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

Optimal. Leaf size=108 \[ -3 b^2 c \text {Li}_2\left (1-\frac {2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )+c \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3+x \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b c \log \left (\frac {2 c}{c-x}\right ) \left (a+b \coth ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b^3 c \text {Li}_3\left (1-\frac {2 c}{c-x}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [F]  time = 0.70, 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 = {} \[ \int \left (a+b \tanh ^{-1}\left (\frac {c}{x}\right )\right )^3 \, dx \]

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*}

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Mathematica [C]  time = 0.28, size = 198, normalized size = 1.83 \[ a^3 x+\frac {3}{2} a^2 b c \log \left (x^2-c^2\right )+3 a^2 b x \tanh ^{-1}\left (\frac {c}{x}\right )-3 a b^2 \left (\tanh ^{-1}\left (\frac {c}{x}\right ) \left ((c-x) \tanh ^{-1}\left (\frac {c}{x}\right )+2 c \log \left (1-e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )-c \text {Li}_2\left (e^{-2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )\right )+\frac {1}{8} b^3 \left (-24 c \tanh ^{-1}\left (\frac {c}{x}\right ) \text {Li}_2\left (e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )+12 c \text {Li}_3\left (e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )+8 c \tanh ^{-1}\left (\frac {c}{x}\right )^3+8 x \tanh ^{-1}\left (\frac {c}{x}\right )^3-24 c \tanh ^{-1}\left (\frac {c}{x}\right )^2 \log \left (1-e^{2 \tanh ^{-1}\left (\frac {c}{x}\right )}\right )-i \pi ^3 c\right ) \]

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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fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} \operatorname {artanh}\left (\frac {c}{x}\right )^{3} + 3 \, a b^{2} \operatorname {artanh}\left (\frac {c}{x}\right )^{2} + 3 \, a^{2} b \operatorname {artanh}\left (\frac {c}{x}\right ) + a^{3}, x\right ) \]

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

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)

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maple [C]  time = 0.36, size = 1756, normalized size = 16.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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+(1+c/x)^2/(1-c^2/x^2)))^2*arcta
nh(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)^2-3/4*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)))^3*arctanh(c/x)^2+3/2*I*c*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+(1+c/x)^2/(1-c^2/x^2)))^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+(1+c/x)^2/(1-c^2/x^2)))^2*arctanh(c/x)^2+3/2*c*a^2*b*ln(1+c/x)-3*c*
b^3*arctanh(c/x)^2*ln(1+(1+c/x)/(1-c^2/x^2)^(1/2))+3*a*b^2*x*arctanh(c/x)^2+3*a^2*b*x*arctanh(c/x)-3*c*b^3*arc
tanh(c/x)^2*ln(2)-3*c*a^2*b*ln(c/x)+3*c*a*b^2*dilog(c/x)+3*c*a*b^2*dilog(1+c/x)-3*c*a*b^2*dilog(1/2+1/2*c/x)-3
*c*b^3*arctanh(c/x)^2*ln(1-(1+c/x)/(1-c^2/x^2)^(1/2))-6*c*b^3*arctanh(c/x)*polylog(2,-(1+c/x)/(1-c^2/x^2)^(1/2
))-3*c*b^3*ln(c/x)*arctanh(c/x)^2-6*c*b^3*arctanh(c/x)*polylog(2,(1+c/x)/(1-c^2/x^2)^(1/2))+3*c*b^3*arctanh(c/
x)^2*ln((1+c/x)^2/(1-c^2/x^2)-1)+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
*arctanh(c/x)^2*ln((1+c/x)/(1-c^2/x^2)^(1/2))+3/4*c*a*b^2*ln(c/x-1)^2-3/4*c*a*b^2*ln(1+c/x)^2+3/2*c*a^2*b*ln(c
/x-1)-3/2*c*a*b^2*ln(c/x-1)*ln(1/2+1/2*c/x)+3/4*I*c*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+b^3*x*arctanh(c/x)^3+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)^3-3
/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)^2-3/2*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+(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)^2-3/2*I*c*b^3*Pi
*csgn(I*((1+c/x)^2/(1-c^2/x^2)-1))*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+(1+c/
x)^2/(1-c^2/x^2)))*arctanh(c/x)^2+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)-3/2*I*c*b^3*Pi*arctanh(c/x)^2-3/2*I*c*b^3*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^3*arctanh(c/x)^2+3/2*I*
c*b^3*Pi*csgn(I/(1+(1+c/x)^2/(1-c^2/x^2)))^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
+(1+c/x)^2/(1-c^2/x^2)))^3*arctanh(c/x)^2+3*c*a*b^2*arctanh(c/x)*ln(c/x-1)+3*c*a*b^2*arctanh(c/x)*ln(1+c/x)-6*
c*a*b^2*arctanh(c/x)*ln(c/x)+3*c*a*b^2*ln(c/x)*ln(1+c/x)+x*a^3

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

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x))^3,x)

[Out]

int((a + b*atanh(c/x))^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c/x))**3, x)

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