3.150 \(\int x^3 (a+b \tanh ^{-1}(\frac {c}{x}))^3 \, dx\)

Optimal. Leaf size=203 \[ -2 b^2 c^4 \log \left (2-\frac {2}{\frac {c}{x}+1}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{4} b^2 c^2 x^2 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{4} c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-b c^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{4} b c^3 x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{4} x^4 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{4} b c x^3 \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+b^3 c^4 \text {Li}_2\left (\frac {2}{\frac {c}{x}+1}-1\right )-\frac {1}{4} b^3 c^4 \coth ^{-1}\left (\frac {x}{c}\right )+\frac {1}{4} b^3 c^3 x \]

[Out]

1/4*b^3*c^3*x-1/4*b^3*c^4*arccoth(x/c)+1/4*b^2*c^2*x^2*(a+b*arccoth(x/c))-b*c^4*(a+b*arccoth(x/c))^2+3/4*b*c^3
*x*(a+b*arccoth(x/c))^2+1/4*b*c*x^3*(a+b*arccoth(x/c))^2-1/4*c^4*(a+b*arccoth(x/c))^3+1/4*x^4*(a+b*arccoth(x/c
))^3-2*b^2*c^4*(a+b*arccoth(x/c))*ln(2-2/(1+c/x))+b^3*c^4*polylog(2,-1+2/(1+c/x))

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Rubi [F]  time = 4.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^3 \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*(a + b*ArcTanh[c/x])^3,x]

[Out]

(3*a^2*b*c^3*x)/8 - (5*a*b^2*c^3*x)/16 + (b^3*c^3*x)/16 - (3*a^2*b*c^2*x^2)/16 + (3*a*b^2*c^2*x^2)/16 + (a^2*b
*c*x^3)/8 + (b^3*c^4*Log[1 - c/x])/32 - (3*a*b^2*c^3*x*Log[1 - c/x])/8 + (3*a*b^2*c^2*x^2*Log[1 - c/x])/16 - (
a*b^2*c*x^3*Log[1 - c/x])/8 + (5*b^2*c^3*(1 - c/x)*x*(2*a - b*Log[1 - c/x]))/32 + (b^2*c^2*x^2*(2*a - b*Log[1
- c/x]))/32 - (5*b*c^4*(2*a - b*Log[1 - c/x])^2)/64 + (3*b*c^3*(1 - c/x)*x*(2*a - b*Log[1 - c/x])^2)/32 + (3*b
*c^2*x^2*(2*a - b*Log[1 - c/x])^2)/64 + (b*c*x^3*(2*a - b*Log[1 - c/x])^2)/32 - (c^4*(2*a - b*Log[1 - c/x])^3)
/32 + (x^4*(2*a - b*Log[1 - c/x])^3)/32 + (3*a*b^2*c^3*x*Log[1 + c/x])/8 + (3*a*b^2*c^2*x^2*Log[1 + c/x])/16 +
(a*b^2*c*x^3*Log[1 + c/x])/8 + (3*a^2*b*x^4*Log[1 + c/x])/8 - (3*a*b^2*x^4*Log[1 - c/x]*Log[1 + c/x])/8 + (5*
a*b^2*c^4*Log[c - x])/16 + (3*a*b^2*c^4*Log[1 + c/x]*Log[c - x])/8 - (3*b*c^4*(2*a - b*Log[1 - c/x])^2*Log[c/x
])/32 + (11*a*b^2*c^4*Log[x])/8 + (3*a*b^2*c^4*Log[c - x]*Log[x/c])/8 - (3*a^2*b*c^4*Log[c + x])/8 + (5*a*b^2*
c^4*Log[c + x])/16 + (3*a*b^2*c^4*Log[1 - c/x]*Log[c + x])/8 - (3*a*b^2*c^4*Log[(c - x)/(2*c)]*Log[c + x])/8 +
(3*a*b^2*c^4*Log[-(x/c)]*Log[c + x])/8 - (3*a*b^2*c^4*Log[c - x]*Log[(c + x)/(2*c)])/8 + (11*a*b^2*c^4*Log[(c
+ x)/x])/16 - (b^3*c^4*Log[(c + x)/x])/32 + (3*a*b^2*c^3*x*Log[(c + x)/x])/8 - (5*b^3*c^3*(1 + c/x)*x*Log[(c
+ x)/x])/32 - (3*a*b^2*c^2*x^2*Log[(c + x)/x])/16 + (b^3*c^2*x^2*Log[(c + x)/x])/32 + (a*b^2*c*x^3*Log[(c + x)
/x])/8 - (3*a*b^2*c^4*Log[(c + x)/x]^2)/16 + (5*b^3*c^4*Log[(c + x)/x]^2)/64 + (3*b^3*c^3*(1 + c/x)*x*Log[(c +
x)/x]^2)/32 - (3*b^3*c^2*x^2*Log[(c + x)/x]^2)/64 + (b^3*c*x^3*Log[(c + x)/x]^2)/32 + (3*a*b^2*x^4*Log[(c + x
)/x]^2)/16 + (3*b^3*c^4*Log[-(c/x)]*Log[(c + x)/x]^2)/32 - (b^3*c^4*Log[(c + x)/x]^3)/32 + (b^3*x^4*Log[(c + x
)/x]^3)/32 + (3*b^2*c^4*(2*a - b*Log[1 - c/x])*PolyLog[2, 1 - c/x])/16 - (3*a*b^2*c^4*PolyLog[2, (c - x)/(2*c)
])/8 - (3*a*b^2*c^4*PolyLog[2, -(c/x)])/8 + (11*b^3*c^4*PolyLog[2, -(c/x)])/32 - (11*b^3*c^4*PolyLog[2, c/x])/
32 - (3*a*b^2*c^4*PolyLog[2, (c + x)/(2*c)])/8 + (3*b^3*c^4*Log[(c + x)/x]*PolyLog[2, (c + x)/x])/16 + (3*a*b^
2*c^4*PolyLog[2, 1 - x/c])/8 + (3*a*b^2*c^4*PolyLog[2, 1 + x/c])/8 + (3*b^3*c^4*PolyLog[3, 1 - c/x])/16 - (3*b
^3*c^4*PolyLog[3, (c + x)/x])/16 + (3*b^3*Defer[Int][x^3*Log[1 - c/x]^2*Log[1 + c/x], x])/8 - (3*b^3*Defer[Int
][x^3*Log[1 - c/x]*Log[1 + c/x]^2, x])/8

