Optimal. Leaf size=141 \[ -\frac{3 b^3 \text{PolyLog}\left (2,1-\frac{2}{1-c x^2}\right )}{4 c^2}-\frac{3 b^2 \log \left (\frac{2}{1-c x^2}\right ) \left (a+b \tanh ^{-1}\left (c x^2\right )\right )}{2 c^2}-\frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3}{4 c^2}+\frac{3 b \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{4 c^2}+\frac{1}{4} x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3+\frac{3 b x^2 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{4 c} \]
[Out]
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Rubi [B] time = 4.21552, antiderivative size = 479, normalized size of antiderivative = 3.4, number of steps used = 155, number of rules used = 30, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.875, Rules used = {6099, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2439, 2416, 2396, 2433, 2374, 6589, 2411, 43, 2334, 12, 14, 2301, 6742, 2430, 2394, 2393, 2391, 2395, 2375, 2317, 2425} \[ -\frac{3 b^3 \text{PolyLog}\left (2,\frac{1}{2} \left (1-c x^2\right )\right )}{8 c^2}+\frac{3 b^3 \text{PolyLog}\left (2,\frac{1}{2} \left (c x^2+1\right )\right )}{8 c^2}-\frac{3 b^2 \log ^2\left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{32 c^2}+\frac{3 b^2 \log \left (\frac{1}{2} \left (c x^2+1\right )\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{8 c^2}+\frac{3}{32} b^2 x^4 \log ^2\left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )+\frac{3 b^2 x^2 \log \left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{8 c}+\frac{\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}-\frac{3 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c^2}-\frac{3 b \log \left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{32 c^2}+\frac{3}{32} b x^4 \log \left (c x^2+1\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac{b^3 \left (c x^2+1\right )^2 \log ^3\left (c x^2+1\right )}{32 c^2}-\frac{b^3 \left (c x^2+1\right ) \log ^3\left (c x^2+1\right )}{16 c^2}+\frac{3 b^3 \left (c x^2+1\right ) \log ^2\left (c x^2+1\right )}{16 c^2}+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log \left (c x^2+1\right )}{8 c^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 6099
Rule 2454
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rule 2439
Rule 2416
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 2411
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rule 6742
Rule 2430
Rule 2394
Rule 2393
Rule 2391
Rule 2395
Rule 2375
Rule 2317
Rule 2425
Rubi steps
\begin{align*} \int x^3 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^3 \, dx &=\int \left (\frac{1}{8} x^3 \left (2 a-b \log \left (1-c x^2\right )\right )^3+\frac{3}{8} b x^3 \left (-2 a+b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )-\frac{3}{8} b^2 x^3 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{1}{8} b^3 x^3 \log ^3\left (1+c x^2\right )\right ) \, dx\\ &=\frac{1}{8} \int x^3 \left (2 a-b \log \left (1-c x^2\right )\right )^3 \, dx+\frac{1}{8} (3 b) \int x^3 \left (-2 a+b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right ) \, dx-\frac{1}{8} \left (3 b^2\right ) \int x^3 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right ) \, dx+\frac{1}{8} b^3 \int x^3 \log ^3\left (1+c x^2\right ) \, dx\\ &=\frac{1}{16} \operatorname{Subst}\left (\int x (2 a-b \log (1-c x))^3 \, dx,x,x^2\right )+\frac{1}{16} (3 b) \operatorname{Subst}\left (\int x (-2 a+b \log (1-c x))^2 \log (1+c x) \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^2\right ) \operatorname{Subst}\left (\int x (-2 a+b \log (1-c x)) \log ^2(1+c x) \, dx,x,x^2\right )+\frac{1}{16} b^3 \operatorname{Subst}\left (\int x \log ^3(1+c x) \, dx,x,x^2\right )\\ &=\frac{3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{1}{16} \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,x^2\right )+\frac{1}{16} 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,x^2\right )-\frac{1}{32} (3 b c) \operatorname{Subst}\left (\int \frac{x^2 (-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^2\right )+\frac{1}{16} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2 (-2 a+b \log (1-c x)) \log (1+c x)}{1-c x} \, dx,x,x^2\right )+\frac{1}{16} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2 (-2 a+b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^2\right )-\frac{1}{32} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \frac{x^2 \log ^2(1+c x)}{1-c x} \, dx,x,x^2\right )\\ &=\frac{3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )+\frac{\operatorname{Subst}\left (\int (2 a-b \log (1-c x))^3 \, dx,x,x^2\right )}{16 c}-\frac{\operatorname{Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^3 \, dx,x,x^2\right )}{16 c}-\frac{b^3 \operatorname{Subst}\left (\int \log ^3(1+c x) \, dx,x,x^2\right )}{16 c}+\frac{b^3 \operatorname{Subst}\left (\int (1+c x) \log ^3(1+c x) \, dx,x,x^2\right )}{16 c}-\frac{1}{32} (3 b c) \operatorname{Subst}\left (\int \left (-\frac{(-2 a+b \log (1-c x))^2}{c^2}+\frac{x (-2 a+b \log (1-c x))^2}{c}+\frac{(-2 a+b \log (1-c x))^2}{c^2 (1+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{16} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c^2}+\frac{x (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^2 (-1+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{16} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{(2 a-b \log (1-c x)) \log (1+c x)}{c^2}-\frac{x (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^2 (1+c x)}\right ) \, dx,x,x^2\right )-\frac{1}{32} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \left (-\frac{\log ^2(1+c x)}{c^2}-\frac{x \log ^2(1+c x)}{c}-\frac{\log ^2(1+c x)}{c^2 (-1+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac{1}{32} (3 b) \operatorname{Subst}\left (\int x (-2 a+b \log (1-c x))^2 \, dx,x,x^2\right )+\frac{1}{32} \left (3 b^3\right ) \operatorname{Subst}\left (\int x \log ^2(1+c x) \, dx,x,x^2\right )-\frac{\operatorname{Subst}\left (\int (2 a-b \log (x))^3 \, dx,x,1-c x^2\right )}{16 c^2}+\frac{\operatorname{Subst}\left (\int x (2 a-b \log (x))^3 \, dx,x,1-c x^2\right )}{16 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \log ^3(x) \, dx,x,1+c x^2\right )}{16 c^2}+\frac{b^3 \operatorname{Subst}\left (\int x \log ^3(x) \, dx,x,1+c x^2\right )}{16 c^2}+\frac{(3 b) \operatorname{Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^2\right )}{32 c}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{(-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^2\right )}{32 c}+2 \frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a-b \log (1-c x)) \log (1+c x) \, dx,x,x^2\right )}{16 c}+\frac{\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,x^2\right )}{16 c}-\frac{\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,x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(1+c x) \, dx,x,x^2\right )}{32 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+c x)}{-1+c x} \, dx,x,x^2\right )}{32 c}\\ &=-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac{\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{32 c^2}+\frac{3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac{b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}-\frac{1}{32} (3 b) \operatorname{Subst}\left (\int \left (\frac{(-2 a+b \log (1-c x))^2}{c}-\frac{(1-c x) (-2 a+b \log (1-c x))^2}{c}\right ) \, dx,x,x^2\right )+\frac{1}{32} \left (3 b^3\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,x^2\right )+2 \left (\frac{3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}-\frac{1}{16} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{x (2 a-b \log (1-c x))}{1+c x} \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{x \log (1+c x)}{1-c x} \, dx,x,x^2\right )\right )+\frac{(3 b) \operatorname{Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-c x^2\right )}{32 c^2}-\frac{(3 b) \operatorname{Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^2\right )}{32 c^2}-\frac{(3 b) \operatorname{Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-c x^2\right )}{16 c^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a-b \log (2-x)) \log (x)}{x} \, dx,x,1+c x^2\right )}{16 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2-x) (2 a-b \log (x))}{x} \, dx,x,1-c x^2\right )}{16 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+c x^2\right )}{32 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+c x^2\right )}{16 c^2}-\frac{\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,x^2\right )}{16 c}-\frac{\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,x^2\right )}{16 c}\\ &=-\frac{9 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{32 c^2}+\frac{3 b \left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2}{64 c^2}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac{\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{32 c^2}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{9 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{32 