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

Optimal. Leaf size=139 \[ \frac{3 b^3 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{c}{x}}\right )}{2 c^2}+\frac{3 b^2 \log \left (\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )}{c^2}-\frac{3 b \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2}{2 c^2}+\frac{\left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3}{2 c^2}-\frac{\left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3}{2 x^2}-\frac{3 b \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2}{2 c x} \]

[Out]

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

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Rubi [F]  time = 2.10214, 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 = {} \[ \int \frac{\left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^3}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTanh[c/x])^3/x^3,x]

[Out]

(-3*b^3*(1 - c/x)^2)/(64*c^2) - (3*a*b^2*(1 + c/x)^2)/(16*c^2) + (3*b^3*(1 + c/x)^2)/(64*c^2) + (3*a^2*b)/(8*x
^2) + (3*a*b^2)/(8*x^2) - (3*a^2*b)/(4*c*x) - (3*b^3)/(2*c*x) + (3*a*b^2*Log[1 - c/x])/(8*c^2) - (3*a*b^2*(1 -
c/x)*Log[1 - c/x])/(4*c^2) - (3*b^3*(1 - c/x)*Log[1 - c/x])/(4*c^2) - (3*a*b^2*Log[1 - c/x])/(8*x^2) - (3*b^2
*(1 - c/x)^2*(2*a - b*Log[1 - c/x]))/(32*c^2) + (3*b*(1 - c/x)*(2*a - b*Log[1 - c/x])^2)/(8*c^2) - (3*b*(1 - c
/x)^2*(2*a - b*Log[1 - c/x])^2)/(32*c^2) + ((1 - c/x)*(2*a - b*Log[1 - c/x])^3)/(8*c^2) - ((1 - c/x)^2*(2*a -
b*Log[1 - c/x])^3)/(16*c^2) + (3*a*b^2*Log[1 - c/x]*Log[1 + c/x])/(4*x^2) - (3*a*b^2*Log[1 + c/x]*Log[c - x])/
(4*c^2) - (3*a*b^2*Log[c - x]*Log[x/c])/(4*c^2) - (3*a*b^2*Log[1 - c/x]*Log[c + x])/(4*c^2) + (3*a*b^2*Log[(c
- x)/(2*c)]*Log[c + x])/(4*c^2) - (3*a*b^2*Log[-(x/c)]*Log[c + x])/(4*c^2) + (3*a*b^2*Log[c - x]*Log[(c + x)/(
2*c)])/(4*c^2) + (3*a^2*b*Log[(c + x)/x])/(4*c^2) + (3*a*b^2*Log[(c + x)/x])/(8*c^2) - (9*a*b^2*(1 + c/x)*Log[
(c + x)/x])/(4*c^2) + (3*b^3*(1 + c/x)*Log[(c + x)/x])/(4*c^2) + (3*a*b^2*(1 + c/x)^2*Log[(c + x)/x])/(8*c^2)
- (3*b^3*(1 + c/x)^2*Log[(c + x)/x])/(32*c^2) - (3*a^2*b*Log[(c + x)/x])/(4*x^2) - (3*a*b^2*Log[(c + x)/x])/(8
*x^2) + (3*a*b^2*(1 + c/x)*Log[(c + x)/x]^2)/(4*c^2) - (3*b^3*(1 + c/x)*Log[(c + x)/x]^2)/(8*c^2) - (3*a*b^2*(
1 + c/x)^2*Log[(c + x)/x]^2)/(8*c^2) + (3*b^3*(1 + c/x)^2*Log[(c + x)/x]^2)/(32*c^2) + (b^3*(1 + c/x)*Log[(c +
x)/x]^3)/(8*c^2) - (b^3*(1 + c/x)^2*Log[(c + x)/x]^3)/(16*c^2) + (3*a*b^2*PolyLog[2, (c - x)/(2*c)])/(4*c^2)
+ (3*a*b^2*PolyLog[2, -(c/x)])/(4*c^2) + (3*a*b^2*PolyLog[2, c/x])/(4*c^2) + (3*a*b^2*PolyLog[2, (c + x)/(2*c)
])/(4*c^2) - (3*a*b^2*PolyLog[2, 1 - x/c])/(4*c^2) - (3*a*b^2*PolyLog[2, 1 + x/c])/(4*c^2) + (3*b^3*Defer[Int]
[(Log[1 - c/x]^2*Log[1 + c/x])/x^3, x])/8 - (3*b^3*Defer[Int][(Log[1 - c/x]*Log[1 + c/x]^2)/x^3, x])/8

Rubi steps

\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^3}{x^3} \, dx &=\int \left (\frac{\left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 x^3}+\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (1+\frac{c}{x}\right )}{8 x^3}+\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (1+\frac{c}{x}\right )}{8 x^3}+\frac{b^3 \log ^3\left (1+\frac{c}{x}\right )}{8 x^3}\right ) \, dx\\ &=\frac{1}{8} \int \frac{\left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{x^3} \, dx+\frac{1}{8} (3 b) \int \frac{\left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (1+\frac{c}{x}\right )}{x^3} \, dx+\frac{1}{8} \left (3 b^2\right ) \int \frac{\left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx+\frac{1}{8} b^3 \int \frac{\log ^3\left (1+\frac{c}{x}\right )}{x^3} \, dx\\ &=-\left (\frac{1}{8} \operatorname{Subst}\left (\int x (2 a-b \log (1-c x))^3 \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{8} (3 b) \int \left (\frac{4 a^2 \log \left (1+\frac{c}{x}\right )}{x^3}-\frac{4 a b \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3}+\frac{b^2 \log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3}\right ) \, dx+\frac{1}{8} \left (3 b^2\right ) \int \left (\frac{2 a \log ^2\left (1+\frac{c}{x}\right )}{x^3}-\frac{b \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3}\right ) \, dx-\frac{1}{8} b^3 \operatorname{Subst}\left (\int x \log ^3(1+c x) \, dx,x,\frac{1}{x}\right )\\ &=-\left (\frac{1}{8} \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x))^3}{c}-\frac{(1-c x) (2 a-b \log (1-c x))^3}{c}\right ) \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{2} \left (3 a^2 b\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{x^3} \, dx+\frac{1}{4} \left (3 a b^2\right ) \int \frac{\log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{2} \left (3 a b^2\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} b^3 \operatorname{Subst}\left (\int \left (-\frac{\log ^3(1+c x)}{c}+\frac{(1+c x) \log ^3(1+c x)}{c}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx\\ &=\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{1}{2} \left (3 a^2 b\right ) \operatorname{Subst}\left (\int x \log (1+c x) \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log ^2(1+c x) \, dx,x,\frac{1}{x}\right )+\frac{1}{2} \left (3 a b^2\right ) \int \frac{c \log \left (1-\frac{c}{x}\right )}{2 x^3 (c+x)} \, dx+\frac{1}{2} \left (3 a b^2\right ) \int \frac{c \log \left (1+\frac{c}{x}\right )}{(2 c-2 x) x^3} \, dx+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{\operatorname{Subst}\left (\int (2 a-b \log (1-c x))^3 \, dx,x,\frac{1}{x}\right )}{8 c}+\frac{\operatorname{Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^3 \, dx,x,\frac{1}{x}\right )}{8 c}+\frac{b^3 \operatorname{Subst}\left (\int \log ^3(1+c x) \, dx,x,\frac{1}{x}\right )}{8 c}-\frac{b^3 \operatorname{Subst}\left (\int (1+c x) \log ^3(1+c x) \, dx,x,\frac{1}{x}\right )}{8 c}\\ &=\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 x^2}-\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int \left (-\frac{\log ^2(1+c x)}{c}+\frac{(1+c x) \log ^2(1+c x)}{c}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx+\frac{\operatorname{Subst}\left (\int (2 a-b \log (x))^3 \, dx,x,1-\frac{c}{x}\right )}{8 c^2}-\frac{\operatorname{Subst}\left (\int x (2 a-b \log (x))^3 \, dx,x,1-\frac{c}{x}\right )}{8 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \log ^3(x) \, dx,x,1+\frac{c}{x}\right )}{8 c^2}-\frac{b^3 \operatorname{Subst}\left (\int x \log ^3(x) \, dx,x,1+\frac{c}{x}\right )}{8 c^2}+\frac{1}{4} \left (3 a^2 b c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c x} \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (3 a b^2 c\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{x^3 (c+x)} \, dx+\frac{1}{2} \left (3 a b^2 c\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{(2 c-2 x) x^3} \, dx\\ &=\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{16 c^2}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 x^2}+\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b^3 \left (1+\frac{c}{x}\right )^2 \log ^3\left (\frac{c+x}{x}\right )}{16 c^2}+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{(3 b) \operatorname{Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-\frac{c}{x}\right )}{16 c^2}+\frac{(3 b) \operatorname{Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-\frac{c}{x}\right )}{8 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac{c}{x}\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+\frac{c}{x}\right )}{8 c^2}+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log ^2(1+c x) \, dx,x,\frac{1}{x}\right )}{4 c}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,\frac{1}{x}\right )}{4 c}+\frac{1}{4} \left (3 a^2 b