Rubi steps

\begin {align*} \int x^3 \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3 \, dx &=\int \left (\frac {1}{8} x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} b x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} b^2 x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \log ^2\left (1+\frac {c}{x}\right )+\frac {1}{8} b^3 x^3 \log ^3\left (1+\frac {c}{x}\right )\right ) \, dx\\ &=\frac {1}{8} \int x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3 \, dx+\frac {1}{8} (3 b) \int x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 b^2\right ) \int x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} b^3 \int x^3 \log ^3\left (1+\frac {c}{x}\right ) \, dx\\ &=-\left (\frac {1}{8} \operatorname {Subst}\left (\int \frac {(2 a-b \log (1-c x))^3}{x^5} \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{8} (3 b) \int \left (4 a^2 x^3 \log \left (1+\frac {c}{x}\right )-4 a b x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+b^2 x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )\right ) \, dx+\frac {1}{8} \left (3 b^2\right ) \int \left (2 a x^3 \log ^2\left (1+\frac {c}{x}\right )-b x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right )\right ) \, dx-\frac {1}{8} b^3 \operatorname {Subst}\left (\int \frac {\log ^3(1+c x)}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{2} \left (3 a^2 b\right ) \int x^3 \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{4} \left (3 a b^2\right ) \int x^3 \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{2} \left (3 a b^2\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{32} (3 b c) \operatorname {Subst}\left (\int \frac {(2 a-b \log (1-c x))^2}{x^4 (1-c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{32} \left (3 b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log ^2(1+c x)}{x^4 (1+c x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} (3 b) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )^4} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{4} \left (3 a b^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(1+c x)}{x^5} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c x^3 \log \left (1-\frac {c}{x}\right )}{4 (-c-x)} \, dx+\frac {1}{2} \left (3 a b^2\right ) \int \frac {c x^3 \log \left (1+\frac {c}{x}\right )}{-4 c+4 x} \, dx-\frac {1}{32} \left (3 b^3\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^4} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a^2 b c\right ) \int \frac {x^2}{1+\frac {c}{x}} \, dx\\ &=\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} (3 b) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\left (\frac {1}{c}-\frac {x}{c}\right )^4} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^4} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{32} (3 b c) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (3 a^2 b c\right ) \int \frac {x^3}{c+x} \, dx+\frac {1}{8} \left (3 a b^2 c\right ) \int \frac {x^3 \log \left (1-\frac {c}{x}\right )}{-c-x} \, dx-\frac {1}{8} \left (3 a b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{x^4 (1+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 a b^2 c\right ) \int \frac {x^3 \log \left (1+\frac {c}{x}\right )}{-4 c+4 x} \, dx+\frac {1}{32} \left (3 b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^3} \, dx,x,1+\frac {c}{x}\right )\\ &=\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{32} (3 b c) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (3 a^2 b c\right ) \int \left (c^2-c x+x^2-\frac {c^3}{c+x}\right ) \, dx+\frac {1}{16} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c\right ) \int \left (-c^2 \log \left (1-\frac {c}{x}\right )+c x \log \left (1-\frac {c}{x}\right )-x^2 \log \left (1-\frac {c}{x}\right )+\frac {c^3 \log \left (1-\frac {c}{x}\right )}{c+x}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {\log (1+c x)}{x^4}-\frac {c \log (1+c x)}{x^3}+\frac {c^2 \log (1+c x)}{x^2}-\frac {c^3 \log (1+c x)}{x}+\frac {c^4 \log (1+c x)}{1+c x}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 a b^2 c\right ) \int \left (\frac {1}{4} c^2 \log \left (1+\frac {c}{x}\right )-\frac {c^3 \log \left (1+\frac {c}{x}\right )}{4 (c-x)}+\frac {1}{4} c x \log \left (1+\frac {c}{x}\right )+\frac {1}{4} x^2 \log \left (1+\frac {c}{x}\right )\right ) \, dx-\frac {1}{16} \left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^3} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^3} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b c^2\right ) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )\\ &=\frac {3}{8} a^2 b c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a^2 b c^4 \log (c+x)-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{16} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^3} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{8} \left (3 a b^2 c\right ) \int x^2 \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c\right ) \int x^2 \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{x^4} \, dx,x,\frac {1}{x}\right )-\frac {1}{16} \left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^3} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b c^2\right ) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{16} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{32} \left (3 b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^2\right ) \int x \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^2\right ) \int x \log \left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{x^3} \, dx,x,\frac {1}{x}\right )+\frac {1}{16} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b c^3\right ) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{8} \left (3 a b^2 c^3\right ) \int \log \left (1-\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^3\right ) \int \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{x^2} \, dx,x,\frac {1}{x}\right )+\frac {1}{32} \left (3 b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (1-\frac {c}{x}\right )}{c+x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (1+\frac {c}{x}\right )}{c-x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 a b^2 c^5\right ) \operatorname {Subst}\left (\int \frac {\log (1+c x)}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{8} a^2 b c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {1}{8} a^2 b c x^3-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{16} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{32} \left (3 b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{\left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (a b^2 c^2\right ) \int \frac {x}{1-\frac {c}{x}} \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \frac {x}{1+\frac {c}{x}} \, dx-\frac {1}{8} \left (a b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 (1+c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{32} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (\frac {1}{c}-\frac {x}{c}\right )^2} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{32} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{16} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\left (-\frac {1}{c}+\frac {x}{c}\right )^2} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b c^3\right ) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{16} \left (b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{32} \left (3 b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x \left (\frac {1}{c}-\frac {x}{c}\right )} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{16} \left (3 b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{16} \left (3 a b^2 c^3\right ) \int \frac {1}{1-\frac {c}{x}} \, dx+\frac {1}{16} \left (3 a b^2 c^3\right ) \int \frac {1}{1+\frac {c}{x}} \, dx+\frac {1}{16} \left (3 a b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (1+c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{16} \left (b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (-\frac {1}{c}+\frac {x}{c}\right )} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{16} \left (3 b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b c^4\right ) \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {1}{\left (1-\frac {c}{x}\right ) x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {1}{\left (1+\frac {c}{x}\right ) x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x (1+c x)} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (3 a b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^5\right ) \int \frac {\log (c-x)}{\left (1+\frac {c}{x}\right ) x^2} \, dx-\frac {1}{8} \left (3 a b^2 c^5\right ) \int \frac {\log (c+x)}{\left (1-\frac {c}{x}\right ) x^2} \, dx\\ &=\frac {3}{8} a^2 b c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {1}{8} a^2 b c x^3-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {3}{8} a b^2 c^4 \log (x)-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {3}{16} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \frac {x^2}{-c+x} \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \frac {x^2}{c+x} \, dx-\frac {1}{8} \left (a b^2 c^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^3}-\frac {c}{x^2}+\frac {c^2}{x}-\frac {c^3}{1+c x}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{32} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \left (\frac {c^2}{(-1+x)^2}-\frac {c^2}{-1+x}+\frac {c^2}{x}\right ) \, dx,x,1-\frac {c}{x}\right )-\frac {1}{32} \left (b^3 c^2\right ) \operatorname {Subst}\left (\int \left (\frac {c^2}{(-1+x)^2}-\frac {c^2}{-1+x}+\frac {c^2}{x}\right ) \, dx,x,1+\frac {c}{x}\right )+\frac {1}{16} \left (b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{32} \left (3 b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{16} \left (3 a b^2 c^3\right ) \int \frac {x}{-c+x} \, dx+\frac {1}{16} \left (3 a b^2 c^3\right ) \int \frac {x}{c+x} \, dx+\frac {1}{16} \left (3 a b^2 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {c}{x}+\frac {c^2}{1+c x}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{16} \left (b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{16} \left (b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{16} \left (b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{32} \left (3 b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{32} \left (3 b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{-\frac {1}{c}+\frac {x}{c}} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{16} \left (3 b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{32} \left (3 c^4\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{16} \left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{32} \left (3 b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {2 a-b \log (x)}{x} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{16} \left (3 b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\log (1-x) (2 a-b \log (x))}{x} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {1}{-c+x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {1}{c+x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{16} \left (b^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^4\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\log \left (\frac {c+x}{x}\right )\right )+\frac {1}{32} \left (3 b^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {c}{x}\right )-\frac {1}{16} \left (3 b^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\log (1-x) \log (x)}{x} \, dx,x,1+\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^5\right ) \int \left (\frac {\log (c-x)}{c x}-\frac {\log (c-x)}{c (c+x)}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c^5\right ) \int \left (-\frac {\log (c+x)}{c (c-x)}-\frac {\log (c+x)}{c x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^5\right ) \operatorname {Subst}\left (\int \frac {1}{1+c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {3}{8} a^2 b c^3 x-\frac {5}{16} a b^2 c^3 x+\frac {1}{16} b^3 c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {1}{16} a b^2 c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {1}{32} b^3 c^4 \log \left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {5}{64} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{32} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a b^2 c^4 \log (c-x)+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {11}{8} a b^2 c^4 \log (x)-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)+\frac {11}{16} a b^2 c^4 \log \left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log \left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {5}{64} b^3 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {3}{16} b^2 c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {11}{32} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {3}{16} b^3 c^4 \text {Li}_2\left (\frac {c}{x}\right )+\frac {3}{16} b^3 c^4 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \left (c-\frac {c^2}{c-x}+x\right ) \, dx+\frac {1}{8} \left (a b^2 c^2\right ) \int \left (-c+x+\frac {c^2}{c+x}\right ) \, dx-\frac {1}{16} \left (3 a b^2 c^3\right ) \int \left (1-\frac {c}{c-x}\right ) \, dx+\frac {1}{16} \left (3 a b^2 c^3\right ) \int \left (1-\frac {c}{c+x}\right ) \, dx-\frac {1}{16} \left (b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{32} \left (3 b^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {1}{c}-\frac {x}{c}} \, dx,x,1-\frac {c}{x}\right )+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log (c-x)}{x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log (c-x)}{c+x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log (c+x)}{c-x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log (c+x)}{x} \, dx+\frac {1}{16} \left (3 b^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-\frac {c}{x}\right )-\frac {1}{16} \left (3 b^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+\frac {c}{x}\right )\\ &=\frac {3}{8} a^2 b c^3 x-\frac {5}{16} a b^2 c^3 x+\frac {1}{16} b^3 c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {3}{16} a b^2 c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {1}{32} b^3 c^4 \log \left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {5}{64} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{32} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {5}{16} a