Optimal. Leaf size=149 \[ -\frac {3 b^2 \log \left (\frac {2}{1+i c x^2}\right ) \left (a+b \tan ^{-1}\left (c x^2\right )\right )}{2 c^2}+\frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^3}{4 c^2}-\frac {3 i b \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{4 c^2}-\frac {3 b x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^3-\frac {3 i b^3 \text {Li}_2\left (1-\frac {2}{i c x^2+1}\right )}{4 c^2} \]
[Out]
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Rubi [B] time = 4.74, antiderivative size = 951, normalized size of antiderivative = 6.38, 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 = {5035, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2439, 2416, 2396, 2433, 2374, 6589, 2411, 43, 2334, 12, 14, 2301, 6742, 2395, 2394, 2393, 2391, 2375, 2317, 2430, 2425} \[ \frac {3}{32} i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (i c x^2+1\right ) x^4+\frac {3}{32} i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (i c x^2+1\right ) x^4+\frac {3 b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (i c x^2+1\right ) x^2}{8 c}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac {i b^3 \left (i c x^2+1\right )^2 \log ^3\left (i c x^2+1\right )}{32 c^2}+\frac {i b^3 \left (i c x^2+1\right ) \log ^3\left (i c x^2+1\right )}{16 c^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}-\frac {3 i 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^2}-\frac {3 i b^3 \left (i c x^2+1\right ) \log ^2\left (i c x^2+1\right )}{16 c^2}-\frac {3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (i c x^2+1\right )}{32 c^2}+\frac {3 i b^2 \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{64 c^2}+\frac {3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}+\frac {3 i b \left (2 i a-b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac {1}{2} \left (i c x^2+1\right )\right )}{32 c^2}+\frac {3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac {1}{2} \left (i c x^2+1\right )\right )}{32 c^2}-\frac {3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (i c x^2+1\right )\right )}{8 c^2}-\frac {3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (i c x^2+1\right )}{32 c^2}-\frac {3 i b^3 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (i c x^2+1\right )}{8 c^2}-\frac {3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \text {PolyLog}\left (2,\frac {1}{2} \left (1-i c x^2\right )\right )}{16 c^2}-\frac {3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \text {PolyLog}\left (2,\frac {1}{2} \left (1-i c x^2\right )\right )}{16 c^2}+\frac {3 i b^3 \text {PolyLog}\left (2,\frac {1}{2} \left (1-i c x^2\right )\right )}{8 c^2}-\frac {3 i b^3 \text {PolyLog}\left (2,\frac {1}{2} \left (i c x^2+1\right )\right )}{8 c^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 14
Rule 43
Rule 2295
Rule 2296
Rule 2301
Rule 2304
Rule 2305
Rule 2317
Rule 2334
Rule 2374
Rule 2375
Rule 2389
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2396
Rule 2401
Rule 2411
Rule 2416
Rule 2425
Rule 2430
Rule 2433
Rule 2439
Rule 2454
Rule 5035
Rule 6589
Rule 6742
Rubi steps
\begin {align*} \int x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^3 \, dx &=\int \left (\frac {1}{8} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3+\frac {3}{8} i b x^3 \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^3 \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^3 \log ^3\left (1+i c x^2\right )\right ) \, dx\\ &=\frac {1}{8} \int x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3 \, dx+\frac {1}{8} (3 i b) \int x^3 \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^3 \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^3 \log ^3\left (1+i c x^2\right ) \, dx\\ &=\frac {1}{16} \operatorname {Subst}\left (\int x (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 x (-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 x (-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 x \log ^3(1+i c x) \, dx,x,x^2\right )\\ &=\frac {3}{32} i b x^4 \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}{32} i b^2 x^4 \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}{16} \operatorname {Subst}\left (\int \left (-\frac {i (2 a+i b \log (1-i c