3.151 \(\int x^2 (a+b \tanh ^{-1}(\frac{c}{x}))^3 \, dx\)

Optimal. Leaf size=217 \[ b^2 c^3 \text{PolyLog}\left (2,\frac{2}{\frac{c}{x}+1}-1\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )+\frac{1}{2} b^3 c^3 \text{PolyLog}\left (3,\frac{2}{\frac{c}{x}+1}-1\right )+b^2 c^2 x \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )-\frac{1}{3} c^3 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3-\frac{1}{2} b c^3 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2-b c^3 \log \left (2-\frac{2}{\frac{c}{x}+1}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{2} b c x^2 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2+\frac{1}{3} x^3 \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3+\frac{1}{2} b^3 c^3 \log \left (1-\frac{c^2}{x^2}\right )+b^3 c^3 \log (x) \]

[Out]

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

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Rubi [F]  time = 3.30156, 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 = {} \[ \int x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*(a + b*ArcTanh[c/x])^3,x]

[Out]

-(a^2*b*c^2*x)/2 + (3*a*b^2*c^2*x)/4 + (a^2*b*c*x^2)/4 + (a*b^2*c^2*x*Log[1 - c/x])/2 - (a*b^2*c*x^2*Log[1 - c
/x])/4 + (b^2*c^2*(1 - c/x)*x*(2*a - b*Log[1 - c/x]))/8 - (b*c^3*(2*a - b*Log[1 - c/x])^2)/16 + (b*c^2*(1 - c/
x)*x*(2*a - b*Log[1 - c/x])^2)/8 + (b*c*x^2*(2*a - b*Log[1 - c/x])^2)/16 - (c^3*(2*a - b*Log[1 - c/x])^3)/24 +
(x^3*(2*a - b*Log[1 - c/x])^3)/24 + (a*b^2*c^2*x*Log[1 + c/x])/2 + (a*b^2*c*x^2*Log[1 + c/x])/4 + (a^2*b*x^3*
Log[1 + c/x])/2 - (a*b^2*x^3*Log[1 - c/x]*Log[1 + c/x])/2 - (a*b^2*c^3*Log[c - x])/4 + (a*b^2*c^3*Log[1 + c/x]
*Log[c - x])/2 - (b*c^3*(2*a - b*Log[1 - c/x])^2*Log[c/x])/8 + (b^3*c^3*Log[x])/4 + (a*b^2*c^3*Log[c - x]*Log[
x/c])/2 + (a^2*b*c^3*Log[c + x])/2 + (a*b^2*c^3*Log[c + x])/4 - (a*b^2*c^3*Log[1 - c/x]*Log[c + x])/2 + (a*b^2
*c^3*Log[(c - x)/(2*c)]*Log[c + x])/2 - (a*b^2*c^3*Log[-(x/c)]*Log[c + x])/2 - (a*b^2*c^3*Log[c - x]*Log[(c +
x)/(2*c)])/2 - (3*a*b^2*c^3*Log[(c + x)/x])/4 - (a*b^2*c^2*x*Log[(c + x)/x])/2 + (b^3*c^2*(1 + c/x)*x*Log[(c +
x)/x])/8 + (a*b^2*c*x^2*Log[(c + x)/x])/4 + (a*b^2*c^3*Log[(c + x)/x]^2)/4 - (b^3*c^3*Log[(c + x)/x]^2)/16 -
(b^3*c^2*(1 + c/x)*x*Log[(c + x)/x]^2)/8 + (b^3*c*x^2*Log[(c + x)/x]^2)/16 + (a*b^2*x^3*Log[(c + x)/x]^2)/4 -
(b^3*c^3*Log[-(c/x)]*Log[(c + x)/x]^2)/8 + (b^3*c^3*Log[(c + x)/x]^3)/24 + (b^3*x^3*Log[(c + x)/x]^3)/24 + (b^
2*c^3*(2*a - b*Log[1 - c/x])*PolyLog[2, 1 - c/x])/4 - (a*b^2*c^3*PolyLog[2, (c - x)/(2*c)])/2 + (a*b^2*c^3*Pol
yLog[2, -(c/x)])/2 - (3*b^3*c^3*PolyLog[2, -(c/x)])/8 - (3*b^3*c^3*PolyLog[2, c/x])/8 + (a*b^2*c^3*PolyLog[2,
(c + x)/(2*c)])/2 - (b^3*c^3*Log[(c + x)/x]*PolyLog[2, (c + x)/x])/4 + (a*b^2*c^3*PolyLog[2, 1 - x/c])/2 - (a*
b^2*c^3*PolyLog[2, 1 + x/c])/2 + (b^3*c^3*PolyLog[3, 1 - c/x])/4 + (b^3*c^3*PolyLog[3, (c + x)/x])/4 + (3*b^3*
Defer[Int][x^2*Log[1 - c/x]^2*Log[1 + c/x], x])/8 - (3*b^3*Defer[Int][x^2*Log[1 - c/x]*Log[1 + c/x]^2, x])/8

