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]
________________________________________________________________________________________
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]
[Out]
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]