b^2 c^4 \log (c-x)+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {11}{8} a b^2 c^4 \log (x)+\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {x}{c}\right )-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {5}{16} a b^2 c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {c+x}{2 c}\right )+\frac {11}{16} a b^2 c^4 \log \left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log \left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {5}{64} b^3 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {3}{16} b^2 c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {11}{32} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {11}{32} b^3 c^4 \text {Li}_2\left (\frac {c}{x}\right )+\frac {3}{16} b^3 c^4 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{16} b^3 c^4 \text {Li}_3\left (1-\frac {c}{x}\right )-\frac {3}{16} b^3 c^4 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (-\frac {-c-x}{2 c}\right )}{c-x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (\frac {c-x}{2 c}\right )}{c+x} \, dx-\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (-\frac {x}{c}\right )}{c+x} \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \int \frac {\log \left (\frac {x}{c}\right )}{c-x} \, dx\\ &=\frac {3}{8} a^2 b c^3 x-\frac {5}{16} a b^2 c^3 x+\frac {1}{16} b^3 c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {3}{16} a b^2 c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {1}{32} b^3 c^4 \log \left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {5}{64} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{32} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {5}{16} a b^2 c^4 \log (c-x)+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {11}{8} a b^2 c^4 \log (x)+\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {x}{c}\right )-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {5}{16} a b^2 c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {c+x}{2 c}\right )+\frac {11}{16} a b^2 c^4 \log \left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log \left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {5}{64} b^3 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {3}{16} b^2 c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {11}{32} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {11}{32} b^3 c^4 \text {Li}_2\left (\frac {c}{x}\right )+\frac {3}{16} b^3 c^4 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^4 \text {Li}_2\left (1-\frac {x}{c}\right )+\frac {3}{8} a b^2 c^4 \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {3}{16} b^3 c^4 \text {Li}_3\left (1-\frac {c}{x}\right )-\frac {3}{16} b^3 c^4 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx+\frac {1}{8} \left (3 a b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c-x\right )+\frac {1}{8} \left (3 a b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c+x\right )\\ &=\frac {3}{8} a^2 b c^3 x-\frac {5}{16} a b^2 c^3 x+\frac {1}{16} b^3 c^3 x-\frac {3}{16} a^2 b c^2 x^2+\frac {3}{16} a b^2 c^2 x^2+\frac {1}{8} a^2 b c x^3+\frac {1}{32} b^3 c^4 \log \left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^3 x \log \left (1-\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1-\frac {c}{x}\right )-\frac {1}{8} a b^2 c x^3 \log \left (1-\frac {c}{x}\right )+\frac {5}{32} b^2 c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )+\frac {1}{32} b^2 c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )-\frac {5}{64} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{32} b c^3 \left (1-\frac {c}{x}\right ) x \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {3}{64} b c^2 x^2 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2+\frac {1}{32} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2-\frac {1}{32} c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {1}{32} x^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^3+\frac {3}{8} a b^2 c^3 x \log \left (1+\frac {c}{x}\right )+\frac {3}{16} a b^2 c^2 x^2 \log \left (1+\frac {c}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (1+\frac {c}{x}\right )+\frac {3}{8} a^2 b x^4 \log \left (1+\frac {c}{x}\right )-\frac {3}{8} a b^2 x^4 \log \left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right )+\frac {5}{16} a b^2 c^4 \log (c-x)+\frac {3}{8} a b^2 c^4 \log \left (1+\frac {c}{x}\right ) \log (c-x)-\frac {3}{32} b c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right )^2 \log \left (\frac {c}{x}\right )+\frac {11}{8} a b^2 c^4 \log (x)+\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {x}{c}\right )-\frac {3}{8} a^2 b c^4 \log (c+x)+\frac {5}{16} a b^2 c^4 \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (1-\frac {c}{x}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log \left (\frac {c-x}{2 c}\right ) \log (c+x)+\frac {3}{8} a b^2 c^4 \log \left (-\frac {x}{c}\right ) \log (c+x)-\frac {3}{8} a b^2 c^4 \log (c-x) \log \left (\frac {c+x}{2 c}\right )+\frac {11}{16} a b^2 c^4 \log \left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log \left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^3 x \log \left (\frac {c+x}{x}\right )-\frac {5}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c^2 x^2 \log \left (\frac {c+x}{x}\right )+\frac {1}{8} a b^2 c x^3 \log \left (\frac {c+x}{x}\right )-\frac {3}{16} a b^2 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {5}{64} b^3 c^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^3 \left (1+\frac {c}{x}\right ) x \log ^2\left (\frac {c+x}{x}\right )-\frac {3}{64} b^3 c^2 x^2 \log ^2\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 c x^3 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{16} a b^2 x^4 \log ^2\left (\frac {c+x}{x}\right )+\frac {3}{32} b^3 c^4 \log \left (-\frac {c}{x}\right ) \log ^2\left (\frac {c+x}{x}\right )-\frac {1}{32} b^3 c^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {1}{32} b^3 x^4 \log ^3\left (\frac {c+x}{x}\right )+\frac {3}{16} b^2 c^4 \left (2 a-b \log \left (1-\frac {c}{x}\right )\right ) \text {Li}_2\left (1-\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (\frac {c-x}{2 c}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (-\frac {c}{x}\right )+\frac {11}{32} b^3 c^4 \text {Li}_2\left (-\frac {c}{x}\right )-\frac {11}{32} b^3 c^4 \text {Li}_2\left (\frac {c}{x}\right )-\frac {3}{8} a b^2 c^4 \text {Li}_2\left (\frac {c+x}{2 c}\right )+\frac {3}{16} b^3 c^4 \log \left (\frac {c+x}{x}\right ) \text {Li}_2\left (\frac {c+x}{x}\right )+\frac {3}{8} a b^2 c^4 \text {Li}_2\left (1-\frac {x}{c}\right )+\frac {3}{8} a b^2 c^4 \text {Li}_2\left (1+\frac {x}{c}\right )+\frac {3}{16} b^3 c^4 \text {Li}_3\left (1-\frac {c}{x}\right )-\frac {3}{16} b^3 c^4 \text {Li}_3\left (\frac {c+x}{x}\right )+\frac {1}{8} \left (3 b^3\right ) \int x^3 \log ^2\left (1-\frac {c}{x}\right ) \log \left (1+\frac {c}{x}\right ) \, dx-\frac {1}{8} \left (3 b^3\right ) \int x^3 \log \left (1-\frac {c}{x}\right ) \log ^2\left (1+\frac {c}{x}\right ) \, dx\\ \end {align*}