Rubi steps

\begin{align*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^3 \, dx &=\int \left (\frac{1}{8} x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{3}{8} b x^2 \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^2 \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^2 \log ^3\left (1+\frac{c}{x}\right )\right ) \, dx\\ &=\frac{1}{8} \int x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3 \, dx+\frac{1}{8} (3 b) \int x^2 \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^2 \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^2 \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^4} \, dx,x,\frac{1}{x}\right )\right )+\frac{1}{8} (3 b) \int \left (4 a^2 x^2 \log \left (1+\frac{c}{x}\right )-4 a b x^2 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+b^2 x^2 \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^2 \log ^2\left (1+\frac{c}{x}\right )-b x^2 \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^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{2} \left (3 a^2 b\right ) \int x^2 \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{4} \left (3 a b^2\right ) \int x^2 \log ^2\left (1+\frac{c}{x}\right ) \, dx-\frac{1}{2} \left (3 a b^2\right ) \int x^2 \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^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx-\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x))^2}{x^3 (1-c x)} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{x^3 (1+c x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{8} b \operatorname{Subst}\left (\int \frac{(2 a-b \log (x))^2}{x \left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{4} \left (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{x^4} \, dx,x,\frac{1}{x}\right )+\frac{1}{2} \left (3 a b^2\right ) \int \frac{c x^2 \log \left (1-\frac{c}{x}\right )}{3 (-c-x)} \, dx+\frac{1}{2} \left (3 a b^2\right ) \int \frac{c x^2 \log \left (1+\frac{c}{x}\right )}{-3 c+3 x} \, dx-\frac{1}{8} b^3 \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x \left (-\frac{1}{c}+\frac{x}{c}\right )^3} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int x^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx+\frac{1}{2} \left (a^2 b c\right ) \int \frac{x}{1+\frac{c}{x}} \, dx\\ &=\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{8} b \operatorname{Subst}\left (\int \frac{(2 a-b \log (x))^2}{\left (\frac{1}{c}-\frac{x}{c}\right )^3} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{8} b^3 \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\left (-\frac{1}{c}+\frac{x}{c}\right )^3} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int x^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx+\frac{1}{8} (b c) \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}{2} \left (a^2 b c\right ) \int \frac{x^2}{c+x} \, dx+\frac{1}{2} \left (a b^2 c\right ) \int \frac{x^2 \log \left (1-\frac{c}{x}\right )}{-c-x} \, dx-\frac{1}{2} \left (a b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3 (1+c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{2} \left (3 a b^2 c\right ) \int \frac{x^2 \log \left (1+\frac{c}{x}\right )}{-3 c+3 x} \, dx+\frac{1}{8} \left (b^3 c\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}{16} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{16} b^3 c x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int x^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx+\frac{1}{8} (b c) \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}{2} \left (a^2 b c\right ) \int \left (-c+x+\frac{c^2}{c+x}\right ) \, dx+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{2} \left (a b^2 c\right ) \int \left (c \log \left (1-\frac{c}{x}\right )-x \log \left (1-\frac{c}{x}\right )-\frac{c^2 \log \left (1-\frac{c}{x}\right )}{c+x}\right ) \, dx-\frac{1}{2} \left (a b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+c x)}{x^3}-\frac{c \log (1+c x)}{x^2}+\frac{c^2 \log (1+c x)}{x}-\frac{c^3 \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}{3} c \log \left (1+\frac{c}{x}\right )-\frac{c^2 \log \left (1+\frac{c}{x}\right )}{3 (c-x)}+\frac{1}{3} x \log \left (1+\frac{c}{x}\right )\right ) \, dx-\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (-\frac{1}{c}+\frac{x}{c}\right )^2} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{8} \left (b^3 c\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 (b c^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (x))^2}{x \left (\frac{1}{c}-\frac{x}{c}\right )} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{8} \left (b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x \left (-\frac{1}{c}+\frac{x}{c}\right )} \, dx,x,1+\frac{c}{x}\right )\\ &=-\frac{1}{2} a^2 b c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{8} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a^2 b c^3 \log (c+x)-\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^3 c x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int x^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx+\frac{1}{8} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{\left (\frac{1}{c}-\frac{x}{c}\right )^2} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{2} \left (a b^2 c\right ) \int x \log \left (1-\frac{c}{x}\right ) \, dx+\frac{1}{2} \left (a b^2 c\right ) \int x \log \left (1+\frac{c}{x}\right ) \, dx-\frac{1}{2} \left (a b^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^3} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\left (-\frac{1}{c}+\frac{x}{c}\right )^2} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{8} \left (b c^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (x))^2}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x \left (\frac{1}{c}-\frac{x}{c}\right )} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{4} \left (b^2 c^2\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}{2} \left (a b^2 c^2\right ) \int \log \left (1-\frac{c}{x}\right ) \, dx+\frac{1}{2} \left (a b^2 c^2\right ) \int \log \left (1+\frac{c}{x}\right ) \, dx+\frac{1}{2} \left (a b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x^2} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x \left (-\frac{1}{c}+\frac{x}{c}\right )} \, dx,x,1+\frac{c}{x}\right )-\frac{1}{8} \left (b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{-\frac{1}{c}+\frac{x}{c}} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{4} \left (b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{-\frac{1}{c}+\frac{x}{c}} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{8} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (x))^2}{x} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (1-\frac{c}{x}\right )}{c+x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (1+\frac{c}{x}\right )}{c-x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{2} \left (a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2} a^2 b c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} a b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{8} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)-\frac{1}{8} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c}{x}\right )+\frac{1}{2} a b^2 c^3 \log (x)+\frac{1}{2} a^2 b c^3 \log (c+x)-\frac{1}{2} a b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{2} a b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^3 c x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^3 \log \left (-\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{2} a b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{4} b^3 c^3 \text{Li}_2\left (-\frac{c}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int x^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx+\frac{1}{8} \left (b^2 c^2\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 (a b^2 c^2\right ) \int \frac{1}{1-\frac{c}{x}} \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \frac{1}{1+\frac{c}{x}} \, dx-\frac{1}{4} \left (a b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (1+c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{8} \left (b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c}+\frac{x}{c}} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{8} \left (b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{-\frac{1}{c}+\frac{x}{c}} \, dx,x,1+\frac{c}{x}\right )-\frac{1}{4} \left (b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{8} c^3 \operatorname{Subst}\left (\int x^2 \, dx,x,2 a-b \log \left (1-\frac{c}{x}\right )\right )+\frac{1}{8} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (x)}{x} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{4} \left (b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x) (2 a-b \log (x))}{x} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{1}{\left (1-\frac{c}{x}\right ) x} \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{1}{\left (1+\frac{c}{x}\right ) x} \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x (1+c x)} \, dx,x,\frac{1}{x}\right )+\frac{1}{2} \left (a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{8} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log \left (\frac{c+x}{x}\right )\right )-\frac{1}{8} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{4} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x) \log (x)}{x} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{2} \left (a b^2 c^4\right ) \int \frac{\log (c-x)}{\left (1+\frac{c}{x}\right ) x^2} \, dx+\frac{1}{2} \left (a b^2 c^4\right ) \int \frac{\log (c+x)}{\left (1-\frac{c}{x}\right ) x^2} \, dx\\ &=-\frac{1}{2} a^2 b c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} a b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2-\frac{1}{24} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)-\frac{1}{8} b c^3 \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^3 \log (x)+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{2} a^2 b c^3 \log (c+x)-\frac{1}{2} a b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{1}{2} a b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^3 c x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^3 \log \left (-\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \text{Li}_2\left (1-\frac{c}{x}\right )+\frac{1}{2} a b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{4} b^3 c^3 \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{4} b^3 c^3 \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^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \frac{x}{-c+x} \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \frac{x}{c+x} \, dx-\frac{1}{4} \left (a b^2 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c}{x}+\frac{c^2}{1+c x}\right ) \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{1}{c}-\frac{x}{c}} \, dx,x,1-\frac{c}{x}\right )-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{1}{-c+x} \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{1}{c+x} \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-\frac{c}{x}\right )+\frac{1}{4} \left (b^3 c^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1+\frac{c}{x}\right )+\frac{1}{2} \left (a b^2 c^4\right ) \int \left (\frac{\log (c-x)}{c x}-\frac{\log (c-x)}{c (c+x)}\right ) \, dx+\frac{1}{2} \left (a b^2 c^4\right ) \int \left (-\frac{\log (c+x)}{c (c-x)}-\frac{\log (c+x)}{c x}\right ) \, dx-\frac{1}{2} \left (a b^2 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+c x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2} a^2 b c^2 x+\frac{1}{4} a b^2 c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} a b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2-\frac{1}{24} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 c^3 \log (c-x)+\frac{1}{2} a b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)-\frac{1}{8} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c}{x}\right )+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{2} a^2 b c^3 \log (c+x)+\frac{1}{2} a b^2 c^3 \log (c+x)-\frac{1}{2} a b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)-\frac{3}{4} a b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{2} a b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^3 c x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^3 \log \left (-\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \text{Li}_2\left (1-\frac{c}{x}\right )+\frac{1}{2} a b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{4} b^3 c^3 \log \left (\frac{c+x}{x}\right ) \text{Li}_2\left (\frac{c+x}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1-\frac{c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (\frac{c+x}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int x^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \left (1-\frac{c}{c-x}\right ) \, dx+\frac{1}{4} \left (a b^2 c^2\right ) \int \left (1-\frac{c}{c+x}\right ) \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log (c-x)}{x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log (c-x)}{c+x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log (c+x)}{c-x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log (c+x)}{x} \, dx\\ &=-\frac{1}{2} a^2 b c^2 x+\frac{3}{4} a b^2 c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} a b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2-\frac{1}{24} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{4} a b^2 c^3 \log (c-x)+\frac{1}{2} a