c\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}+\frac{x}{c}+\frac{1}{c^2 (1+c x)}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (3 a b^2 c\right ) \int \left (\frac{\log \left (1-\frac{c}{x}\right )}{c x^3}-\frac{\log \left (1-\frac{c}{x}\right )}{c^2 x^2}+\frac{\log \left (1-\frac{c}{x}\right )}{c^3 x}-\frac{\log \left (1-\frac{c}{x}\right )}{c^3 (c+x)}\right ) \, dx+\frac{1}{2} \left (3 a b^2 c\right ) \int \left (\frac{\log \left (1+\frac{c}{x}\right )}{2 c^3 (c-x)}+\frac{\log \left (1+\frac{c}{x}\right )}{2 c x^3}+\frac{\log \left (1+\frac{c}{x}\right )}{2 c^2 x^2}+\frac{\log \left (1+\frac{c}{x}\right )}{2 c^3 x}\right ) \, dx\\ &=\frac{3 a^2 b}{8 x^2}-\frac{3 a^2 b}{4 c x}+\frac{3 b \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}-\frac{3 b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{32 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{16 c^2}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}+\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 x^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{32 c^2}+\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b^3 \left (1+\frac{c}{x}\right )^2 \log ^3\left (\frac{c+x}{x}\right )}{16 c^2}+\frac{1}{4} \left (3 a b^2\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{x^3} \, dx+\frac{1}{4} \left (3 a b^2\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{x^3} \, dx+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-\frac{c}{x}\right )}{16 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-\frac{c}{x}\right )}{4 c^2}+\frac{\left (3 a b^2\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{x} \, dx}{4 c^2}-\frac{\left (3 a b^2\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{c+x} \, dx}{4 c^2}+\frac{\left (3 a b^2\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{c-x} \, dx}{4 c^2}+\frac{\left (3 a b^2\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{x} \, dx}{4 c^2}+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+\frac{c}{x}\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+\frac{c}{x}\right )}{4 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+\frac{c}{x}\right )}{16 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{c}{x}\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{x^2} \, dx}{4 c}+\frac{\left (3 a b^2\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{x^2} \, dx}{4 c}\\ &=-\frac{3 b^3 \left (1-\frac{c}{x}\right )^2}{64 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2}{64 c^2}+\frac{3 a^2 b}{8 x^2}-\frac{3 a^2 b}{4 c x}-\frac{3 a b^2}{2 c x}-\frac{3 b^3}{4 c x}-\frac{3 b^2 \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{32 c^2}+\frac{3 b \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}-\frac{3 b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{32 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{16 c^2}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{32 c^2}-\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 x^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{32 c^2}+\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b^3 \left (1+\frac{c}{x}\right )^2 \log ^3\left (\frac{c+x}{x}\right )}{16 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}-\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log (1-c x) \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log (1+c x) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+\frac{c}{x}\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{c}{x}\right )}{2 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{c}{x}\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \int \frac{\log (c-x)}{\left (1+\frac{c}{x}\right ) x^2} \, dx}{4 c}+\frac{\left (3 a b^2\right ) \int \frac{\log (c+x)}{\left (1-\frac{c}{x}\right ) x^2} \, dx}{4 c}+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log (1-c x) \, dx,x,\frac{1}{x}\right )}{4 c}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log (1+c x) \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=-\frac{3 b^3 \left (1-\frac{c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2}{64 c^2}+\frac{3 a^2 b}{8 x^2}-\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}-\frac{3 b^3 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{32 c^2}+\frac{3 b \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}-\frac{3 b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{32 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{16 c^2}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{2 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{32 c^2}-\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{32 c^2}+\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b^3 \left (1+\frac{c}{x}\right )^2 \log ^3\left (\frac{c+x}{x}\right )}{16 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{c}{x}\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{c}{x}\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \int \left (\frac{\log (c-x)}{c x}-\frac{\log (c-x)}{c (c+x)}\right ) \, dx}{4 c}+\frac{\left (3 a b^2\right ) \int \left (-\frac{\log (c+x)}{c (c-x)}-\frac{\log (c+x)}{c x}\right ) \, dx}{4 c}-\frac{1}{8} \left (3 a b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-c x} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 a b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 b^3 \left (1-\frac{c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2}{64 c^2}+\frac{3 a^2 b}{8 x^2}-\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}-\frac{3 a b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{32 c^2}+\frac{3 b \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}-\frac{3 b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{32 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{16 c^2}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 c^2}-\frac{9 a b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{32 c^2}-\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{32 c^2}+\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b^3 \left (1+\frac{c}{x}\right )^2 \log ^3\left (\frac{c+x}{x}\right )}{16 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{\left (3 a b^2\right ) \int \frac{\log (c-x)}{x} \, dx}{4 c^2}+\frac{\left (3 a b^2\right ) \int \frac{\log (c-x)}{c+x} \, dx}{4 c^2}-\frac{\left (3 a b^2\right ) \int \frac{\log (c+x)}{c-x} \, dx}{4 c^2}-\frac{\left (3 a b^2\right ) \int \frac{\log (c+x)}{x} \, dx}{4 c^2}-\frac{1}{8} \left (3 a b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{x}{c}-\frac{1}{c^2 (-1+c x)}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 a b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}+\frac{x}{c}+\frac{1}{c^2 (1+c x)}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 b^3 \left (1-\frac{c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2}{64 c^2}+\frac{3 a^2 b}{8 x^2}+\frac{3 a b^2}{8 x^2}-\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{32 c^2}+\frac{3 b \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}-\frac{3 b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{32 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{16 c^2}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{3 a b^2 \log (c-x) \log \left (\frac{x}{c}\right )}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{3 a b^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac{3 a b^2 \log \left (-\frac{x}{c}\right ) \log (c+x)}{4 c^2}+\frac{3 a b^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )}{4 c^2}+\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{9 a b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{32 c^2}-\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{32 