-398)
integrate(asin(x)*sqrt(1-x^2), x)

\label{eq7}\frac{{2 \  x \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\arcsin \left({x}\right)}^{2}}-{{x}^{2}}}{4}(7)
Type: Union(Expression(Integer),...)
fricas
integrate(acos(x)*sqrt(1-x^2), x)

\label{eq8}\frac{{2 \  x \ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}-{{\arccos \left({x}\right)}^{2}}+{{x}^{2}}}{4}(8)
Type: Union(Expression(Integer),...)
fricas
integrate(x*acos(x)*sqrt(1-x^2), x)

\label{eq9}\frac{{{\left({3 \ {{x}^{2}}}- 3 \right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{x}^{3}}-{3 \  x}}{9}(9)
Type: Union(Expression(Integer),...)
fricas
integrate((1-x^2)^(3/2)*asin(x), x)

\label{eq10}\frac{{{\left(-{4 \ {{x}^{3}}}+{{10}\  x}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{3 \ {{\arcsin \left({x}\right)}^{2}}}+{{x}^{4}}-{5 \ {{x}^{2}}}}{16}(10)
Type: Union(Expression(Integer),...)
fricas
integrate(x*(1-x^2)^(3/2)*asin(x), x)

\label{eq11}\frac{{{\left(-{{15}\ {{x}^{4}}}+{{30}\ {{x}^{2}}}-{15}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{3 \ {{x}^{5}}}-{{1
0}\ {{x}^{3}}}+{{15}\  x}}{75}(11)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*(1-x^2)^(3/2)*acos(x), x)

\label{eq12}\frac{{{\left(-{{525}\ {{x}^{6}}}+{{840}\ {{x}^{4}}}-{{105}\ {{x}^{2}}}-{210}\right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}-{{75}\ {{x}^{7}}}+{{168}\ {{x}^{5}}}-{{35}\ {{x}^{3}}}-{{210}\  x}}{3675}(12)
Type: Union(Expression(Integer),...)
fricas
integrate((1-x^2)^(3/2)*acos(x)/x, x)

\label{eq13}\int^{
\displaystyle
x}{{\frac{{\left(-{{\%A}^{2}}+ 1 \right)}\ {\arccos \left({\%A}\right)}\ {\sqrt{-{{\%A}^{2}}+ 1}}}{\%A}}\ {d \%A}}(13)
Type: Union(Expression(Integer),...)
fricas
integrate((1-x^2)^(3/2)*asin(x)/x^6, x)

\label{eq14}\frac{{{\left(-{4 \ {{x}^{4}}}+{8 \ {{x}^{2}}}- 4 \right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{4 \ {{x}^{5}}\ {\log \left({x}\right)}}+{4 \ {{x}^{3}}}- x}{{20}\ {{x}^{5}}}(14)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*asin(x)/sqrt(1-x^2), x)

\label{eq15}\frac{-{2 \  x \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\arcsin \left({x}\right)}^{2}}+{{x}^{2}}}{4}(15)
Type: Union(Expression(Integer),...)
fricas
integrate(x^4*asin(x)/sqrt(1-x^2), x)

\label{eq16}\frac{{{\left(-{4 \ {{x}^{3}}}-{6 \  x}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{3 \ {{\arcsin \left({x}\right)}^{2}}}+{{x}^{4}}+{3 \ {{x}^{2}}}}{16}(16)
Type: Union(Expression(Integer),...)
fricas
integrate(x*asin(x)/(1-x^2)^(3/2), x)

\label{eq17}\frac{-{2 \ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\left(-{{x}^{2}}+ 1 \right)}\ {\log \left({x + 1}\right)}}+{{\left({{x}^{2}}- 1 \right)}\ {\log \left({x - 1}\right)}}}{{2 \ {{x}^{2}}}- 2}(17)
Type: Union(Expression(Integer),...)
fricas
integrate(x*acos(x)/(1-x^2)^(3/2), x)

\label{eq18}\frac{-{2 \ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\left({{x}^{2}}- 1 \right)}\ {\log \left({x + 1}\right)}}+{{\left(-{{x}^{2}}+ 1 \right)}\ {\log \left({x - 1}\right)}}}{{2 \ {{x}^{2}}}- 2}(18)
Type: Union(Expression(Integer),...)
fricas
integrate(asin(x)/(1-x^2)^(5/2), x)

