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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),...)