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Mathematica [A]  time = 0.68, size = 286, normalized size = 1.41 \[ \frac {1}{8} \left (2 a^3 x^4+2 b \tanh ^{-1}\left (\frac {c}{x}\right ) \left (3 a^2 x^4+2 a b c x \left (3 c^2+x^2\right )-8 b^2 c^4 \log \left (1-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )+b^2 \left (c^2 x^2-c^4\right )\right )+3 a^2 b c^4 \log \left (1-\frac {c}{x}\right )-3 a^2 b c^4 \log \left (\frac {c+x}{x}\right )+6 a^2 b c^3 x+2 a^2 b c x^3-2 a b^2 c^4+2 a b^2 c^2 x^2-16 a b^2 c^4 \log \left (\frac {c}{x \sqrt {1-\frac {c^2}{x^2}}}\right )+2 b^2 \tanh ^{-1}\left (\frac {c}{x}\right )^2 \left (3 a \left (x^4-c^4\right )+b c \left (-4 c^3+3 c^2 x+x^3\right )\right )+8 b^3 c^4 \text {Li}_2\left (e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )+2 b^3 \left (x^4-c^4\right ) \tanh ^{-1}\left (\frac {c}{x}\right )^3+2 b^3 c^3 x\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3*(a + b*ArcTanh[c/x])^3,x]