\label{eq19}\frac{{{\left(-{4 \ {{x}^{3}}}+{6 \  x}\right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\left({2 \ {{x}^{4}}}-{4 \ {{x}^{2}}}+ 2 \right)}\ {\log \left({{{x}^{2}}- 1}\right)}}+{{x}^{2}}- 1}{{6 \ {{x}^{4}}}-{{1
2}\ {{x}^{2}}}+ 6}(19)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*asin(x)/(1-x^2)^(3/2), x)

\label{eq20}\frac{{{\left({2 \ {{x}^{2}}}- 4 \right)}\ {\arcsin \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\left(-{{x}^{2}}+ 1 \right)}\ {\log \left({x + 1}\right)}}+{{\left({{x}^{2}}- 1 \right)}\ {\log \left({x - 1}\right)}}-{2 \ {{x}^{3}}}+{2 \  x}}{{2 \ {{x}^{2}}}- 2}(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}-{{\frac{\arcsin \left({\%A}\right)}{{\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}\frac{{{\left(-{4 \ {{x}^{2}}}- 2 \right)}\ {\arccos \left({x}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}-{4 \ {{x}^{3}}\ {\log \left({x}\right)}}+ x}{6 \ {{x}^{3}}}(22)
Type: Union(Expression(Integer),...)
fricas
integrate(x*acos(x)^2*sqrt(1-x^2), x)

\label{eq23}\frac{{{\left({{\left({9 \ {{x}^{2}}}- 9 \right)}\ {{\arccos \left({x}\right)}^{2}}}-{2 \ {{x}^{2}}}+{14}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\left({6 \ {{x}^{3}}}-{{18}\  x}\right)}\ {\arccos \left({x}\right)}}}{27}(23)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*asin(x)^3/sqrt(1-x^2), x)

\label{eq24}\frac{{{\left(-{4 \  x \ {{\arcsin \left({x}\right)}^{3}}}+{6 \  x \ {\arcsin \left({x}\right)}}\right)}\ {\sqrt{-{{x}^{2}}+ 1}}}+{{\arcsin \left({x}\right)}^{4}}+{{\left({6 \ {{x}^{2}}}- 3 \right)}\ {{\arcsin \left({x}\right)}^{2}}}-{3 \ {{x}^{2}}}}{8}(24)
Type: Union(Expression(Integer),...)
fricas
-- Problems 23 - 26 (p. 404-405)
integrate(x*atan(x)/(1+x^2)^2, x)

\label{eq25}\frac{{{\left({{x}^{2}}- 1 \right)}\ {\arctan \left({x}\right)}}+ x}{{4 \ {{x}^{2}}}+ 4}(25)
Type: Union(Expression(Integer),...)
fricas
integrate(x*atan(x)/(1+x^2)^3, x)

\label{eq26}\frac{{{\left({3 \ {{x}^{4}}}+{6 \ {{x}^{2}}}- 5 \right)}\ {\arctan \left({x}\right)}}+{3 \ {{x}^{3}}}+{5 \  x}}{{{32}\ {{x}^{4}}}+{{64}\ {{x}^{2}}}+{32}}(26)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*atan(x)/(1+x^2), x)

\label{eq27}\frac{-{\log \left({{{x}^{2}}+ 1}\right)}-{{\arctan \left({x}\right)}^{2}}+{2 \  x \ {\arctan \left({x}\right)}}}{2}(27)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*atan(x)/(1+x^2), x)

\label{eq28}\int^{
\displaystyle
x}{{\frac{{{\%A}^{3}}\ {\arctan \left({\%A}\right)}}{{{\%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}\frac{{{\left({{x}^{2}}+ 1 \right)}\ {{\arctan \left({x}\right)}^{2}}}-{2 \  x \ {\arctan \left({x}\right)}}- 1}{{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}{{\frac{{{\%A}^{3}}\ {\arctan \left({\%A}\right)}}{{{\%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}{{\frac{{{\%A}^{5}}\ {\arctan \left({\%A}\right)}}{{{\%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}\frac{-{x \ {\log \left({{{x}^{2}}+ 1}\right)}}+{x \ {\log \left({x}\right)}}+{{\left({{x}^{2}}- 1 \right)}\ {\arctan \left({x}\right)}}}{x}(32)
Type: Union(Expression(Integer),...)
fricas
integrate((1+x^2)*atan(x)/x^5, x)