b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)-\frac{1}{8} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c}{x}\right )+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{2} a b^2 c^3 \log (c-x) \log \left (\frac{x}{c}\right )+\frac{1}{2} a^2 b c^3 \log (c+x)+\frac{1}{4} a b^2 c^3 \log (c+x)-\frac{1}{2} a b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{2} a b^2 c^3 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)-\frac{1}{2} a b^2 c^3 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{2} a b^2 c^3 \log (c-x) \log \left (\frac{c+x}{2 c}\right )-\frac{3}{4} a b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{2} a b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^3 c x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^3 \log \left (-\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \text{Li}_2\left (1-\frac{c}{x}\right )+\frac{1}{2} a b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{4} b^3 c^3 \log \left (\frac{c+x}{x}\right ) \text{Li}_2\left (\frac{c+x}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1-\frac{c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (\frac{c+x}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int x^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (-\frac{-c-x}{2 c}\right )}{c-x} \, dx-\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (\frac{c-x}{2 c}\right )}{c+x} \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (-\frac{x}{c}\right )}{c+x} \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \int \frac{\log \left (\frac{x}{c}\right )}{c-x} \, dx\\ &=-\frac{1}{2} a^2 b c^2 x+\frac{3}{4} a b^2 c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} a b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2-\frac{1}{24} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{4} a b^2 c^3 \log (c-x)+\frac{1}{2} a b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)-\frac{1}{8} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c}{x}\right )+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{2} a b^2 c^3 \log (c-x) \log \left (\frac{x}{c}\right )+\frac{1}{2} a^2 b c^3 \log (c+x)+\frac{1}{4} a b^2 c^3 \log (c+x)-\frac{1}{2} a b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{2} a b^2 c^3 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)-\frac{1}{2} a b^2 c^3 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{2} a b^2 c^3 \log (c-x) \log \left (\frac{c+x}{2 c}\right )-\frac{3}{4} a b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{2} a b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^3 c x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^3 \log \left (-\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \text{Li}_2\left (1-\frac{c}{x}\right )+\frac{1}{2} a b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (\frac{c}{x}\right )-\frac{1}{4} b^3 c^3 \log \left (\frac{c+x}{x}\right ) \text{Li}_2\left (\frac{c+x}{x}\right )+\frac{1}{2} a b^2 c^3 \text{Li}_2\left (1-\frac{x}{c}\right )-\frac{1}{2} a b^2 c^3 \text{Li}_2\left (1+\frac{x}{c}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1-\frac{c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (\frac{c+x}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int x^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx+\frac{1}{2} \left (a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c-x\right )-\frac{1}{2} \left (a b^2 c^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 c}\right )}{x} \, dx,x,c+x\right )\\ &=-\frac{1}{2} a^2 b c^2 x+\frac{3}{4} a b^2 c^2 x+\frac{1}{4} a^2 b c x^2+\frac{1}{2} a b^2 c^2 x \log \left (1-\frac{c}{x}\right )-\frac{1}{4} a b^2 c x^2 \log \left (1-\frac{c}{x}\right )+\frac{1}{8} b^2 c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )-\frac{1}{16} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{8} b c^2 \left (1-\frac{c}{x}\right ) x \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2+\frac{1}{16} b c x^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2-\frac{1}{24} c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{24} x^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3+\frac{1}{2} a b^2 c^2 x \log \left (1+\frac{c}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (1+\frac{c}{x}\right )+\frac{1}{2} a^2 b x^3 \log \left (1+\frac{c}{x}\right )-\frac{1}{2} a b^2 x^3 \log \left (1-\frac{c}{x}\right ) \log \left (1+\frac{c}{x}\right )-\frac{1}{4} a b^2 c^3 \log (c-x)+\frac{1}{2} a b^2 c^3 \log \left (1+\frac{c}{x}\right ) \log (c-x)-\frac{1}{8} b c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c}{x}\right )+\frac{1}{4} b^3 c^3 \log (x)+\frac{1}{2} a b^2 c^3 \log (c-x) \log \left (\frac{x}{c}\right )+\frac{1}{2} a^2 b c^3 \log (c+x)+\frac{1}{4} a b^2 c^3 \log (c+x)-\frac{1}{2} a b^2 c^3 \log \left (1-\frac{c}{x}\right ) \log (c+x)+\frac{1}{2} a b^2 c^3 \log \left (\frac{c-x}{2 c}\right ) \log (c+x)-\frac{1}{2} a b^2 c^3 \log \left (-\frac{x}{c}\right ) \log (c+x)-\frac{1}{2} a b^2 c^3 \log (c-x) \log \left (\frac{c+x}{2 c}\right )-\frac{3}{4} a b^2 c^3 \log \left (\frac{c+x}{x}\right )-\frac{1}{2} a b^2 c^2 x \log \left (\frac{c+x}{x}\right )+\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c x^2 \log \left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{16} b^3 c^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^2 \left (1+\frac{c}{x}\right ) x \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{16} b^3 c x^2 \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{4} a b^2 x^3 \log ^2\left (\frac{c+x}{x}\right )-\frac{1}{8} b^3 c^3 \log \left (-\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 c^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{24} b^3 x^3 \log ^3\left (\frac{c+x}{x}\right )+\frac{1}{4} b^2 c^3 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \text{Li}_2\left (1-\frac{c}{x}\right )-\frac{1}{2} a b^2 c^3 \text{Li}_2\left (\frac{c-x}{2 c}\right )+\frac{1}{2} a b^2 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (-\frac{c}{x}\right )-\frac{3}{8} b^3 c^3 \text{Li}_2\left (\frac{c}{x}\right )+\frac{1}{2} a b^2 c^3 \text{Li}_2\left (\frac{c+x}{2 c}\right )-\frac{1}{4} b^3 c^3 \log \left (\frac{c+x}{x}\right ) \text{Li}_2\left (\frac{c+x}{x}\right )+\frac{1}{2} a b^2 c^3 \text{Li}_2\left (1-\frac{x}{c}\right )-\frac{1}{2} a b^2 c^3 \text{Li}_2\left (1+\frac{x}{c}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (1-\frac{c}{x}\right )+\frac{1}{4} b^3 c^3 \text{Li}_3\left (\frac{c+x}{x}\right )+\frac{1}{8} \left (3 b^3\right ) \int x^2 \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^2 \log \left (1-\frac{c}{x}\right ) \log ^2\left (1+\frac{c}{x}\right ) \, dx\\ \end{align*}