[Out]

(-2*a*b^2*c^4 + 6*a^2*b*c^3*x + 2*b^3*c^3*x + 2*a*b^2*c^2*x^2 + 2*a^2*b*c*x^3 + 2*a^3*x^4 + 2*b^2*(b*c*(-4*c^3
+ 3*c^2*x + x^3) + 3*a*(-c^4 + x^4))*ArcTanh[c/x]^2 + 2*b^3*(-c^4 + x^4)*ArcTanh[c/x]^3 + 2*b*ArcTanh[c/x]*(3
*a^2*x^4 + 2*a*b*c*x*(3*c^2 + x^2) + b^2*(-c^4 + c^2*x^2) - 8*b^2*c^4*Log[1 - E^(-2*ArcTanh[c/x])]) + 3*a^2*b*
c^4*Log[1 - c/x] - 16*a*b^2*c^4*Log[c/(Sqrt[1 - c^2/x^2]*x)] - 3*a^2*b*c^4*Log[(c + x)/x] + 8*b^3*c^4*PolyLog[
2, E^(-2*ArcTanh[c/x])])/8

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{3} \operatorname {artanh}\left (\frac {c}{x}\right )^{3} + 3 \, a b^{2} x^{3} \operatorname {artanh}\left (\frac {c}{x}\right )^{2} + 3 \, a^{2} b x^{3} \operatorname {artanh}\left (\frac {c}{x}\right ) + a^{3} x^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c/x))^3,x, algorithm="fricas")

[Out]

integral(b^3*x^3*arctanh(c/x)^3 + 3*a*b^2*x^3*arctanh(c/x)^2 + 3*a^2*b*x^3*arctanh(c/x) + a^3*x^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {artanh}\left (\frac {c}{x}\right ) + a\right )}^{3} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c/x))^3,x, algorithm="giac")