\label{eq33}\frac{{{\left(-{3 \ {{x}^{4}}}-{6 \ {{x}^{2}}}- 3 \right)}\ {\arctan \left({x}\right)}}-{3 \ {{x}^{3}}}- x}{{12}\ {{x}^{4}}}(33)
Type: Union(Expression(Integer),...)
fricas
integrate((1+x^2)^2*atan(x)/x^5, x)

\label{eq34}\int^{
\displaystyle
x}{{\frac{{\left({{\%A}^{4}}+{2 \ {{\%A}^{2}}}+ 1 \right)}\ {\arctan \left({\%A}\right)}}{{\%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}\frac{-{x \ {\log \left({{{x}^{2}}+ 1}\right)}}+{2 \  x \ {\log \left({x}\right)}}-{x \ {{\arctan \left({x}\right)}^{2}}}-{2 \ {\arctan \left({x}\right)}}}{2 \  x}(35)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(x)^2/x^3, x)

\label{eq36}\frac{-{{{x}^{2}}\ {\log \left({{{x}^{2}}+ 1}\right)}}+{2 \ {{x}^{2}}\ {\log \left({x}\right)}}+{{\left(-{{x}^{2}}- 1 \right)}\ {{\arctan \left({x}\right)}^{2}}}-{2 \  x \ {\arctan \left({x}\right)}}}{2 \ {{x}^{2}}}(36)
Type: Union(Expression(Integer),...)
fricas
integrate((1+x^2)*atan(x)^2/x^5, x)

\label{eq37}\frac{-{2 \ {{x}^{4}}\ {\log \left({{{x}^{2}}+ 1}\right)}}+{4 \ {{x}^{4}}\ {\log \left({x}\right)}}+{{\left(-{3 \ {{x}^{4}}}-{6 \ {{x}^{2}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+{{\left(-{6 \ {{x}^{3}}}-{2 \  x}\right)}\ {\arctan \left({x}\right)}}-{{x}^{2}}}{{12}\ {{x}^{4}}}(37)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*atan(x)^2/(1+x^2)^3, x)

\label{eq38}\frac{{{\left({5 \ {{x}^{4}}}-{6 \ {{x}^{2}}}- 3 \right)}\ {{\arctan \left({x}\right)}^{2}}}+{{\left({{10}\ {{x}^{3}}}+{6 \  x}\right)}\ {\arctan \left({x}\right)}}+{5 \ {{x}^{2}}}+ 4}{{{32}\ {{x}^{4}}}+{{64}\ {{x}^{2}}}+{32}}(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}{{\frac{{asec \left({\%A}\right)}\ {\sqrt{{{\%A}^{2}}- 1}}}{{\%A}^{2}}}\ {d \%A}}(39)
Type: Union(Expression(Integer),...)
fricas
integrate((-1+x^2)^(5/2)*acsc(x)/x^3, x)

\label{eq40}\frac{{{\left({4 \ {{x}^{4}}}-{{28}\ {{x}^{2}}}- 6 \right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}-{{28}\ {{x}^{2}}\ {\log \left({x}\right)}}-{{15}\ {{x}^{2}}\ {{acsc \left({x}\right)}^{2}}}+{2 \ {{x}^{4}}}+ 3}{{12}\ {{x}^{2}}}(40)
Type: Union(Expression(Integer),...)
fricas
integrate(asec(x)*sqrt(-1+x^2)/x^4, x)

\label{eq41}\frac{{{\left({3 \ {{x}^{2}}}- 3 \right)}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{3 \ {{x}^{2}}}- 1}{9 \ {{x}^{3}}}(41)
Type: Union(Expression(Integer),...)
fricas
integrate(asec(x)/(-1+x^2)^(5/2), x)

