Optimal. Leaf size=126 \[ \frac{3 b^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )}{c}-\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1-\frac{c}{x}}\right )}{2 c}-\frac{\left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3}{c}-\frac{\left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^3}{x}+\frac{3 b \log \left (\frac{2}{1-\frac{c}{x}}\right ) \left (a+b \coth ^{-1}\left (\frac{x}{c}\right )\right )^2}{c} \]
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Rubi [B] time = 2.23691, antiderivative size = 387, normalized size of antiderivative = 3.07, number of steps used = 82, number of rules used = 23, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.438, Rules used = {6099, 2454, 2389, 2296, 2295, 6715, 2430, 2416, 2396, 2433, 2374, 6589, 2411, 2346, 2301, 6742, 43, 2394, 2393, 2391, 2375, 2317, 2425} \[ \frac{3 b^2 \text{PolyLog}\left (2,-\frac{c-x}{2 x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{2 c}+\frac{3 b^3 \text{PolyLog}\left (3,-\frac{c-x}{2 x}\right )}{2 c}+\frac{3 b^3 \text{PolyLog}\left (3,\frac{c+x}{2 x}\right )}{2 c}-\frac{3 b^3 \log \left (\frac{c+x}{x}\right ) \text{PolyLog}\left (2,\frac{c+x}{2 x}\right )}{2 c}-\frac{3 b^2 \log ^2\left (\frac{c+x}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 c}-\frac{3 b^2 \log ^2\left (\frac{c+x}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )}{8 x}-\frac{3 b \log \left (\frac{c+x}{2 x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{4 c}+\frac{3 b \log \left (\frac{c+x}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c}-\frac{3 b \log \left (\frac{c+x}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 x}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c}-\frac{b^3 \left (\frac{c}{x}+1\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b^3 \log \left (-\frac{c-x}{2 x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c} \]
Warning: Unable to verify antiderivative.
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Rule 6099
Rule 2454
Rule 2389
Rule 2296
Rule 2295
Rule 6715
Rule 2430
Rule 2416
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 2411
Rule 2346
Rule 2301
Rule 6742
Rule 43
Rule 2394
Rule 2393
Rule 2391
Rule 2375
Rule 2317
Rule 2425
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right )^3}{x^2} \, dx &=\int \left (\frac{\left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 x^2}+\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^2}+\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^2}+\frac{b^3 \log ^3\left (1+\frac{c}{x}\right )}{8 x^2}\right ) \, dx\\ &=\frac{1}{8} \int \frac{\left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{x^2} \, 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^2} \, 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^2} \, dx+\frac{1}{8} b^3 \int \frac{\log ^3\left (1+\frac{c}{x}\right )}{x^2} \, dx\\ &=-\left (\frac{1}{8} \operatorname{Subst}\left (\int (2 a-b \log (1-c x))^3 \, dx,x,\frac{1}{x}\right )\right )-\frac{1}{8} (3 b) \operatorname{Subst}\left (\int (2 a-b \log (1-c x))^2 \log (1+c x) \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a-b \log (1-c x)) \log ^2(1+c x) \, dx,x,\frac{1}{x}\right )-\frac{1}{8} b^3 \operatorname{Subst}\left (\int \log ^3(1+c x) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 x}+\frac{\operatorname{Subst}\left (\int (2 a-b \log (x))^3 \, dx,x,1-\frac{c}{x}\right )}{8 c}-\frac{b^3 \operatorname{Subst}\left (\int \log ^3(x) \, dx,x,1+\frac{c}{x}\right )}{8 c}+\frac{1}{8} (3 b c) \operatorname{Subst}\left (\int \frac{x (2 a-b \log (1-c x))^2}{1+c x} \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x (2 a-b \log (1-c x)) \log (1+c x)}{1-c x} \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x (2 a-b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \frac{x \log ^2(1+c x)}{1-c x} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 x}-\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c}+\frac{(3 b) \operatorname{Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-\frac{c}{x}\right )}{8 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+\frac{c}{x}\right )}{8 c}+\frac{1}{8} (3 b c) \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x))^2}{c}-\frac{(2 a-b \log (1-c x))^2}{c (1+c x)}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c}-\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c (-1+c x)}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c}-\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c (1+c x)}\right ) \, dx,x,\frac{1}{x}\right )+\frac{1}{8} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log ^2(1+c x)}{c}-\frac{\log ^2(1+c x)}{c (-1+c x)}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{3 b \left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2}{8 c}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 x}+\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 x}-\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c}+\frac{1}{8} (3 b) \operatorname{Subst}\left (\int (2 a-b \log (1-c x))^2 \, dx,x,\frac{1}{x}\right )-\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x))^2}{1+c x} \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x)) \log (1+c x)}{-1+c x} \, dx,x,\frac{1}{x}\right )-\frac{1}{4} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(1+c x) \, dx,x,\frac{1}{x}\right )-\frac{1}{8} \left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{-1+c x} \, dx,x,\frac{1}{x}\right )+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-\frac{c}{x}\right )}{4 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{c}{x}\right )}{4 c}\\ &=-\frac{3 a b^2}{2 x}+\frac{3 b^3}{4 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}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{2 x}\right )}{8 c}-\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 x}+\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^3 \log \left (-\frac{c-x}{2 x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c}-\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c}+\frac{1}{4} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (1-c x)) \log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,\frac{1}{x}\right )+\frac{1}{4} \left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right ) \log (1+c x)}{1+c x} \, dx,x,\frac{1}{x}\right )-\frac{(3 b) \operatorname{Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-\frac{c}{x}\right )}{8 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (2-x)) \log (x)}{x} \, dx,x,1+\frac{c}{x}\right )}{4 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2-x) (2 a-b \log (x))}{x} \, dx,x,1-\frac{c}{x}\right )}{4 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+\frac{c}{x}\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{c}{x}\right )}{4 c}\\ &=-\frac{3 a b^2}{2 x}-\frac{3 b^3 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{2 x}\right )}{8 c}-\frac{3 b^3 \left (1+\frac{c}{x}\right ) \log \left (\frac{c+x}{x}\right )}{4 c}+\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^3 \log \left (-\frac{c-x}{2 x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c}-\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{(2 a-b \log (x))^2}{2-x} \, dx,x,1-\frac{c}{x}\right )}{8 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-\frac{c}{x}\right )}{4 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2-x}{2}\right ) (2 a-b \log (x))}{x} \, dx,x,1-\frac{c}{x}\right )}{4 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{2-x} \, dx,x,1+\frac{c}{x}\right )}{8 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+\frac{c}{x}\right )}{4 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2-x}{2}\right ) \log (x)}{x} \, dx,x,1+\frac{c}{x}\right )}{4 c}\\ &=-\frac{3 b^3}{4 x}-\frac{3 b^3 \left (1-\frac{c}{x}\right ) \log \left (1-\frac{c}{x}\right )}{4 c}+\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{2 x}\right )}{4 c}+\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^3 \log \left (-\frac{c-x}{2 x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c}-\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c}+\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \text{Li}_2\left (-\frac{c-x}{2 x}\right )}{4 c}-\frac{3 b^3 \log \left (\frac{c+x}{x}\right ) \text{Li}_2\left (\frac{c+x}{2 x}\right )}{4 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) (2 a-b \log (x))}{x} \, dx,x,1-\frac{c}{x}\right )}{4 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-\frac{c}{x}\right )}{4 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) \log (x)}{x} \, dx,x,1+\frac{c}{x}\right )}{4 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-\frac{c}{x}\right )}{4 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+\frac{c}{x}\right )}{4 c}\\ &=\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{2 x}\right )}{4 c}+\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^3 \log \left (-\frac{c-x}{2 x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c}-\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c}+\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \text{Li}_2\left (-\frac{c-x}{2 x}\right )}{2 c}-\frac{3 b^3 \log \left (\frac{c+x}{x}\right ) \text{Li}_2\left (\frac{c+x}{2 x}\right )}{2 c}+\frac{3 b^3 \text{Li}_3\left (-\frac{c-x}{2 x}\right )}{4 c}+\frac{3 