c^2}+\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b^3 \left (1+\frac{c}{x}\right )^2 \log ^3\left (\frac{c+x}{x}\right )}{16 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx+\frac{\left (3 a b^2\right ) \int \frac{\log \left (-\frac{-c-x}{2 c}\right )}{c-x} \, dx}{4 c^2}-\frac{\left (3 a b^2\right ) \int \frac{\log \left (\frac{c-x}{2 c}\right )}{c+x} \, dx}{4 c^2}+\frac{\left (3 a b^2\right ) \int \frac{\log \left (-\frac{x}{c}\right )}{c+x} \, dx}{4 c^2}-\frac{\left (3 a b^2\right ) \int \frac{\log \left (\frac{x}{c}\right )}{c-x} \, dx}{4 c^2}\\ &=-\frac{3 b^3 \left (1-\frac{c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2}{64 c^2}+\frac{3 a^2 b}{8 x^2}+\frac{3 a b^2}{8 x^2}-\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{32 c^2}+\frac{3 b \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}-\frac{3 b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{32 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{16 c^2}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{3 a b^2 \log (c-x) \log \left (\frac{x}{c}\right )}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{3 a b^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac{3 a b^2 \log \left (-\frac{x}{c}\right ) \log (c+x)}{4 c^2}+\frac{3 a b^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )}{4 c^2}+\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{9 a b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{32 c^2}-\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{32 c^2}+\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b^3 \left (1+\frac{c}{x}\right )^2 \log ^3\left (\frac{c+x}{x}\right )}{16 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}-\frac{3 a b^2 \text{Li}_2\left (1-\frac{x}{c}\right )}{4 c^2}-\frac{3 a b^2 \text{Li}_2\left (1+\frac{x}{c}\right )}{4 c^2}+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-x\right )}{4 c^2}-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+x\right )}{4 c^2}\\ &=-\frac{3 b^3 \left (1-\frac{c}{x}\right )^2}{64 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2}{16 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2}{64 c^2}+\frac{3 a^2 b}{8 x^2}+\frac{3 a b^2}{8 x^2}-\frac{3 a^2 b}{4 c x}-\frac{3 b^3}{2 c x}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right )}{8 x^2}-\frac{3 b^2 \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{32 c^2}+\frac{3 b \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c^2}-\frac{3 b \left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{32 c^2}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c^2}-\frac{\left (1-\frac{c}{x}\right )^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{16 c^2}+\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (1+\frac{c}{x}\right ) \log (c-x)}{4 c^2}-\frac{3 a b^2 \log (c-x) \log \left (\frac{x}{c}\right )}{4 c^2}-\frac{3 a b^2 \log \left (1-\frac{c}{x}\right ) \log (c+x)}{4 c^2}+\frac{3 a b^2 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)}{4 c^2}-\frac{3 a b^2 \log \left (-\frac{x}{c}\right ) \log (c+x)}{4 c^2}+\frac{3 a b^2 \log (c-x) \log \left (\frac{c+x}{2 c}\right )}{4 c^2}+\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 a b^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{9 a b^2 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log \left (\frac{c+x}{x}\right )}{32 c^2}-\frac{3 a^2 b \log \left (\frac{c+x}{x}\right )}{4 x^2}-\frac{3 a b^2 \log \left (\frac{c+x}{x}\right )}{8 x^2}+\frac{3 a b^2 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c^2}-\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{3 a b^2 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{8 c^2}+\frac{3 b^3 \left (1+\frac{c}{x}\right )^2 \log ^2\left (\frac{c+x}{x}\right )}{32 c^2}+\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c^2}-\frac{b^3 \left (1+\frac{c}{x}\right )^2 \log ^3\left (\frac{c+x}{x}\right )}{16 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c-x}{2 c}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (-\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c}{x}\right )}{4 c^2}+\frac{3 a b^2 \text{Li}_2\left (\frac{c+x}{2 c}\right )}{4 c^2}-\frac{3 a b^2 \text{Li}_2\left (1-\frac{x}{c}\right )}{4 c^2}-\frac{3 a b^2 \text{Li}_2\left (1+\frac{x}{c}\right )}{4 c^2}+\frac{1}{8} \left (3 b^3\right ) \int \frac{\log ^2\left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )}{x^3} \, dx-\frac{1}{8} \left (3 b^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right )}{x^3} \, dx\\ \end{align*}

Mathematica [A]  time = 0.334353, size = 195, normalized size = 1.4 \[ \frac{-6 b^3 x^2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+a \left (12 b^2 x^2 \log \left (\frac{1}{\sqrt{1-\frac{c^2}{x^2}}}\right )-a \left (2 a c^2+3 b x^2 \log \left (1-\frac{c}{x}\right )-3 b x^2 \log \left (\frac{c+x}{x}\right )+6 b c x\right )\right )+6 b \tanh ^{-1}\left (\frac{c}{x}\right ) \left (2 b^2 x^2 \log \left (e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}+1\right )-a c (a c+2 b x)\right )+6 b^2 (x-c) \tanh ^{-1}\left (\frac{c}{x}\right )^2 (a (c+x)+b x)+2 b^3 \left (x^2-c^2\right ) \tanh ^{-1}\left (\frac{c}{x}\right )^3}{4 c^2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.266, size = 6645, normalized size = 47.8 \begin{align*} \text{output 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^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(c*(log(c + x)/c^3 - log(-c + x)/c^3 - 2/(c^2*x)) - 2*arctanh(c/x)/x^2)*a^2*b - 3/8*(c^2*((log(c + x)^2 -
2*(log(c + x) - 2)*log(-c + x) + log(-c + x)^2 + 4*log(c + x))/c^4 - 8*log(x)/c^4) - 4*c*(log(c + x)/c^3 - log
(-c + x)/c^3 - 2/(c^2*x))*arctanh(c/x))*a*b^2 + 1/64*(32*c^4*integrate(-1/4*log(x)^3/(c^4*x^3 - c^2*x^5), x) -
3*c^3*(log(c + x)/c^5 - log(-c + x)/c^5 - 2/(c^4*x)) + 48*c^3*integrate(-1/4*x*log(x)^2/(c^4*x^3 - c^2*x^5),
x) + 48*c^3*integrate(-1/4*x*log(x)/(c^4*x^3 - c^2*x^5), x) - 6*c*(2*log(-c + x)/c^3 - 2*log(x)/c^3 + (c + 2*x
)/(c^2*x^2))*log(-c/x + 1)^2 + 21*c^2*(log(c + x)/c^4 + log(-c + x)/c^4 - 2*log(x)/c^4) - 32*c^2*integrate(-1/
4*x^2*log(x)^3/(c^4*x^3 - c^2*x^5), x) + 48*c^2*integrate(-1/4*x^2*log(x)^2/(c^4*x^3 - c^2*x^5), x) - 384*c^2*
integrate(-1/4*x^2*log(c + x)/(c^4*x^3 - c^2*x^5), x) + 144*c^2*integrate(-1/4*x^2*log(x)/(c^4*x^3 - c^2*x^5),
x) - 18*c*(log(c + x)/c^3 - log(-c + x)/c^3) + c*(6*(2*x^2*log(-c + x)^2 + 2*x^2*log(x)^2 - 6*x^2*log(x) + c^
2 + 6*c*x - 2*(2*x^2*log(x) - 3*x^2)*log(-c + x))*log(-c/x + 1)/(c^3*x^2) - (4*x^2*log(-c + x)^3 - 4*x^2*log(x
)^3 + 18*x^2*log(x)^2 - 6*(2*x^2*log(x) - 3*x^2)*log(-c + x)^2 - 42*x^2*log(x) + 3*c^2 + 42*c*x + 6*(2*x^2*log
(x)^2 - 6*x^2*log(x) + 7*x^2)*log(-c + x))/(c^3*x^2)) - 48*c*integrate(-1/4*x^3*log(x)^2/(c^4*x^3 - c^2*x^5),
x) - 192*c*integrate(-1/4*x^3*log(c + x)/(c^4*x^3 - c^2*x^5), x) + 336*c*integrate(-1/4*x^3*log(x)/(c^4*x^3 -
c^2*x^5), x) + 4*log(-c/x + 1)^3/x^2 - 2*(12*c*x*log(c + x)^2 + 2*(c^2 - x^2)*log(c + x)^3 - 3*(c^2 - 2*c*x +
x^2 - 2*(c^2 - x^2)*log(c + x) + 2*(c^2 - x^2)*log(x))*log(-c + x)^2 - 3*(2*(c^2 - x^2)*log(c + x)^2 - 2*(c^2
- x^2)*log(x)^2 - c^2 - 6*c*x + 8*(c*x + x^2)*log(c + x) - 2*(c^2 + 2*c*x + 5*x^2)*log(x))*log(-c + x))/(c^2*x
^2) - 48*integrate(-1/4*x^4*log(x)^2/(c^4*x^3 - c^2*x^5), x) - 192*integrate(-1/4*x^4*log(c + x)/(c^4*x^3 - c^
2*x^5), x) + 240*integrate(-1/4*x^4*log(x)/(c^4*x^3 - c^2*x^5), x))*b^3 - 3/2*a*b^2*arctanh(c/x)^2/x^2 - 1/2*a
^3/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x))^3/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^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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