\label{eq42}\frac{{{\left({8 \ {{x}^{3}}}-{{12}\  x}\right)}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{{\left({5 \ {{x}^{4}}}-{{10}\ {{x}^{2}}}+ 5 \right)}\ {\log \left({x + 1}\right)}}+{{\left(-{5 \ {{x}^{4}}}+{{10}\ {{x}^{2}}}- 5 \right)}\ {\log \left({x - 1}\right)}}-{2 \ {{x}^{3}}}+{2 \  x}}{{{12}\ {{x}^{4}}}-{{24}\ {{x}^{2}}}+{12}}(42)
Type: Union(Expression(Integer),...)
fricas
integrate(x^2*asec(x)/(-1+x^2)^(5/2), x)

\label{eq43}\frac{-{4 \ {{x}^{3}}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{{\left(-{{x}^{4}}+{2 \ {{x}^{2}}}- 1 \right)}\ {\log \left({x + 1}\right)}}+{{\left({{x}^{4}}-{2 \ {{x}^{2}}}+ 1 \right)}\ {\log \left({x - 1}\right)}}-{2 \ {{x}^{3}}}+{2 \  x}}{{{12}\ {{x}^{4}}}-{{24}\ {{x}^{2}}}+{12}}(43)
Type: Union(Expression(Integer),...)
fricas
integrate(x^3*asec(x)/(-1+x^2)^(5/2), x)

\label{eq44}\frac{{{\left(-{6 \ {{x}^{2}}}+ 4 \right)}\ {asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{{\left({2 \ {{x}^{4}}}-{4 \ {{x}^{2}}}+ 2 \right)}\ {\log \left({{{x}^{2}}- 1}\right)}}+{{\left(-{4 \ {{x}^{4}}}+{8 \ {{x}^{2}}}- 4 \right)}\ {\log \left({x}\right)}}-{{x}^{2}}+ 1}{{6 \ {{x}^{4}}}-{{12}\ {{x}^{2}}}+ 6}(44)
Type: Union(Expression(Integer),...)
fricas
integrate(x^6*asec(x)/(-1+x^2)^(5/2), x)

\label{eq45}\int^{
\displaystyle
x}{{\frac{{{\%A}^{6}}\ {asec \left({\%A}\right)}}{{\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}\frac{{{asec \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+ 1}{x}(46)
Type: Union(Expression(Integer),...)
fricas
integrate(acsc(x)/(x^2*(-1+x^2)^(5/2)), x)

\label{eq47}\frac{{{\left({{32}\ {{x}^{4}}}-{{48}\ {{x}^{2}}}+{12}\right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{{\left(-{{11}\ {{x}^{5}}}+{{22}\ {{x}^{3}}}-{{11}\  x}\right)}\ {\log \left({x + 1}\right)}}+{{\left({{11}\ {{x}^{5}}}-{{22}\ {{x}^{3}}}+{{11}\  x}\right)}\ {\log \left({x - 1}\right)}}-{{10}\ {{x}^{4}}}+{{2
2}\ {{x}^{2}}}-{12}}{{{12}\ {{x}^{5}}}-{{24}\ {{x}^{3}}}+{{12}\  x}}(47)
Type: Union(Expression(Integer),...)
fricas
integrate(acsc(x)^4/(x^2*sqrt(-1+x^2)), x)

\label{eq48}\frac{{{\left({{acsc \left({x}\right)}^{4}}-{{12}\ {{acsc \left({x}\right)}^{2}}}+{24}\right)}\ {\sqrt{{{x}^{2}}- 1}}}-{4 \ {{acsc \left({x}\right)}^{3}}}+{{24}\ {acsc \left({x}\right)}}}{x}(48)
Type: Union(Expression(Integer),...)
fricas
integrate((-1+x^2)^(3/2)*asec(x)^2/x^5, x)

\label{eq49}\frac{{{\left({{\left(-{{40}\ {{x}^{2}}}+{16}\right)}\ {{asec \left({x}\right)}^{2}}}+{{17}\ {{x}^{2}}}- 2 \right)}\ {\sqrt{{{x}^{2}}- 1}}}+{8 \ {{x}^{4}}\ {{asec \left({x}\right)}^{3}}}+{{\left({{1
7}\ {{x}^{4}}}-{{40}\ {{x}^{2}}}+ 8 \right)}\ {asec \left({x}\right)}}}{{6
4}\ {{x}^{4}}}(49)
Type: Union(Expression(Integer),...)
fricas
integrate(asec(x)^3*sqrt(-1+x^2)/x^4, x)

