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

Optimal. Leaf size=135 \[ \frac{3}{2} b^3 c^2 \text{PolyLog}\left (2,\frac{2}{\frac{c}{x}+1}-1\right )-3 b^2 c^2 \log \left (2-\frac{2}{\frac{c}{x}+1}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )-\frac{3}{2} b c^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2-\frac{1}{2} c^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3+\frac{1}{2} x^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3+\frac{3}{2} b c x \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2 \]

[Out]

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

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Rubi [F]  time = 2.24482, 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 = {} \[ \int x \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*(a + b*ArcTanh[c/x])^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.317704, size = 193, normalized size = 1.43 \[ \frac{1}{4} \left (6 b^3 c^2 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+a \left (a \left (2 a x^2-3 b c^2 \log \left (\frac{c+x}{x}\right )+6 b c x\right )+3 a b c^2 \log \left (1-\frac{c}{x}\right )-12 b^2 c^2 \log \left (\frac{c}{x \sqrt{1-\frac{c^2}{x^2}}}\right )\right )+6 b \tanh ^{-1}\left (\frac{c}{x}\right ) \left (a x (a x+2 b c)-2 b^2 c^2 \log \left (1-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )\right )+6 b^2 (x-c) \tanh ^{-1}\left (\frac{c}{x}\right )^2 (a (c+x)+b c)+2 b^3 \left (x^2-c^2\right ) \tanh ^{-1}\left (\frac{c}{x}\right )^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(a + b*ArcTanh[c/x])^3,x]

[Out]

(6*b^2*(-c + x)*(b*c + a*(c + x))*ArcTanh[c/x]^2 + 2*b^3*(-c^2 + x^2)*ArcTanh[c/x]^3 + 6*b*ArcTanh[c/x]*(a*x*(
2*b*c + a*x) - 2*b^2*c^2*Log[1 - E^(-2*ArcTanh[c/x])]) + a*(3*a*b*c^2*Log[1 - c/x] - 12*b^2*c^2*Log[c/(Sqrt[1
- c^2/x^2]*x)] + a*(6*b*c*x + 2*a*x^2 - 3*b*c^2*Log[(c + x)/x])) + 6*b^3*c^2*PolyLog[2, E^(-2*ArcTanh[c/x])])/
4

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Maple [C]  time = 0.278, size = 5536, normalized size = 41. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{2} \, a b^{2} x^{2} \operatorname{artanh}\left (\frac{c}{x}\right )^{2} + \frac{1}{2} \, a^{3} x^{2} + \frac{3}{4} \,{\left (2 \, x^{2} \operatorname{artanh}\left (\frac{c}{x}\right ) -{\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c\right )} a^{2} b + \frac{3}{8} \,{\left ({\left (\log \left (c + x\right )^{2} - 2 \,{\left (\log \left (c + x\right ) - 2\right )} \log \left (-c + x\right ) + \log \left (-c + x\right )^{2} + 4 \, \log \left (c + x\right )\right )} c^{2} - 4 \,{\left (c \log \left (c + x\right ) - c \log \left (-c + x\right ) - 2 \, x\right )} c \operatorname{artanh}\left (\frac{c}{x}\right )\right )} a b^{2} + \frac{1}{16} \,{\left (6 \, c x \log \left (c + x\right )^{2} -{\left (c^{2} - x^{2}\right )} \log \left (c + x\right )^{3} +{\left (c^{2} - x^{2}\right )} \log \left (-c + x\right )^{3} - 3 \,{\left (2 \, c^{2} - 2 \, c x +{\left (c^{2} - x^{2}\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )^{2} + 3 \,{\left ({\left (c^{2} - x^{2}\right )} \log \left (c + x\right )^{2} - 4 \,{\left (c^{2} + c x\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right ) + 2 \, \int -\frac{6 \,{\left (c^{3} + 3 \, c^{2} x\right )} \log \left (c + x\right )}{c^{2} - x^{2}}\,{d x}\right )} b^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a*b^2*x^2*arctanh(c/x)^2 + 1/2*a^3*x^2 + 3/4*(2*x^2*arctanh(c/x) - (c*log(c + x) - c*log(-c + x) - 2*x)*c)
*a^2*b + 3/8*((log(c + x)^2 - 2*(log(c + x) - 2)*log(-c + x) + log(-c + x)^2 + 4*log(c + x))*c^2 - 4*(c*log(c
+ x) - c*log(-c + x) - 2*x)*c*arctanh(c/x))*a*b^2 + 1/16*(6*c*x*log(c + x)^2 - (c^2 - x^2)*log(c + x)^3 + (c^2
- x^2)*log(-c + x)^3 - 3*(2*c^2 - 2*c*x + (c^2 - x^2)*log(c + x))*log(-c + x)^2 + 3*((c^2 - x^2)*log(c + x)^2
- 4*(c^2 + c*x)*log(c + x))*log(-c + x) + 2*integrate(-6*(c^3 + 3*c^2*x)*log(c + x)/(c^2 - x^2), x))*b^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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(x*(a+b*atanh(c/x))**3,x)

[Out]

Integral(x*(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} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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