x))^3}{c}+\frac {i (1-i c x) (2 a+i b \log (1-i c x))^3}{c}\right ) \, dx,x,x^2\right )+\frac {1}{16} \left (i b^3\right ) \operatorname {Subst}\left (\int \left (\frac {i \log ^3(1+i c x)}{c}-\frac {i (1+i c x) \log ^3(1+i c x)}{c}\right ) \, dx,x,x^2\right )+\frac {1}{32} (3 b c) \operatorname {Subst}\left (\int \frac {x^2 (-2 i a+b \log (1-i c x))^2}{1+i c x} \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2 (-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^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2 (-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}{32} \left (3 b^3 c\right ) \operatorname {Subst}\left (\int \frac {x^2 \log ^2(1+i c x)}{1-i c x} \, dx,x,x^2\right )\\ &=\frac {3}{32} i b x^4 \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}{32} i b^2 x^4 \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 (1-i c x))^3 \, dx,x,x^2\right )}{16 c}+\frac {i \operatorname {Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^3 \, dx,x,x^2\right )}{16 c}-\frac {b^3 \operatorname {Subst}\left (\int \log ^3(1+i c x) \, dx,x,x^2\right )}{16 c}+\frac {b^3 \operatorname {Subst}\left (\int (1+i c x) \log ^3(1+i c x) \, dx,x,x^2\right )}{16 c}+\frac {1}{32} (3 b c) \operatorname {Subst}\left (\int \left (\frac {(-2 i a+b \log (1-i c x))^2}{c^2}-\frac {i x (-2 i a+b \log (1-i c x))^2}{c}+\frac {i (-2 i a+b \log (1-i c x))^2}{c^2 (-i+c x)}\right ) \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2 c\right ) \operatorname {Subst}\left (\int \left (-\frac {x (2 a+i b \log (1-i c x)) \log (1+i c x)}{c}+\frac {(2 a+i b \log (1-i c x)) \log (1+i c x)}{c^2 (-i+c x)}+\frac {(-2 i a+b \log (1-i c x)) \log (1+i c x)}{c^2}\right ) \, dx,x,x^2\right )-\frac {1}{16} \left (3 b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {x (2 a+i b \log (1-i c x)) \log (1+i c x)}{c}-\frac {(2 a+i b \log (1-i c x)) \log (1+i c x)}{c^2 (i+c x)}+\frac {(-2 i a+b \log (1-i c x)) \log (1+i c x)}{c^2}\right ) \, dx,x,x^2\right )+\frac {1}{32} \left (3 b^3 c\right ) \operatorname {Subst}\left (\int \left (\frac {\log ^2(1+i c x)}{c^2}+\frac {i x \log ^2(1+i c x)}{c}-\frac {i \log ^2(1+i c x)}{c^2 (i+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {3}{32} i b x^4 \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}{32} i b^2 x^4 \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}{32} (3 i b) \operatorname {Subst}\left (\int x (-2 i a+b \log (1-i c x))^2 \, dx,x,x^2\right )+\frac {1}{32} \left (3 i b^3\right ) \operatorname {Subst}\left (\int x \log ^2(1+i c x) \, dx,x,x^2\right )+\frac {\operatorname {Subst}\left (\int (2 a+i b \log (x))^3 \, dx,x,1-i c x^2\right )}{16 c^2}-\frac {\operatorname {Subst}\left (\int x (2 a+i b \log (x))^3 \, dx,x,1-i c x^2\right )}{16 c^2}+\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \log ^3(x) \, dx,x,1+i c x^2\right )}{16 c^2}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int x \log ^3(x) \, dx,x,1+i c x^2\right )}{16 c^2}+\frac {(3 i b) \operatorname {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x))^2}{-i+c x} \, dx,x,x^2\right )}{32 c}+\frac {(3 b) \operatorname {Subst}\left (\int (-2 i a+b \log (1-i c x))^2 \, dx,x,x^2\right )}{32 c}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {(2 a+i b \log (1-i c x)) \log (1+i c x)}{-i+c x} \, dx,x,x^2\right )}{16 c}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {(2 a+i b \log (1-i c x)) \log (1+i c x)}{i+c x} \, dx,x,x^2\right )}{16 c}-2 \frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^2\right )}{16 c}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\log ^2(1+i c x)}{i+c x} \, dx,x,x^2\right )}{32 c}+\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^2\right )}{32 c}\\ &=\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac {3 i 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 )}{32 c^2}+\frac {3}{32} i b x^4 \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 i b^3 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac {3}{32} i b^2 x^4 \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 b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac {i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac {1}{32} (3 i b) \operatorname {Subst}\left (\int \left (-\frac {i (-2 i a+b \log (1-i c x))^2}{c}+\frac {i (1-i c x) (-2 i a+b \log (1-i c x))^2}{c}\right ) \, dx,x,x^2\right )+\frac {1}{32} \left (3 i b^3\right ) \operatorname {Subst}\left (\int \left (\frac {i \log ^2(1+i c x)}{c}-\frac {i (1+i c x) \log ^2(1+i c x)}{c}\right ) \, dx,x,x^2\right )-2 \left (-\frac {3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}-\frac {1}{16} \left (3 i b^2\right ) \operatorname {Subst}\left (\int \frac {x (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^2\right )+\frac {1}{16} \left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {x \log (1+i c x)}{1-i c x} \, dx,x,x^2\right )\right )+\frac {(3 i b) \operatorname {Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-i c x^2\right )}{32 c^2}+\frac {(3 i b) \operatorname {Subst}\left (\int (-2 i a+b \log (x))^2 \, dx,x,1-i c x^2\right )}{32 c^2}-\frac {(3 i b) \operatorname {Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-i c x^2\right )}{16 c^2}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {(2 a+i b \log (2-x)) \log (x)}{x} \, dx,x,1+i c x^2\right )}{16 c^2}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (2-x) (2 a+i b \log (x))}{x} \, dx,x,1-i c x^2\right )}{16 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{32 c^2}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{16 c^2}-\frac {\left (3 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 )}{16 c}-\frac {\left (3 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 )}{16 c}\\ &=\frac {3 i b \left (1-i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{32 c^2}-\frac {3 i 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^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac {3 i 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 )}{32 c^2}+\frac {3}{32} i b x^4 \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 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac {9 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac {3 i b^3 \left (1+i c x^2\right )^2 \log ^2\left (1+i c x^2\right )}{64 c^2}-\frac {3 i b^3 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac {3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac {3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac {i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac {i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-2 \left (-\frac {3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}-\frac {1}{16} \left (3 i b^2\right ) \operatorname {Subst}\left (\int \left (-\frac {i (-2 i a+b \log (1-i c x))}{c}+\frac {-2 i a+b \log (1-i c x)}{c (-i+c x)}\right ) \, dx,x,x^2\right )+\frac {1}{16} \left (3 i b^3\right ) \operatorname {Subst}\left (\int \left (\frac {i \log (1+i c x)}{c}+\frac {\log (1+i c x)}{c (i+c x)}\right ) \, dx,x,x^2\right )\right )-\frac {(3 i b) \operatorname {Subst}\left (\int \frac {(2 a+i b \log (x))^2}{2-x} \, dx,x,1-i c x^2\right )}{32 c^2}-\frac {\left (3 i b^2\right ) \operatorname {Subst}\left (\int (-2 i a+b \log (x)) \, dx,x,1-i c x^2\right )}{16 c^2}-\frac {\left (3 i 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 )}{16 c^2}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-i c x^2\right )}{32 c^2}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-i c x^2\right )}{8 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int x \log (x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\log ^2(x)}{2-x} \, dx,x,1+i c x^2\right )}{32 c^2}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{16 c^2}+\frac {\left (3 i 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 )}{16 c^2}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{8 c^2}-\frac {(3 b) \operatorname {Subst}\left (\int (-2 i a+b \log (1-i c x))^2 \, dx,x,x^2\right )}{32 c}+\frac {(3 b) \operatorname {Subst}\left (\int (1-i c x) (-2 i a+b \log (1-i c x))^2 \, dx,x,x^2\right )}{32 c}-\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^2\right )}{32 c}+\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,x^2\right )}{32 c}\\ &=\frac {9 i a b^2 x^2}{8 c}+\frac {9 b^3 x^2}{16 c}-\frac {3 i b^3 \left (1-i c x^2\right )^2}{128 c^2}+\frac {3 i b^3 \left (1+i c x^2\right )^2}{128 c^2}+\frac {3 i b \left (1-i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{32 c^2}+\frac {3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}-\frac {3 i 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^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac {3 i 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 )}{32 c^2}+\frac {3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac {9 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac {3 i b^3 \left (1+i c x^2\right )^2 \log \left (1+i c x^2\right )}{64 c^2}+\frac {3}{32} i b x^4 \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 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac {9 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac {3 i b^3 \left (1+i c x^2\right )^2 \log ^2\left (1+i c x^2\right )}{64 c^2}+\frac {3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac {3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac {i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac {i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac {3 i 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 )}{16 c^2}-\frac {3 i 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 )}{16 c^2}-\frac {(3 i b) \operatorname {Subst}\left (\int (-2 i a+b \log (x))^2 \, dx,x,1-i c x^2\right )}{32 c^2}+\frac {(3 i b) \operatorname {Subst}\left (\int x (-2 i a+b \log (x))^2 \, dx,x,1-i 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+i b \log (x))}{x} \, dx,x,1-i c x^2\right )}{16 c^2}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{16 c^2}-\frac {\left (3 i 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 )}{16 c^2}-\frac {\left (3 i 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 )}{16 c^2}+\frac {\left (3 i 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 )}{16 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{8 c^2}-2 \left (-\frac {3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}-\frac {\left (3 i b^2\right ) \operatorname {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^2\right )}{16 c}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )}{16 c}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{i+c x} \, dx,x,x^2\right )}{16 c}-\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int \log (1+i c x) \, dx,x,x^2\right )}{16 c}\right )\\ &=\frac {9 i a b^2 x^2}{8 c}+\frac {9 b^3 x^2}{8 c}-\frac {3 i b^3 \left (1-i c x^2\right )^2}{128 c^2}+\frac {3 i b^3 \left (1+i c x^2\right )^2}{128 c^2}-\frac {9 i b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{16 c^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac {3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}-\frac {3 i 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^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac {3 i 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 )}{32 c^2}+\frac {3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac {9 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac {3 i b^3 \left (1+i c x^2\right )^2 \log \left (1+i c x^2\right )}{64 c^2}+\frac {3}{32} i b x^4 \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 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac {3 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{16 c^2}+\frac {3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac {3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac {i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac {i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac {3 i 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 )}{16 c^2}-\frac {3 b^2 \left (2 a+i 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 )}{16 c^2}-\frac {3 i b^3 \text {Li}_3\left (\frac {1}{2} \left (1-i c x^2\right )\right )}{16 c^2}+\frac {3 i b^3 \text {Li}_3\left (\frac {1}{2} \left (1+i c x^2\right )\right )}{16 c^2}-\frac {\left (3 i b^2\right ) \operatorname {Subst}\left (\int x (-2 i a+b \log (x)) \, dx,x,1-i c x^2\right )}{32 c^2}+\frac {\left (3 i b^2\right ) \operatorname {Subst}\left (\int (-2 i a+b \log (x)) \, dx,x,1-i c x^2\right )}{16 c^2}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int x \log (x) \, dx,x,1+i c x^2\right )}{32 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{16 