Mathematica [C]  time = 0.744349, size = 316, normalized size = 1.46 \[ \frac{1}{6} \left (6 a b^2 \left (c^3 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+\left (x^3-c^3\right ) \tanh ^{-1}\left (\frac{c}{x}\right )^2+c \tanh ^{-1}\left (\frac{c}{x}\right ) \left (-2 c^2 \log \left (1-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )-c^2+x^2\right )+c^2 x\right )+\frac{1}{4} b^3 \left (-24 c^3 \tanh ^{-1}\left (\frac{c}{x}\right ) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+12 c^3 \text{PolyLog}\left (3,e^{2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )-24 c^3 \log \left (\frac{c}{x \sqrt{1-\frac{c^2}{x^2}}}\right )+8 c^3 \tanh ^{-1}\left (\frac{c}{x}\right )^3-12 c^3 \tanh ^{-1}\left (\frac{c}{x}\right )^2+24 c^2 x \tanh ^{-1}\left (\frac{c}{x}\right )-24 c^3 \tanh ^{-1}\left (\frac{c}{x}\right )^2 \log \left (1-e^{2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )-i \pi ^3 c^3+12 c x^2 \tanh ^{-1}\left (\frac{c}{x}\right )^2+8 x^3 \tanh ^{-1}\left (\frac{c}{x}\right )^3\right )+3 a^2 b c^3 \log \left (x^2-c^2\right )+3 a^2 b c x^2+6 a^2 b x^3 \tanh ^{-1}\left (\frac{c}{x}\right )+2 a^3 x^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcTanh[c/x])^3,x]

