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

Optimal. Leaf size=108 \[ -3 b^2 c \text{PolyLog}\left (2,1-\frac{2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{3}{2} b^3 c \text{PolyLog}\left (3,1-\frac{2 c}{c-x}\right )+c \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3+x \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3-3 b c \log \left (\frac{2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2 \]

[Out]

c*(a + b*ArcCoth[x/c])^3 + x*(a + b*ArcCoth[x/c])^3 - 3*b*c*(a + b*ArcCoth[x/c])^2*Log[(2*c)/(c - x)] - 3*b^2*
c*(a + b*ArcCoth[x/c])*PolyLog[2, 1 - (2*c)/(c - x)] + (3*b^3*c*PolyLog[3, 1 - (2*c)/(c - x)])/2

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Rubi [F]  time = 0.700751, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c/x])^3,x]

[Out]

a^3*x - (3*a^2*b*x*Log[1 - c/x])/2 - (3*a*b^2*(c - x)*Log[1 - c/x]^2)/4 + (b^3*(c - x)*Log[1 - c/x]^3)/8 + (3*
a^2*b*x*Log[1 + c/x])/2 - (3*a*b^2*x*Log[1 - c/x]*Log[1 + c/x])/2 + (3*a*b^2*(c + x)*Log[1 + c/x]^2)/4 + (b^3*
(c + x)*Log[1 + c/x]^3)/8 - (3*a*b^2*c*Log[1 - c/x]*Log[-c - x])/2 + (3*a^2*b*c*Log[c - x])/2 + (3*a*b^2*c*Log
[-c - x]*Log[(c - x)/(2*c)])/2 - (3*b^3*c*Log[1 - c/x]^2*Log[c/x])/8 - (3*a*b^2*c*Log[-c - x]*Log[-(x/c)])/2 +
(3*a*b^2*c*Log[1 + c/x]*Log[-c + x])/2 + (3*a*b^2*c*Log[x/c]*Log[-c + x])/2 + (3*a^2*b*c*Log[c + x])/2 - (3*a
*b^2*c*Log[-c + x]*Log[(c + x)/(2*c)])/2 - (3*b^3*c*Log[-(c/x)]*Log[(c + x)/x]^2)/8 - (3*b^3*c*Log[1 - c/x]*Po
lyLog[2, 1 - c/x])/4 - (3*a*b^2*c*PolyLog[2, (c - x)/(2*c)])/2 + (3*a*b^2*c*PolyLog[2, -(c/x)])/2 - (3*a*b^2*c
*PolyLog[2, c/x])/2 + (3*a*b^2*c*PolyLog[2, (c + x)/(2*c)])/2 - (3*b^3*c*Log[(c + x)/x]*PolyLog[2, (c + x)/x])
/4 + (3*a*b^2*c*PolyLog[2, 1 - x/c])/2 - (3*a*b^2*c*PolyLog[2, 1 + x/c])/2 + (3*b^3*c*PolyLog[3, 1 - c/x])/4 +
(3*b^3*c*PolyLog[3, (c + x)/x])/4 + (3*b^3*Defer[Int][Log[1 - c/x]^2*Log[1 + c/x], x])/8 - (3*b^3*Defer[Int][
Log[1 - c/x]*Log[1 + c/x]^2, x])/8

Rubi steps

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

Mathematica [C]  time = 0.263006, size = 198, normalized size = 1.83 \[ -3 a b^2 \left (\tanh ^{-1}\left (\frac{c}{x}\right ) \left ((c-x) \tanh ^{-1}\left (\frac{c}{x}\right )+2 c \log \left (1-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )\right )-c \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )\right )+\frac{1}{8} b^3 \left (-24 c \tanh ^{-1}\left (\frac{c}{x}\right ) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+12 c \text{PolyLog}\left (3,e^{2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+8 c \tanh ^{-1}\left (\frac{c}{x}\right )^3+8 x \tanh ^{-1}\left (\frac{c}{x}\right )^3-24 c \tanh ^{-1}\left (\frac{c}{x}\right )^2 \log \left (1-e^{2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )-i \pi ^3 c\right )+\frac{3}{2} a^2 b c \log \left (x^2-c^2\right )+3 a^2 b x \tanh ^{-1}\left (\frac{c}{x}\right )+a^3 x \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTanh[c/x])^3,x]