[Out]

integrate((b*arctanh(c/x) + a)^3*x^3, x)

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maple [C]  time = 0.71, size = 1410, normalized size = 6.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arctanh(c/x))^3,x)

[Out]

3/16*I*c^4*b^3*Pi*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/(1-c^2/x^2)))^3*arctanh(c/x)^2+3/16*I*c^4*b^3*Pi*
csgn(I*(1+c/x)^2/(-1+c^2/x^2))^3*arctanh(c/x)^2+c^4*b^3*arctanh(c/x)^2-3/16*I*c^4*b^3*Pi*csgn(I/(1+(1+c/x)^2/(
1-c^2/x^2)))*csgn(I*(1+c/x)^2/(-1+c^2/x^2))*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/(1-c^2/x^2)))*arctanh(c
/x)^2+3/4*a*b^2*x^4*arctanh(c/x)^2+3/8*c^4*b^3*arctanh(c/x)^2*ln(c/x-1)+3/8*c^4*a^2*b*ln(c/x-1)-3/8*c^4*a^2*b*
ln(1+c/x)-2*c^4*b^3*arctanh(c/x)*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))+3/4*c^4*b^3*arctanh(c/x)^2*ln((1+c/x)/(1-c^2/
x^2)^(1/2))+3/16*c^4*a*b^2*ln(c/x-1)^2+1/4*c^4*b^3/(c/x+1-(1-c^2/x^2)^(1/2))*(1-c^2/x^2)^(1/2)-2*c^4*a*b^2*ln(
c/x)+3/16*c^4*a*b^2*ln(1+c/x)^2+1/4*c*b^3*arctanh(c/x)^2*x^3+3/4*c^3*b^3*arctanh(c/x)^2*x+1/4*c^2*b^3*arctanh(
c/x)*x^2-3/8*c^4*b^3*arctanh(c/x)^2*ln(1+c/x)-1/4*c^4*b^3/((1-c^2/x^2)^(1/2)+c/x+1)*(1-c^2/x^2)^(1/2)+c^4*a*b^
2*ln(c/x-1)+c^4*a*b^2*ln(1+c/x)+3/4*c^3*x*a^2*b+1/4*c*a^2*b*x^3+1/4*c^2*a*b^2*x^2+3/4*a^2*b*x^4*arctanh(c/x)+3
/16*I*c^4*b^3*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))^2*csgn(I*(1+c/x)^2/(-1+c^2/x^2))*arctanh(c/x)^2-3/8*I*c^4*b
^3*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^3*arctanh(c/x)^2+3/8*I*c^4*b^3*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^2*
arctanh(c/x)^2+3/16*I*c^4*b^3*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/(1+(1+c/x)^2/
(1-c^2/x^2)))^2*arctanh(c/x)^2+3/8*I*c^4*b^3*Pi*csgn(I*(1+c/x)/(1-c^2/x^2)^(1/2))*csgn(I*(1+c/x)^2/(-1+c^2/x^2
))^2*arctanh(c/x)^2+1/4*b^3*x^4*arctanh(c/x)^3-2*c^4*b^3*dilog(1+(1+c/x)/(1-c^2/x^2)^(1/2))-1/4*c^4*b^3*arctan
h(c/x)^3-1/4*c^4*b^3*arctanh(c/x)-3/16*I*c^4*b^3*Pi*csgn(I*(1+c/x)^2/(-1+c^2/x^2))*csgn(I*(1+c/x)^2/(-1+c^2/x^
2)/(1+(1+c/x)^2/(1-c^2/x^2)))^2*arctanh(c/x)^2+1/4*x^4*a^3-3/4*c^4*a*b^2*arctanh(c/x)*ln(1+c/x)-3/8*I*c^4*b^3*
Pi*arctanh(c/x)^2+2*c^4*b^3*dilog((1+c/x)/(1-c^2/x^2)^(1/2))-3/8*c^4*a*b^2*ln(c/x-1)*ln(1/2+1/2*c/x)+1/2*c*a*b
^2*arctanh(c/x)*x^3+3/2*c^3*a*b^2*arctanh(c/x)*x-3/8*c^4*a*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)+3/8*c^4*a*b^2*ln(-1/
2*c/x+1/2)*ln(1/2+1/2*c/x)+3/4*c^4*a*b^2*arctanh(c/x)*ln(c/x-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3}{4} \, a b^{2} x^{4} \operatorname {artanh}\left (\frac {c}{x}\right )^{2} + \frac {1}{4} \, a^{3} x^{4} + \frac {1}{8} \, {\left (6 \, x^{4} \operatorname {artanh}\left (\frac {c}{x}\right ) - {\left (3 \, c^{3} \log \left (c + x\right ) - 3 \, c^{3} \log \left (-c + x\right ) - 6 \, c^{2} x - 2 \, x^{3}\right )} c\right )} a^{2} b + \frac {1}{16} \, {\left ({\left (3 \, c^{2} \log \left (c + x\right )^{2} + 3 \, c^{2} \log \left (-c + x\right )^{2} + 16 \, c^{2} \log \left (c + x\right ) + 4 \, x^{2} - 2 \, {\left (3 \, c^{2} \log \left (c + x\right ) - 8 \, c^{2}\right )} \log \left (-c + x\right )\right )} c^{2} - 4 \, {\left (3 \, c^{3} \log \left (c + x\right ) - 3 \, c^{3} \log \left (-c + x\right ) - 6 \, c^{2} x - 2 \, x^{3}\right )} c \operatorname {artanh}\left (\frac {c}{x}\right )\right )} a b^{2} + \frac {1}{32} \, {\left (16 \, c^{5} \int -\frac {\log \left (c + x\right )}{c^{2} - x^{2}}\,{d x} + 40 \, c^{4} \int -\frac {x \log \left (c + x\right )}{c^{2} - x^{2}}\,{d x} - 2 \, {\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c^{3} - {\left (c^{4} - x^{4}\right )} \log \left (c + x\right )^{3} + {\left (c^{4} - x^{4}\right )} \log \left (-c + x\right )^{3} + 2 \, {\left (c^{2} \log \left (-c^{2} + x^{2}\right ) + x^{2}\right )} c^{2} + 8 \, c^{2} \int -\frac {x^{3} \log \left (c + x\right )}{c^{2} - x^{2}}\,{d x} + 2 \, {\left (3 \, c^{3} x + c x^{3}\right )} \log \left (c + x\right )^{2} - {\left (8 \, c^{4} - 6 \, c^{3} x - 2 \, c x^{3} + 3 \, {\left (c^{4} - x^{4}\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )^{2} - {\left (4 \, c^{2} x^{2} - 3 \, {\left (c^{4} - x^{4}\right )} \log \left (c + x\right )^{2} + 4 \, {\left (4 \, c^{4} + 3 \, c^{3} x + c x^{3}\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )\right )} b^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arctanh(c/x))^3,x, algorithm="maxima")