\label{eq50}\frac{{{\left({{\left({9 \ {{x}^{2}}}- 9 \right)}\ {{asec \left({x}\right)}^{3}}}+{{\left(-{{42}\ {{x}^{2}}}+ 6 \right)}\ {asec \left({x}\right)}}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{{\left({{27}\ {{x}^{2}}}- 9 \right)}\ {{asec \left({x}\right)}^{2}}}-{{42}\ {{x}^{2}}}+ 2}{{27}\ {{x}^{3}}}(50)
Type: Union(Expression(Integer),...)
fricas
-- Problems 49 - 56 (p. 422)
integrate(asin(sqrt((-a+x)/(a+x))), x)

\label{eq51}\begin{array}{@{}l}
\displaystyle
{{\left(x + a \right)}\ {\arcsin \left({\sqrt{\frac{x - a}{x + a}}}\right)}}+ 
\
\
\displaystyle
{{\left(- x - a \right)}\ {\sqrt{\frac{2 \  a}{x + a}}}\ {\sqrt{\frac{x - a}{x + a}}}}
(51)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(sqrt((-a+x)/(a+x))), x)

\label{eq52}\frac{-{a \ {\log \left({{\sqrt{\frac{x - a}{x + a}}}+ 1}\right)}}+{a \ {\log \left({{\sqrt{\frac{x - a}{x + a}}}- 1}\right)}}+{2 \  x \ {\arctan \left({\sqrt{\frac{x - a}{x + a}}}\right)}}}{2}(52)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(x)/(1+x)^3, x)

\label{eq53}\frac{{{\left(-{{x}^{2}}-{2 \  x}- 1 \right)}\ {\log \left({{{x}^{2}}+ 1}\right)}}+{{\left({2 \ {{x}^{2}}}+{4 \  x}+ 2 \right)}\ {\log \left({x + 1}\right)}}-{4 \ {\arctan \left({x}\right)}}-{2 \  x}- 2}{{8 \ {{x}^{2}}}+{{16}\  x}+ 8}(53)
Type: Union(Expression(Integer),...)
fricas
integrate(atan(-a+x)/(a+x), x)

\label{eq54}\int^{
\displaystyle
x}-{{\frac{\arctan \left({a - \%A}\right)}{a + \%A}}\ {d \%A}}(54)
Type: Union(Expression(Integer),...)
fricas
integrate(asin(sqrt(1-x^2))/sqrt(1-x^2), x)

\label{eq55}-{\frac{{\arcsin \left({\sqrt{-{{x}^{2}}+ 1}}\right)}^{2}}{2}}(55)
Type: Union(Expression(Integer),...)
fricas
integrate(x*atan(sqrt(1+x^2))/sqrt(1+x^2), x)

\label{eq56}\frac{{2 \ {\sqrt{{{x}^{2}}+ 1}}\ {\arctan \left({\sqrt{{{x}^{2}}+ 1}}\right)}}-{\log \left({{{x}^{2}}+ 2}\right)}}{2}(56)
Type: Union(Expression(Integer),...)
fricas
integrate(asin(x)/(1-x)^(5/2), x)

\label{eq57}\frac{{{\left({{x}^{2}}-{2 \  x}+ 1 \right)}\ {\log \left({\frac{-{4 \ {\sqrt{-{{x}^{2}}+ 1}}\ {\sqrt{- x + 1}}}+{{\left(-{{x}^{2}}-{2 \  x}+ 3 \right)}\ {\sqrt{2}}}}{{{x}^{2}}-{2 \  x}+ 1}}\right)}}+{{\left(-{2 \ {\sqrt{2}}\ {\sqrt{-{{x}^{2}}+ 1}}}+{4 \ {\sqrt{2}}\ {\arcsin \left({x}\right)}}\right)}\ {\sqrt{- x + 1}}}}{{\left({6 \ {{x}^{2}}}-{{12}\  x}+ 6 \right)}\ {\sqrt{2}}}(57)
Type: Union(Expression(Integer),...)
fricas
integrate((-1+x)^(5/2)*acsc(x), x)