[Out]

a^3*x + 3*a^2*b*x*ArcTanh[c/x] + (3*a^2*b*c*Log[-c^2 + x^2])/2 - 3*a*b^2*(ArcTanh[c/x]*((c - x)*ArcTanh[c/x] +
2*c*Log[1 - E^(-2*ArcTanh[c/x])]) - c*PolyLog[2, E^(-2*ArcTanh[c/x])]) + (b^3*((-I)*c*Pi^3 + 8*c*ArcTanh[c/x]
^3 + 8*x*ArcTanh[c/x]^3 - 24*c*ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - 24*c*ArcTanh[c/x]*PolyLog[2, E^(2*
ArcTanh[c/x])] + 12*c*PolyLog[3, E^(2*ArcTanh[c/x])]))/8

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Maple [C]  time = 0.121, size = 1756, normalized size = 16.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*c*a*b^2*ln(c/x)*arctanh(c/x)+3*c*a*b^2*ln(c/x)*ln(1+c/x)+3/2*c*a*b^2*ln(-1/2*c/x+1/2)*ln(1+c/x)-3/2*c*a*b^2
*ln(-1/2*c/x+1/2)*ln(1/2+1/2*c/x)+3*c*a*b^2*arctanh(c/x)*ln(c/x-1)+3*c*a*b^2*arctanh(c/x)*ln(1+c/x)-3/2*c*a*b^
2*ln(c/x-1)*ln(1/2+1/2*c/x)-3/2*I*c*b^3*Pi*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1))*csgn(I/((1+c/x)^2/(1-c^2/x^2)+1))
*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1)/((1+c/x)^2/(1-c^2/x^2)+1))*arctanh(c/x)^2+6*c*b^3*polylog(3,-(1+c/x)/(1-c^2/
x^2)^(1/2))+6*c*b^3*polylog(3,(1+c/x)/(1-c^2/x^2)^(1/2))+c*b^3*arctanh(c/x)^3+b^3*x*arctanh(c/x)^3+3/4*I*c*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+c/x)^2/(1-c^2/x^2)+1))^2*arctanh(c/x)^2+3
/2*I*c*b^3*Pi*csgn(I/((1+c/x)^2/(1-c^2/x^2)+1))^2*arctanh(c/x)^2-3/2*I*c*b^3*Pi*csgn(I*((1+c/x)^2/(1-c^2/x^2)-
1)/((1+c/x)^2/(1-c^2/x^2)+1))^3*arctanh(c/x)^2-3/2*I*c*b^3*Pi*csgn(I/((1+c/x)^2/(1-c^2/x^2)+1))^3*arctanh(c/x)
^2-3/4*I*c*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)^2-3/4*I*c*b^
3*Pi*csgn(I/((1+c/x)^2/(1-c^2/x^2)+1))*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/((1+c/x)^2/(1-c^2/x^2)+1))^2*arctanh(c/x)
^2-3/2*I*c*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+3/4*I*c*b^
3*Pi*csgn(I/((1+c/x)^2/(1-c^2/x^2)+1))*csgn(I*(1+c/x)^2/(-1+c^2/x^2))*csgn(I*(1+c/x)^2/(-1+c^2/x^2)/((1+c/x)^2
/(1-c^2/x^2)+1))*arctanh(c/x)^2+x*a^3-6*c*b^3*arctanh(c/x)*polylog(2,(1+c/x)/(1-c^2/x^2)^(1/2))-6*c*b^3*arctan
h(c/x)*polylog(2,-(1+c/x)/(1-c^2/x^2)^(1/2))-3*c*b^3*arctanh(c/x)^2*ln(1-(1+c/x)/(1-c^2/x^2)^(1/2))+3*c*a*b^2*
dilog(c/x)+3/2*c*a^2*b*ln(c/x-1)+3/2*c*a^2*b*ln(1+c/x)+3/4*c*a*b^2*ln(c/x-1)^2-3/4*c*a*b^2*ln(1+c/x)^2-3*c*b^3
*arctanh(c/x)^2*ln(2)-3*c*a^2*b*ln(c/x)+3*c*a*b^2*dilog(1+c/x)-3*c*a*b^2*dilog(1/2+1/2*c/x)+3*a*b^2*x*arctanh(
c/x)^2+3*a^2*b*x*arctanh(c/x)+3/2*c*b^3*arctanh(c/x)^2*ln(c/x-1)+3/2*c*b^3*arctanh(c/x)^2*ln(1+c/x)-3*c*b^3*ar
