Optimal. Leaf size=144 \[ \frac{3 i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x^2}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{2 c}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x^2}\right )}{4 c}+\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^3+\frac{i \left (a+b \tan ^{-1}\left (c x^2\right )\right )^3}{2 c}+\frac{3 b \log \left (\frac{2}{1+i c x^2}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{2 c} \]
[Out]
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Rubi [B] time = 2.545, antiderivative size = 545, normalized size of antiderivative = 3.78, number of steps used = 82, number of rules used = 23, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.643, Rules used = {5035, 2454, 2389, 2296, 2295, 6715, 2430, 2416, 2396, 2433, 2374, 6589, 2411, 2346, 2301, 6742, 43, 2394, 2393, 2391, 2375, 2317, 2425} \[ -\frac{3 b^2 \text{PolyLog}\left (2,\frac{1}{2} \left (1-i c x^2\right )\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{PolyLog}\left (3,\frac{1}{2} \left (1-i c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{PolyLog}\left (3,\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}+\frac{3 b^3 \log \left (1+i c x^2\right ) \text{PolyLog}\left (2,\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}+\frac{3}{16} i b^2 x^2 \log ^2\left (1+i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac{3 b^2 \log ^2\left (1+i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )}{16 c}+\frac{3 b \left (1-i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{3 b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{3 b \log \left (\frac{1}{2} \left (1+i c x^2\right )\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{8 c}+\frac{3}{16} i b x^2 \log \left (1+i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2-\frac{3 b \log \left (1+i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{i \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c}+\frac{b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c}+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{8 c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 5035
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 x \left (a+b \tan ^{-1}\left (c x^2\right )\right )^3 \, dx &=\int \left (\frac{1}{8} x \left (2 a+i b \log \left (1-i c x^2\right )\right )^3+\frac{3}{8} i b x \left (-2 i a+b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3}{8} i b^2 x \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{1}{8} i b^3 x \log ^3\left (1+i c x^2\right )\right ) \, dx\\ &=\frac{1}{8} \int x \left (2 a+i b \log \left (1-i c x^2\right )\right )^3 \, dx+\frac{1}{8} (3 i b) \int x \left (-2 i a+b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right ) \, dx-\frac{1}{8} \left (3 i b^2\right ) \int x \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right ) \, dx+\frac{1}{8} \left (i b^3\right ) \int x \log ^3\left (1+i c x^2\right ) \, dx\\ &=\frac{1}{16} \operatorname{Subst}\left (\int (2 a+i b \log (1-i c x))^3 \, dx,x,x^2\right )+\frac{1}{16} (3 i b) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x))^2 \log (1+i c x) \, dx,x,x^2\right )-\frac{1}{16} \left (3 i b^2\right ) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x)) \log ^2(1+i c x) \, dx,x,x^2\right )+\frac{1}{16} \left (i b^3\right ) \operatorname{Subst}\left (\int \log ^3(1+i c x) \, dx,x,x^2\right )\\ &=\frac{3}{16} i b x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )+\frac{3}{16} i b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{i \operatorname{Subst}\left (\int (2 a+i b \log (x))^3 \, dx,x,1-i c x^2\right )}{16 c}+\frac{b^3 \operatorname{Subst}\left (\int \log ^3(x) \, dx,x,1+i c x^2\right )}{16 c}+\frac{1}{16} (3 b c) \operatorname{Subst}\left (\int \frac{x (-2 i a+b \log (1-i c x))^2}{1+i c x} \, dx,x,x^2\right )-\frac{1}{8} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x (-2 i a+b \log (1-i c x)) \log (1+i c x)}{1-i c x} \, dx,x,x^2\right )-\frac{1}{8} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{x (-2 i a+b \log (1-i c x)) \log (1+i c x)}{1+i c x} \, dx,x,x^2\right )+\frac{1}{16} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \frac{x \log ^2(1+i c x)}{1-i c x} \, dx,x,x^2\right )\\ &=\frac{i \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c}+\frac{3}{16} i b x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )+\frac{3}{16} i b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c}+\frac{(3 b) \operatorname{Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-i c x^2\right )}{16 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{16 c}+\frac{1}{16} (3 b c) \operatorname{Subst}\left (\int \left (-\frac{i (-2 i a+b \log (1-i c x))^2}{c}+\frac{(-2 i a+b \log (1-i c x))^2}{c (-i+c x)}\right ) \, dx,x,x^2\right )-\frac{1}{8} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (-\frac{(2 a+i b \log (1-i c x)) \log (1+i c x)}{c}+\frac{(-2 i a+b \log (1-i c x)) \log (1+i c x)}{c (-i+c x)}\right ) \, dx,x,x^2\right )-\frac{1}{8} \left (3 b^2 c\right ) \operatorname{Subst}\left (\int \left (\frac{(2 a+i b \log (1-i c x)) \log (1+i c x)}{c}+\frac{(-2 i a+b \log (1-i c x)) \log (1+i c x)}{c (i+c x)}\right ) \, dx,x,x^2\right )+\frac{1}{16} \left (3 b^3 c\right ) \operatorname{Subst}\left (\int \left (\frac{i \log ^2(1+i c x)}{c}+\frac{\log ^2(1+i c x)}{c (i+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{3 b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{i \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c}+\frac{3}{16} i b x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3 b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c}-\frac{1}{16} (3 i b) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x))^2 \, dx,x,x^2\right )+\frac{1}{16} (3 b) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x))^2}{-i+c x} \, dx,x,x^2\right )-\frac{1}{8} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log (1+i c x)}{-i+c x} \, dx,x,x^2\right )-\frac{1}{8} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log (1+i c x)}{i+c x} \, dx,x,x^2\right )+\frac{1}{16} \left (3 i b^3\right ) \operatorname{Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^2\right )+\frac{1}{16} \left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{i+c x} \, dx,x,x^2\right )-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-i c x^2\right )}{8 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{8 c}\\ &=-\frac{3}{4} a b^2 x^2-\frac{3}{8} i b^3 x^2+\frac{3 b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{i \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c}+\frac{3 b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c}+\frac{3}{16} i b x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )-\frac{3 b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{16 c}+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c}+\frac{1}{8} \left (3 i b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log \left (\frac{1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^2\right )-\frac{1}{8} \left (3 i b^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+i c x) \log \left (-\frac{1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^2\right )+\frac{(3 b) \operatorname{Subst}\left (\int (-2 i a+b \log (x))^2 \, dx,x,1-i c x^2\right )}{16 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (2-x)) \log (x)}{x} \, dx,x,1+i c x^2\right )}{8 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2-x) (-2 i a+b \log (x))}{x} \, dx,x,1-i c x^2\right )}{8 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{16 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{8 c}\\ &=-\frac{3}{4} a b^2 x^2+\frac{3 b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{8 c}+\frac{3 b \left (1-i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{3 b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{i \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{16 c}+\frac{3 b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c}-\frac{3 b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{16 c}+\frac{3 b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (x))^2}{2-x} \, dx,x,1-i c x^2\right )}{16 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (-2 i a+b \log (x)) \, dx,x,1-i c x^2\right )}{8 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{1}{2} i (-2 i+i x)\right ) (-2 i a+b \log (x))}{x} \, dx,x,1-i c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{2-x} \, dx,x,1+i c x^2\right )}{16 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{1}{2} i (2 i-i x)\right ) \log (x)}{x} \, dx,x,1+i c x^2\right )}{8 c}\\ &=\frac{3}{8} i b^3 x^2+\frac{3 b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{8 c}+\frac{3 b \left (1-i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{3 b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{i \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{8 c}-\frac{3 b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{8 c}+\frac{3 b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c}-\frac{3 b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{8 c}+\frac{3 b^3 \log \left (1+i c x^2\right ) \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{8 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) (-2 i a+b \log (x))}{x} \, dx,x,1-i c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{8 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+i c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{8 c}\\ &=\frac{3 b \left (1-i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{3 b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{i \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{8 c}-\frac{3 b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{8 c}+\frac{3 b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c}-\frac{3 b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{4 c}+\frac{3 b^3 \log \left (1+i c x^2\right ) \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{8 c}-\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{8 