Optimal. Leaf size=144 \[ \frac {3 i b^2 \text {Li}_2\left (1-\frac {2}{i c x^2+1}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{2 c}+\frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^3+\frac {i \left (a+b \tan ^{-1}\left (c x^2\right )\right )^3}{2 c}+\frac {3 b \log \left (\frac {2}{1+i c x^2}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{2 c}+\frac {3 b^3 \text {Li}_3\left (1-\frac {2}{i c x^2+1}\right )}{4 c} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 2.54, 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 43
Rule 2295
Rule 2296
Rule 2301
Rule 2317
Rule 2346
Rule 2374
Rule 2375
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2396
Rule 2411
Rule 2416
Rule 2425
Rule 2430
Rule 2433
Rule 2454
Rule 5035
Rule 6589
Rule 6715
Rule 6742
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 = 0.13, size = 224, normalized size = 1.56 \[ \frac {2 a^3 c x^2-3 a^2 b \log \left (c^2 x^4+1\right )+6 a^2 b c x^2 \tan ^{-1}\left (c x^2\right )-6 i b^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}\left (c x^2\right )}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )-6 i a b^2 \tan ^{-1}\left (c x^2\right )^2+6 a b^2 c x^2 \tan ^{-1}\left (c x^2\right )^2+12 a b^2 \tan ^{-1}\left (c x^2\right ) \log \left (1+e^{2 i \tan ^{-1}\left (c x^2\right )}\right )+3 b^3 \text {Li}_3\left (-e^{2 i \tan ^{-1}\left (c x^2\right )}\right )-2 i b^3 \tan ^{-1}\left (c x^2\right )^3+2 b^3 c x^2 \tan ^{-1}\left (c x^2\right )^3+6 b^3 \tan ^{-1}\left (c x^2\right )^2 \log \left (1+e^{2 i \tan ^{-1}\left (c x^2\right )}\right )}{4 c} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x \arctan \left (c x^{2}\right )^{3} + 3 \, a b^{2} x \arctan \left (c x^{2}\right )^{2} + 3 \, a^{2} b x \arctan \left (c x^{2}\right ) + a^{3} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{3} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.24, size = 306, normalized size = 2.12 \[ \frac {x^{2} a^{3}}{2}-\frac {i b^{3} \arctan \left (c \,x^{2}\right )^{3}}{2 c}+\frac {b^{3} x^{2} \arctan \left (c \,x^{2}\right )^{3}}{2}+\frac {3 b^{3} \arctan \left (c \,x^{2}\right )^{2} \ln \left (\frac {\left (i c \,x^{2}+1\right )^{2}}{c^{2} x^{4}+1}+1\right )}{2 c}-\frac {3 i b^{3} \arctan \left (c \,x^{2}\right ) \polylog \left (2, -\frac {\left (i c \,x^{2}+1\right )^{2}}{c^{2} x^{4}+1}\right )}{2 c}+\frac {3 b^{3} \polylog \left (3, -\frac {\left (i c \,x^{2}+1\right )^{2}}{c^{2} x^{4}+1}\right )}{4 c}-\frac {3 i \arctan \left (c \,x^{2}\right )^{2} a \,b^{2}}{2 c}+\frac {3 x^{2} a \,b^{2} \arctan \left (c \,x^{2}\right )^{2}}{2}+\frac {3 \ln \left (\frac {\left (i c \,x^{2}+1\right )^{2}}{c^{2} x^{4}+1}+1\right ) \arctan \left (c \,x^{2}\right ) a \,b^{2}}{c}-\frac {3 i \polylog \left (2, -\frac {\left (i c \,x^{2}+1\right )^{2}}{c^{2} x^{4}+1}\right ) a \,b^{2}}{2 c}+\frac {3 x^{2} a^{2} b \arctan \left (c \,x^{2}\right )}{2}-\frac {3 a^{2} b \ln \left (c^{2} x^{4}+1\right )}{4 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{16} \, b^{3} x^{2} \arctan \left (c x^{2}\right )^{3} - \frac {3}{64} \, b^{3} x^{2} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )^{2} + \frac {7 \, b^{3} \arctan \left (c x^{2}\right )^{4}}{64 \, c} + 28 \, b^{3} c^{2} \int \frac {x^{5} \arctan \left (c x^{2}\right )^{3}}{32 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + 3 \, b^{3} c^{2} \int \frac {x^{5} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )^{2}}{32 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + 96 \, a b^{2} c^{2} \int \frac {x^{5} \arctan \left (c x^{2}\right )^{2}}{32 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + 12 \, b^{3} c^{2} \int \frac {x^{5} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )}{32 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + \frac {1}{2} \, a^{3} x^{2} + \frac {a b^{2} \arctan \left (c x^{2}\right )^{3}}{2 \, c} - 12 \, b^{3} c \int \frac {x^{3} \arctan \left (c x^{2}\right )^{2}}{32 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + 3 \, b^{3} c \int \frac {x^{3} \log \left (c^{2} x^{4} + 1\right )^{2}}{32 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + 3 \, b^{3} \int \frac {x \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )^{2}}{32 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x} + \frac {3 \, {\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} a^{2} b}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________