[Out]

(3*a^2*b*c*x^2 + 2*a^3*x^3 + 6*a^2*b*x^3*ArcTanh[c/x] + 3*a^2*b*c^3*Log[-c^2 + x^2] + 6*a*b^2*(c^2*x + (-c^3 +
x^3)*ArcTanh[c/x]^2 + c*ArcTanh[c/x]*(-c^2 + x^2 - 2*c^2*Log[1 - E^(-2*ArcTanh[c/x])]) + c^3*PolyLog[2, E^(-2
*ArcTanh[c/x])]) + (b^3*((-I)*c^3*Pi^3 + 24*c^2*x*ArcTanh[c/x] - 12*c^3*ArcTanh[c/x]^2 + 12*c*x^2*ArcTanh[c/x]
^2 + 8*c^3*ArcTanh[c/x]^3 + 8*x^3*ArcTanh[c/x]^3 - 24*c^3*ArcTanh[c/x]^2*Log[1 - E^(2*ArcTanh[c/x])] - 24*c^3*
Log[c/(Sqrt[1 - c^2/x^2]*x)] - 24*c^3*ArcTanh[c/x]*PolyLog[2, E^(2*ArcTanh[c/x])] + 12*c^3*PolyLog[3, E^(2*Arc
Tanh[c/x])]))/4)/6

________________________________________________________________________________________

Maple [C]  time = 0.02, size = 2033, normalized size = 9.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{3} x^{3} + \frac{1}{2} \,{\left (2 \, x^{3} \operatorname{artanh}\left (\frac{c}{x}\right ) +{\left (c^{2} \log \left (-c^{2} + x^{2}\right ) + x^{2}\right )} c\right )} a^{2} b + \frac{1}{24} \,{\left (b^{3} c^{3} - b^{3} x^{3}\right )} \log \left (-c + x\right )^{3} + \frac{1}{8} \,{\left (b^{3} c x^{2} + 2 \, a b^{2} x^{3} +{\left (b^{3} c^{3} + b^{3} x^{3}\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )^{2} - \int -\frac{{\left (b^{3} c x^{2} - b^{3} x^{3}\right )} \log \left (c + x\right )^{3} + 6 \,{\left (a b^{2} c x^{2} - a b^{2} x^{3}\right )} \log \left (c + x\right )^{2} +{\left (2 \, b^{3} c x^{2} + 4 \, a b^{2} x^{3} - 3 \,{\left (b^{3} c x^{2} - b^{3} x^{3}\right )} \log \left (c + x\right )^{2} + 2 \,{\left (b^{3} c^{3} - 6 \, a b^{2} c x^{2} +{\left (6 \, a b^{2} + b^{3}\right )} x^{3}\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )}{8 \,{\left (c - x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c/x))^3,x, algorithm="maxima")

[Out]

1/3*a^3*x^3 + 1/2*(2*x^3*arctanh(c/x) + (c^2*log(-c^2 + x^2) + x^2)*c)*a^2*b + 1/24*(b^3*c^3 - b^3*x^3)*log(-c
+ x)^3 + 1/8*(b^3*c*x^2 + 2*a*b^2*x^3 + (b^3*c^3 + b^3*x^3)*log(c + x))*log(-c + x)^2 - integrate(-1/8*((b^3*
c*x^2 - b^3*x^3)*log(c + x)^3 + 6*(a*b^2*c*x^2 - a*b^2*x^3)*log(c + x)^2 + (2*b^3*c*x^2 + 4*a*b^2*x^3 - 3*(b^3
*c*x^2 - b^3*x^3)*log(c + x)^2 + 2*(b^3*c^3 - 6*a*b^2*c*x^2 + (6*a*b^2 + b^3)*x^3)*log(c + x))*log(-c + x))/(c
- x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c/x))^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \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**2*(a+b*atanh(c/x))**3,x)

[Out]

Integral(x**2*(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^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c/x))^3,x, algorithm="giac")

[Out]

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