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{8 c}\\ &=\frac{3 b \left (1-i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{3 b \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c}+\frac{i \left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c}+\frac{3 b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^2\right )\right )}{8 c}-\frac{3 b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )+\frac{3 b^3 \log \left (\frac{1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{8 c}+\frac{3 b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{16 c}+\frac{3}{16} i b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )+\frac{b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c}-\frac{3 b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{4 c}+\frac{3 b^3 \log \left (1+i c x^2\right ) \text{Li}_2\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1-i c x^2\right )\right )}{4 c}-\frac{3 b^3 \text{Li}_3\left (\frac{1}{2} \left (1+i c x^2\right )\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.102436, size = 224, normalized size = 1.56 \[ \frac{-6 i b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}\left (c x^2\right )}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )+3 b^3 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}\left (c x^2\right )}\right )-3 a^2 b \log \left (c^2 x^4+1\right )+6 a^2 b c x^2 \tan ^{-1}\left (c x^2\right )+2 a^3 c x^2-6 i a b^2 \tan ^{-1}\left (c x^2\right )^2+6 a b^2 c x^2 \tan ^{-1}\left (c x^2\right )^2+12 a b^2 \tan ^{-1}\left (c x^2\right ) \log \left (1+e^{2 i \tan ^{-1}\left (c x^2\right )}\right )-2 i b^3 \tan ^{-1}\left (c x^2\right )^3+2 b^3 c x^2 \tan ^{-1}\left (c x^2\right )^3+6 b^3 \tan ^{-1}\left (c x^2\right )^2 \log \left (1+e^{2 i \tan ^{-1}\left (c x^2\right )}\right )}{4 c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.125, size = 306, normalized size = 2.1 \begin{align*}{\frac{{x}^{2}{a}^{3}}{2}}-{\frac{{\frac{i}{2}}{b}^{3} \left ( \arctan \left ( c{x}^{2} \right ) \right ) ^{3}}{c}}+{\frac{{b}^{3} \left ( \arctan \left ( c{x}^{2} \right ) \right ) ^{3}{x}^{2}}{2}}+{\frac{3\,{b}^{3} \left ( \arctan \left ( c{x}^{2} \right ) \right ) ^{2}}{2\,c}\ln \left ({\frac{ \left ( 1+ic{x}^{2} \right ) ^{2}}{{c}^{2}{x}^{4}+1}}+1 \right ) }-{\frac{{\frac{3\,i}{2}}{b}^{3}\arctan \left ( c{x}^{2} \right ) }{c}{\it polylog} \left ( 2,-{\frac{ \left ( 1+ic{x}^{2} \right ) ^{2}}{{c}^{2}{x}^{4}+1}} \right ) }+{\frac{3\,{b}^{3}}{4\,c}{\it polylog} \left ( 3,-{\frac{ \left ( 1+ic{x}^{2} \right ) ^{2}}{{c}^{2}{x}^{4}+1}} \right ) }-{\frac{{\frac{3\,i}{2}} \left ( \arctan \left ( c{x}^{2} \right ) \right ) ^{2}a{b}^{2}}{c}}+{\frac{3\, \left ( \arctan \left ( c{x}^{2} \right ) \right ) ^{2}{x}^{2}a{b}^{2}}{2}}+3\,{\frac{\arctan \left ( c{x}^{2} \right ) a{b}^{2}}{c}\ln \left ({\frac{ \left ( 1+ic{x}^{2} \right ) ^{2}}{{c}^{2}{x}^{4}+1}}+1 \right ) }-{\frac{{\frac{3\,i}{2}}a{b}^{2}}{c}{\it polylog} \left ( 2,-{\frac{ \left ( 1+ic{x}^{2} \right ) ^{2}}{{c}^{2}{x}^{4}+1}} \right ) }+{\frac{3\,{a}^{2}b{x}^{2}\arctan \left ( c{x}^{2} \right ) }{2}}-{\frac{3\,{a}^{2}b\ln \left ({c}^{2}{x}^{4}+1 \right ) }{4\,c}} \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{1}{16} \, b^{3} x^{2} \arctan \left (c x^{2}\right )^{3} - \frac{3}{64} \, b^{3} x^{2} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )^{2} + \frac{7 \, b^{3} \arctan \left (c x^{2}\right )^{4}}{64 \, c} + 28 \, b^{3} c^{2} \int \frac{x^{5} \arctan \left (c x^{2}\right )^{3}}{32 \,{\left (c^{2} x^{4} + 1\right )}}\,{d x} + 3 \, b^{3} c^{2} \int \frac{x^{5} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )^{2}}{32 \,{\left (c^{2} x^{4} + 1\right )}}\,{d x} + 96 \, a b^{2} c^{2} \int \frac{x^{5} \arctan \left (c x^{2}\right )^{2}}{32 \,{\left (c^{2} x^{4} + 1\right )}}\,{d x} + 12 \, b^{3} c^{2} \int \frac{x^{5} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )}{32 \,{\left (c^{2} x^{4} + 1\right )}}\,{d x} + \frac{1}{2} \, a^{3} x^{2} + \frac{a b^{2} \arctan \left (c x^{2}\right )^{3}}{2 \, c} - 12 \, b^{3} c \int \frac{x^{3} \arctan \left (c x^{2}\right )^{2}}{32 \,{\left (c^{2} x^{4} + 1\right )}}\,{d x} + 3 \, b^{3} c \int \frac{x^{3} \log \left (c^{2} x^{4} + 1\right )^{2}}{32 \,{\left (c^{2} x^{4} + 1\right )}}\,{d x} + 3 \, b^{3} \int \frac{x \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )^{2}}{32 \,{\left (c^{2} x^{4} + 1\right )}}\,{d x} + \frac{3 \,{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} a^{2} b}{4 \, c} \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 \arctan \left (c x^{2}\right )^{3} + 3 \, a b^{2} x \arctan \left (c x^{2}\right )^{2} + 3 \, a^{2} b x \arctan \left (c x^{2}\right ) + a^{3} x, 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 \left (a + b \operatorname{atan}{\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 \arctan \left (c x^{2}\right ) + a\right )}^{3} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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