ctanh(c/x)^2*ln((1+c/x)/(1-c^2/x^2)^(1/2))-3*c*b^3*arctanh(c/x)^2*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))-3*c*b^3*ln(c
/x)*arctanh(c/x)^2+3*c*b^3*arctanh(c/x)^2*ln((1+c/x)^2/(1-c^2/x^2)-1)-3/4*I*c*b^3*Pi*csgn(I*(1+c/x)^2/(-1+c^2/
x^2)/((1+c/x)^2/(1-c^2/x^2)+1))^3*arctanh(c/x)^2-3/4*I*c*b^3*Pi*csgn(I*(1+c/x)^2/(-1+c^2/x^2))^3*arctanh(c/x)^
2-3/2*I*c*b^3*Pi*arctanh(c/x)^2+3/2*I*c*b^3*Pi*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1))*csgn(I*((1+c/x)^2/(1-c^2/x^2)
-1)/((1+c/x)^2/(1-c^2/x^2)+1))^2*arctanh(c/x)^2+3/2*I*c*b^3*Pi*csgn(I/((1+c/x)^2/(1-c^2/x^2)+1))*csgn(I*((1+c/
x)^2/(1-c^2/x^2)-1)/((1+c/x)^2/(1-c^2/x^2)+1))^2*arctanh(c/x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{2} \,{\left (2 \, x \operatorname{artanh}\left (\frac{c}{x}\right ) + c \log \left (-c^{2} + x^{2}\right )\right )} a^{2} b + a^{3} x + \frac{1}{8} \,{\left (b^{3} c - b^{3} x\right )} \log \left (-c + x\right )^{3} + \frac{3}{8} \,{\left (2 \, a b^{2} x +{\left (b^{3} c + b^{3} x\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )^{2} - \int -\frac{{\left (b^{3} c - b^{3} x\right )} \log \left (c + x\right )^{3} + 6 \,{\left (a b^{2} c - a b^{2} x\right )} \log \left (c + x\right )^{2} + 3 \,{\left (4 \, a b^{2} x -{\left (b^{3} c - b^{3} x\right )} \log \left (c + x\right )^{2} - 2 \,{\left (2 \, a b^{2} c - b^{3} c -{\left (2 \, a b^{2} + b^{3}\right )} x\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )}{8 \,{\left (c - x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(2*x*arctanh(c/x) + c*log(-c^2 + x^2))*a^2*b + a^3*x + 1/8*(b^3*c - b^3*x)*log(-c + x)^3 + 3/8*(2*a*b^2*x
+ (b^3*c + b^3*x)*log(c + x))*log(-c + x)^2 - integrate(-1/8*((b^3*c - b^3*x)*log(c + x)^3 + 6*(a*b^2*c - a*b^
2*x)*log(c + x)^2 + 3*(4*a*b^2*x - (b^3*c - b^3*x)*log(c + x)^2 - 2*(2*a*b^2*c - b^3*c - (2*a*b^2 + b^3)*x)*lo
g(c + x))*log(-c + x))/(c - x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} \operatorname{artanh}\left (\frac{c}{x}\right )^{3} + 3 \, a b^{2} \operatorname{artanh}\left (\frac{c}{x}\right )^{2} + 3 \, a^{2} b \operatorname{artanh}\left (\frac{c}{x}\right ) + a^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atanh}{\left (\frac{c}{x} \right )}\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c/x))**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{artanh}\left (\frac{c}{x}\right ) + a\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctanh(c/x) + a)^3, x)