b^3 \text{Li}_3\left (\frac{c+x}{2 x}\right )}{4 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-\frac{c}{x}\right )}{4 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+\frac{c}{x}\right )}{4 c}\\ &=\frac{\left (1-\frac{c}{x}\right ) \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^3}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{2 x}\right )}{4 c}+\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b \left (2 a-b \log \left (1-\frac{c}{x}\right )\right )^2 \log \left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 c}-\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \log ^2\left (\frac{c+x}{x}\right )}{8 x}-\frac{3 b^3 \log \left (-\frac{c-x}{2 x}\right ) \log ^2\left (\frac{c+x}{x}\right )}{4 c}-\frac{b^3 \left (1+\frac{c}{x}\right ) \log ^3\left (\frac{c+x}{x}\right )}{8 c}+\frac{3 b^2 \left (2 a-b \log \left (1-\frac{c}{x}\right )\right ) \text{Li}_2\left (-\frac{c-x}{2 x}\right )}{2 c}-\frac{3 b^3 \log \left (\frac{c+x}{x}\right ) \text{Li}_2\left (\frac{c+x}{2 x}\right )}{2 c}+\frac{3 b^3 \text{Li}_3\left (-\frac{c-x}{2 x}\right )}{2 c}+\frac{3 b^3 \text{Li}_3\left (\frac{c+x}{2 x}\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.122466, size = 205, normalized size = 1.63 \[ -\frac{3 a b^2 \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+\tanh ^{-1}\left (\frac{c}{x}\right ) \left (\frac{c \tanh ^{-1}\left (\frac{c}{x}\right )}{x}-\tanh ^{-1}\left (\frac{c}{x}\right )-2 \log \left (e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}+1\right )\right )\right )}{c}-\frac{b^3 \left (3 \tanh ^{-1}\left (\frac{c}{x}\right ) \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+\frac{3}{2} \text{PolyLog}\left (3,-e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}\right )+\tanh ^{-1}\left (\frac{c}{x}\right )^2 \left (\frac{c \tanh ^{-1}\left (\frac{c}{x}\right )}{x}-\tanh ^{-1}\left (\frac{c}{x}\right )-3 \log \left (e^{-2 \tanh ^{-1}\left (\frac{c}{x}\right )}+1\right )\right )\right )}{c}-\frac{3 a^2 b \log \left (1-\frac{c^2}{x^2}\right )}{2 c}-\frac{3 a^2 b \tanh ^{-1}\left (\frac{c}{x}\right )}{x}-\frac{a^3}{x} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.004, size = 298, normalized size = 2.4 \begin{align*} -{\frac{{a}^{3}}{x}}-{\frac{{b}^{3}}{x} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{3}}-{\frac{{b}^{3}}{c} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{3}}+3\,{\frac{{b}^{3}}{c} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}\ln \left ({ \left ( 1+{\frac{c}{x}} \right ) ^{2} \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) ^{-1}}+1 \right ) }+3\,{\frac{{b}^{3}}{c}{\it Artanh} \left ({\frac{c}{x}} \right ){\it polylog} \left ( 2,-{ \left ( 1+{\frac{c}{x}} \right ) ^{2} \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) ^{-1}} \right ) }-{\frac{3\,{b}^{3}}{2\,c}{\it polylog} \left ( 3,-{ \left ( 1+{\frac{c}{x}} \right ) ^{2} \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) ^{-1}} \right ) }-3\,{\frac{a{b}^{2}}{x} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}}+6\,{\frac{a{b}^{2}}{c}\ln \left ({ \left ( 1+{\frac{c}{x}} \right ) ^{2} \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) ^{-1}}+1 \right ){\it Artanh} \left ({\frac{c}{x}} \right ) }-3\,{\frac{a{b}^{2}}{c} \left ({\it Artanh} \left ({\frac{c}{x}} \right ) \right ) ^{2}}+3\,{\frac{a{b}^{2}}{c}{\it polylog} \left ( 2,-{ \left ( 1+{\frac{c}{x}} \right ) ^{2} \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) ^{-1}} \right ) }-3\,{\frac{{a}^{2}b}{x}{\it Artanh} \left ({\frac{c}{x}} \right ) }-{\frac{3\,{a}^{2}b}{2\,c}\ln \left ( 1-{\frac{{c}^{2}}{{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{3 \, a^{2} b{\left (\frac{2 \, c \operatorname{artanh}\left (\frac{c}{x}\right )}{x} + \log \left (-\frac{c^{2}}{x^{2}} + 1\right )\right )}}{2 \, c} - \frac{a^{3}}{x} + \frac{{\left (b^{3} c - b^{3} x\right )} \log \left (-c + x\right )^{3} - 3 \,{\left (2 \, a b^{2} c +{\left (b^{3} c + b^{3} x\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )^{2}}{8 \, c x} - \int -\frac{{\left (b^{3} c^{2} - b^{3} c x\right )} \log \left (c + x\right )^{3} + 6 \,{\left (a b^{2} c^{2} - a b^{2} c x\right )} \log \left (c + x\right )^{2} - 3 \,{\left (4 \, a b^{2} c x +{\left (b^{3} c^{2} - b^{3} c x\right )} \log \left (c + x\right )^{2} + 2 \,{\left (2 \, a b^{2} c^{2} + b^{3} x^{2} -{\left (2 \, a b^{2} c - b^{3} c\right )} x\right )} \log \left (c + x\right )\right )} \log \left (-c + x\right )}{8 \,{\left (c^{2} x^{2} - c x^{3}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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