c^2}-\frac{3 b^3 \left (1+c x^2\right )^2 \log ^2\left (1+c x^2\right )}{64 c^2}+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}-\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac{b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+2 \left (\frac{3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}-\frac{1}{16} \left (3 b^2\right ) \operatorname{Subst}\left (\int \left (\frac{2 a-b \log (1-c x)}{c}-\frac{2 a-b \log (1-c x)}{c (1+c x)}\right ) \, dx,x,x^2\right )-\frac{1}{16} \left (3 b^3\right ) \operatorname{Subst}\left (\int \left (-\frac{\log (1+c x)}{c}-\frac{\log (1+c x)}{c (-1+c x)}\right ) \, dx,x,x^2\right )\right )-\frac{(3 b) \operatorname{Subst}\left (\int \frac{(2 a-b \log (x))^2}{2-x} \, dx,x,1-c x^2\right )}{32 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-c x^2\right )}{32 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^2\right )}{16 c^2}+\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-c x^2\right )}{16 c^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-c x^2\right )}{8 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{2-x} \, dx,x,1+c x^2\right )}{32 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2-x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{8 c^2}-\frac{(3 b) \operatorname{Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^2\right )}{32 c}+\frac{(3 b) \operatorname{Subst}\left (\int (1-c x) (-2 a+b \log (1-c x))^2 \, dx,x,x^2\right )}{32 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(1+c x) \, dx,x,x^2\right )}{32 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,x^2\right )}{32 c}\\ &=\frac{9 a b^2 x^2}{8 c}+\frac{9 b^3 x^2}{16 c}+\frac{3 b^3 \left (1-c x^2\right )^2}{128 c^2}-\frac{3 b^3 \left (1+c x^2\right )^2}{128 c^2}+\frac{3 b^2 \left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{64 c^2}-\frac{9 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{32 c^2}+\frac{3 b \left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^2}{64 c^2}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac{\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac{9 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac{3 b^3 \left (1+c x^2\right )^2 \log \left (1+c x^2\right )}{64 c^2}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{9 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{32 c^2}-\frac{3 b^3 \left (1+c x^2\right )^2 \log ^2\left (1+c x^2\right )}{64 c^2}-\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac{b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-c x^2\right )\right )}{16 c^2}+\frac{3 b^3 \log \left (1+c x^2\right ) \text{Li}_2\left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac{(3 b) \operatorname{Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^2\right )}{32 c^2}-\frac{(3 b) \operatorname{Subst}\left (\int x (-2 a+b \log (x))^2 \, dx,x,1-c x^2\right )}{32 c^2}+\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-c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{16 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^2\right )}{16 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{16 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{8 c^2}+2 \left (\frac{3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (2 a-b \log (1-c x)) \, dx,x,x^2\right )}{16 c}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{2 a-b \log (1-c x)}{1+c x} \, dx,x,x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (1+c x) \, dx,x,x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+c x)}{-1+c x} \, dx,x,x^2\right )}{16 c}\right )\\ &=\frac{9 a b^2 x^2}{8 c}+\frac{9 b^3 x^2}{8 c}+\frac{3 b^3 \left (1-c x^2\right )^2}{128 c^2}-\frac{3 b^3 \left (1+c x^2\right )^2}{128 c^2}+\frac{9 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{16 c^2}+\frac{3 b^2 \left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{64 c^2}-\frac{3 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c^2}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac{\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac{9 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac{3 b^3 \left (1+c x^2\right )^2 \log \left (1+c x^2\right )}{64 c^2}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{16 c^2}-\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac{b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1-c x^2\right )\right )}{16 c^2}-\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int