[Out]

3/4*a*b^2*x^4*arctanh(c/x)^2 + 1/4*a^3*x^4 + 1/8*(6*x^4*arctanh(c/x) - (3*c^3*log(c + x) - 3*c^3*log(-c + x) -
6*c^2*x - 2*x^3)*c)*a^2*b + 1/16*((3*c^2*log(c + x)^2 + 3*c^2*log(-c + x)^2 + 16*c^2*log(c + x) + 4*x^2 - 2*(
3*c^2*log(c + x) - 8*c^2)*log(-c + x))*c^2 - 4*(3*c^3*log(c + x) - 3*c^3*log(-c + x) - 6*c^2*x - 2*x^3)*c*arct
anh(c/x))*a*b^2 + 1/32*(16*c^5*integrate(-log(c + x)/(c^2 - x^2), x) + 40*c^4*integrate(-x*log(c + x)/(c^2 - x
^2), x) - 2*(c*log(c + x) - c*log(-c + x) - 2*x)*c^3 - (c^4 - x^4)*log(c + x)^3 + (c^4 - x^4)*log(-c + x)^3 +
2*(c^2*log(-c^2 + x^2) + x^2)*c^2 + 8*c^2*integrate(-x^3*log(c + x)/(c^2 - x^2), x) + 2*(3*c^3*x + c*x^3)*log(
c + x)^2 - (8*c^4 - 6*c^3*x - 2*c*x^3 + 3*(c^4 - x^4)*log(c + x))*log(-c + x)^2 - (4*c^2*x^2 - 3*(c^4 - x^4)*l
og(c + x)^2 + 4*(4*c^4 + 3*c^3*x + c*x^3)*log(c + x))*log(-c + x))*b^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*atanh(c/x))^3,x)

[Out]

int(x^3*(a + b*atanh(c/x))^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a + b \operatorname {atanh}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*atanh(c/x))**3,x)

[Out]

Integral(x**3*(a + b*atanh(c/x))**3, x)

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