\label{eq58}\frac{{{30}\ {\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}\ {\log \left({\frac{{{\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}}+{{x}^{2}}- 1}{{{x}^{2}}- 1}}\right)}}-{{30}\ {\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}\ {\log \left({\frac{{{\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}}-{{x}^{2}}+ 1}{{{x}^{2}}- 1}}\right)}}+{{\left({{30}\ {{x}^{4}}}-{{120}\ {{x}^{3}}}+{{180}\ {{x}^{2}}}-{{120}\  x}+{30}\right)}\ {acsc \left({x}\right)}\ {\sqrt{{{x}^{2}}- 1}}}+{{12}\ {{x}^{4}}}-{{76}\ {{x}^{3}}}+{{320}\ {{x}^{2}}}+{{76}\  x}-{332}}{{105}\ {\sqrt{x - 1}}\ {\sqrt{{{x}^{2}}- 1}}}(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 4 sinh(x) + 24 cosh(x)sinh(x) + (60 cosh(x) + 12)sinh(x) + 3 3 (80 cosh(x) + 48 cosh(x))sinh(x) + 4 2 2 (60 cosh(x) + 72 cosh(x) + 12)sinh(x) + 5 3 6 (24 cosh(x) + 48 cosh(x) + 24 cosh(x))sinh(x) + 4 cosh(x) + 4 2 12 cosh(x) + 12 cosh(x) + 4 * 2 2 3 sinh(x) + 6 cosh(x)sinh(x) + 3 cosh(x) - 1 atan(----------------------------------------------) +--------------------------------------+ | 2 2 | - 2 sinh(x) - 2 cosh(x) + 6 |-------------------------------------- | 2 2 \|sinh(x) - 2 cosh(x)sinh(x) + cosh(x) + 2 2 (- 24 sinh(x) - 48 cosh(x)sinh(x) - 24 cosh(x) - 8) * 2 2 sinh(x) + 2 cosh(x)sinh(x) + cosh(x) - 1 atan(------------------------------------------) +--------------------------------------+ | 2 2 | - 2 sinh(x) - 2 cosh(x) + 6 |-------------------------------------- | 2 2 \|sinh(x) - 2 cosh(x)sinh(x) + cosh(x) + 4 3 2 2 sinh(x) + 4 cosh(x)sinh(x) + (6 cosh(x) + 2)sinh(x) + 3 4 2 (4 cosh(x) + 4 cosh(x))sinh(x) + cosh(x) + 2 cosh(x) + 1 * +--------------------------------------+ | 2 2 | - 2 sinh(x) - 2 cosh(x) + 6 |-------------------------------------- | 2 2 \|sinh(x) - 2 cosh(x)sinh(x) + cosh(x) / 6 5 2 4 6 sinh(x) + 36 cosh(x)sinh(x) + (90 cosh(x) + 18)sinh(x) + 3 3 (120 cosh(x) + 72 cosh(x))sinh(x) + 4 2 2 (90 cosh(x) + 108 cosh(x) + 18)sinh(x) + 5 3 6 (36 cosh(x) + 72 cosh(x) + 36 cosh(x))sinh(x) + 6 cosh(x) + 4 2 18 cosh(x) + 18 cosh(x) + 6
Type: Union(Expression(Integer),...)
fricas
integrate(acot(cosh(x))*cosh(x)/sinh(x)^4, x)
(60) 6 5 2 4 sinh(x) + 6 cosh(x)sinh(x) + (15 cosh(x) - 3)sinh(x) + 3 3 (20 cosh(x) - 12 cosh(x))sinh(x) + 4 2 2 (15 cosh(x) - 18 cosh(x) + 3)sinh(x) + 5 3 6 4 (6 cosh(x) - 12 cosh(x) + 6 cosh(x))sinh(x) + cosh(x) - 3 cosh(x) + 2 3 cosh(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) * 2 sinh(x) + 2 cosh(x) atan(------------------------------------------) 2 2 sinh(x) + 2 cosh(x)sinh(x) + cosh(x) + 1 + +-+ 4 +-+ 3 4 \|2 sinh(x) + 16 \|2 cosh(x)sinh(x) + +-+ 2 +-+ 2 (24 \|2 cosh(x) - 8 \|2 )sinh(x) + +-+ 3 +-+ +-+ 4 (16 \|2 cosh(x) - 16 \|2 cosh(x))sinh(x) + 4 \|2 cosh(x) + +-+ 2 +-+ - 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({-{\frac{2 \ {\cosh \left({x}\right)}}{{\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),...)




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