x (-2 a+b \log (x)) \, dx,x,1-c x^2\right )}{32 c^2}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+c x^2\right )}{32 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{16 c^2}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{16 c^2}+2 \left (-\frac{3 a b^2 x^2}{8 c}+\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac{3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (1-c x) \, dx,x,x^2\right )}{16 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^2\right )}{16 c}\right )\\ &=\frac{3 a b^2 x^2}{4 c}+\frac{15 b^3 x^2}{16 c}+\frac{9 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{16 c^2}-\frac{3 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c^2}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac{\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac{3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{8 c^2}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{16 c^2}-\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac{b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+2 \left (-\frac{3 a b^2 x^2}{8 c}-\frac{3 b^3 x^2}{16 c}+\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac{3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac{3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+c x^2\right )}{16 c^2}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{16 c^2}\right )-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-c x^2\right )}{16 c^2}\\ &=\frac{3 a b^2 x^2}{4 c}+\frac{3 b^3 x^2}{4 c}+\frac{3 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{8 c^2}-\frac{3 b \left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^2}{16 c^2}-\frac{\left (1-c x^2\right ) \left (2 a-b \log \left (1-c x^2\right )\right )^3}{16 c^2}+\frac{\left (1-c x^2\right )^2 \left (2 a-b \log \left (1-c x^2\right )\right )^3}{32 c^2}-\frac{3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{8 c^2}-\frac{3 b \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \log \left (1+c x^2\right )+\frac{3 b^3 \left (1+c x^2\right ) \log ^2\left (1+c x^2\right )}{16 c^2}-\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )}{32 c^2}+\frac{3}{32} b^2 x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \log ^2\left (1+c x^2\right )-\frac{b^3 \left (1+c x^2\right ) \log ^3\left (1+c x^2\right )}{16 c^2}+\frac{b^3 \left (1+c x^2\right )^2 \log ^3\left (1+c x^2\right )}{32 c^2}+2 \left (-\frac{3 a b^2 x^2}{8 c}-\frac{3 b^3 x^2}{8 c}-\frac{3 b^3 \left (1-c x^2\right ) \log \left (1-c x^2\right )}{16 c^2}+\frac{3 b^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^2}+\frac{3 b^3 \left (1+c x^2\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c^2}+\frac{3 b^2 x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{16 c}-\frac{3 b^3 \text{Li}_2\left (\frac{1}{2} \left (1-c x^2\right )\right )}{16 c^2}+\frac{3 b^3 \text{Li}_2\left (\frac{1}{2} \left (1+c x^2\right )\right )}{16 c^2}\right )\\ \end{align*}
Mathematica [A] time = 0.464082, size = 185, normalized size = 1.31 \[ \frac{6 b^3 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c x^2\right )}\right )+a \left (2 a^2 c^2 x^4+6 a b c x^2+3 a b \log \left (1-c x^2\right )-3 a b \log \left (c x^2+1\right )+6 b^2 \log \left (1-c^2 x^4\right )\right )+6 b^2 \left (c x^2-1\right ) \tanh ^{-1}\left (c x^2\right )^2 \left (a c x^2+a+b\right )+6 b \tanh ^{-1}\left (c x^2\right ) \left (a c x^2 \left (a c x^2+2 b\right )-2 b^2 \log \left (e^{-2 \tanh ^{-1}\left (c x^2\right )}+1\right )\right )+2 b^3 \left (c^2 x^4-1\right ) \tanh ^{-1}\left (c x^2\right )^3}{8 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.269, size = 751, normalized size = 5.3 \begin{align*} \text{result too large to display} \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*} \text{result too large to display} \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 (b^{3} x^{3} \operatorname{artanh}\left (c x^{2}\right )^{3} + 3 \, a b^{2} x^{3} \operatorname{artanh}\left (c x^{2}\right )^{2} + 3 \, a^{2} b x^{3} \operatorname{artanh}\left (c x^{2}\right ) + a^{3} x^{3}, 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 x^{3} \left (a + b \operatorname{atanh}{\left (c x^{2} \right )}\right )^{3}\, 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{\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )}^{3} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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