c^2}+\frac {\left (3 i 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 )}{16 c^2}-\frac {\left (3 i 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 )}{16 c^2}-2 \left (\frac {3 i a b^2 x^2}{8 c}+\frac {3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{16 c^2}+\frac {3 i b^3 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac {3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{16 c^2}-\frac {\left (3 b^3\right ) \operatorname {Subst}\left (\int \log (1-i 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} i (-i+c x)\right )}{1-i 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} i (i+c x)\right )}{1+i c x} \, dx,x,x^2\right )}{16 c}\right )\\ &=\frac {3 i a b^2 x^2}{4 c}+\frac {15 b^3 x^2}{16 c}-\frac {9 i b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{16 c^2}+\frac {3 i b^2 \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{64 c^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac {3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}-\frac {3 i 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^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac {3 i 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 )}{32 c^2}+\frac {3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac {3 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c^2}+\frac {3}{32} i b x^4 \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 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac {3 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{16 c^2}+\frac {3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac {3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac {i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac {i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac {3 i 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 )}{16 c^2}-\frac {3 b^2 \left (2 a+i 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 )}{16 c^2}-2 \left (\frac {3 i a b^2 x^2}{8 c}+\frac {3 b^3 x^2}{16 c}+\frac {3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{16 c^2}+\frac {3 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^2}+\frac {3 i b^3 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac {3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{16 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{16 c^2}-\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{16 c^2}\right )+\frac {\left (3 i b^3\right ) \operatorname {Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{16 c^2}\\ &=\frac {3 i a b^2 x^2}{4 c}+\frac {3 b^3 x^2}{4 c}-\frac {3 i b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{8 c^2}+\frac {3 i b^2 \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )}{64 c^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 i a-b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac {3 b^2 \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{64 c^2}-\frac {3 i 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^2}+\frac {3 i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{64 c^2}+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{16 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^3}{32 c^2}+\frac {3 i 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 )}{32 c^2}+\frac {3 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{32 c^2}+\frac {3 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c^2}+\frac {3}{32} i b x^4 \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 i b \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \log \left (1+i c x^2\right )}{32 c^2}-\frac {3 i b^3 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{16 c^2}+\frac {3}{32} i b^2 x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )-\frac {3 b^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \log ^2\left (1+i c x^2\right )}{32 c^2}+\frac {i b^3 \left (1+i c x^2\right ) \log ^3\left (1+i c x^2\right )}{16 c^2}-\frac {i b^3 \left (1+i c x^2\right )^2 \log ^3\left (1+i c x^2\right )}{32 c^2}-\frac {3 i 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 )}{16 c^2}-\frac {3 b^2 \left (2 a+i 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 )}{16 c^2}-2 \left (\frac {3 i a b^2 x^2}{8 c}+\frac {3 b^3 x^2}{8 c}-\frac {3 i b^3 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{16 c^2}+\frac {3 i b^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{16 c^2}+\frac {3 i b^3 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{16 c^2}+\frac {3 i b^3 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c^2}-\frac {3 b^2 x^2 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{16 c}-\frac {3 i b^3 \text {Li}_2\left (\frac {1}{2} \left (1-i c x^2\right )\right )}{16 c^2}+\frac {3 i b^3 \text {Li}_2\left (\frac {1}{2} \left (1+i c x^2\right )\right )}{16 c^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.17, size = 170, normalized size = 1.14 \[ \frac {a \left (a c x^2 \left (a c x^2-3 b\right )+3 b^2 \log \left (c^2 x^4+1\right )\right )+3 b^2 \tan ^{-1}\left (c x^2\right )^2 \left (a c^2 x^4+a+b \left (-c x^2+i\right )\right )+3 b \tan ^{-1}\left (c x^2\right ) \left (a \left (a c^2 x^4+a-2 b c x^2\right )-2 b^2 \log \left (1+e^{2 i \tan ^{-1}\left (c x^2\right )}\right )\right )+b^3 \left (c^2 x^4+1\right ) \tan ^{-1}\left (c x^2\right )^3+3 i b^3 \text {Li}_2\left (-e^{2 i \tan ^{-1}\left (c x^2\right )}\right )}{4 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{3} \arctan \left (c x^{2}\right )^{3} + 3 \, a b^{2} x^{3} \arctan \left (c x^{2}\right )^{2} + 3 \, a^{2} b x^{3} \arctan \left (c x^{2}\right ) + a^{3} x^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arctan \left (c x^{2}\right ) + a\right )}^{3} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.56, size = 690, normalized size = 4.63 \[ -\frac {3 a \,b^{2} x^{4} \ln \left (-i c \,x^{2}+1\right )^{2}}{16}+\frac {3 a \,b^{2} \ln \left (c^{2} x^{4}+1\right )}{4 c^{2}}-\frac {3 a \,b^{2} \ln \left (-i c \,x^{2}+1\right )^{2}}{16 c^{2}}-\frac {3 b^{2} \left (i x^{4} b \ln \left (-i c \,x^{2}+1\right ) c^{2}+2 a \,c^{2} x^{4}-2 b c \,x^{2}+i b \ln \left (-i c \,x^{2}+1\right )+2 i b +2 a \right ) \ln \left (i c \,x^{2}+1\right )^{2}}{32 c^{2}}+\frac {3 i a^{2} b \,x^{4} \ln \left (-i c \,x^{2}+1\right )}{8}-\frac {i b^{3} x^{4} \ln \left (-i c \,x^{2}+1\right )^{3}}{32}+\frac {i b^{3} \left (c^{2} x^{4}+1\right ) \ln \left (i c \,x^{2}+1\right )^{3}}{32 c^{2}}-\frac {3 i a \,b^{2} x^{2} \ln \left (-i c \,x^{2}+1\right )}{4 c}+\frac {3 a^{2} b \arctan \left (c \,x^{2}\right )}{4 c^{2}}+\frac {3 b^{3} x^{2} \ln \left (-i c \,x^{2}+1\right )^{2}}{16 c}-\frac {i b^{3} \ln \left (-i c \,x^{2}+1\right )^{3}}{32 c^{2}}+\frac {3 i b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{2}-\RootOf \left (\textit {\_Z}^{2}+1, \mathit {index} =1\right )\right )}{\sum }\frac {\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (-i c \,x^{2}+1\right )+2 c \left (-\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (i \sqrt {\frac {i}{c}}+\sqrt {\frac {i}{c}}+x -\underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {\frac {i}{c}}}\right )+\ln \left (\frac {\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i \sqrt {\frac {i}{c}}-\sqrt {\frac {i}{c}}-x +\underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {\frac {i}{c}}}\right )\right )}{2 c}-\frac {\dilog \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (i \sqrt {\frac {i}{c}}+\sqrt {\frac {i}{c}}+x -\underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {\frac {i}{c}}}\right )+\dilog \left (\frac {\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i \sqrt {\frac {i}{c}}-\sqrt {\frac {i}{c}}-x +\underline {\hspace {1.25 ex}}\alpha \right )}{\sqrt {\frac {i}{c}}}\right )}{2 c}\right )\right ) b}{c}\right )}{4 c}+\left (\frac {3 i b^{3} \left (c^{2} x^{4}+1\right ) \ln \left (-i c \,x^{2}+1\right )^{2}}{32 c^{2}}+\frac {3 b^{2} x^{2} \left (a \,x^{2} c -b \right ) \ln \left (-i c \,x^{2}+1\right )}{8 c}-\frac {3 i b \left (a^{2} c^{2} x^{4}-2 a b c \,x^{2}+b^{2} \ln \left (-i c \,x^{2}+1\right )+i \ln \left (-i c \,x^{2}+1\right ) a b \right )}{8 c^{2}}\right ) \ln \left (i c \,x^{2}+1\right )-\frac {3 b \,x^{2} a^{2}}{4 c}+\frac {x^{4} a^{3}}{4}+\frac {3 i b^{3} \ln \left (-i c \,x^{2}+1\right )^{2}}{16 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3}{4} \, a b^{2} x^{4} \arctan \left (c x^{2}\right )^{2} + \frac {1}{4} \, a^{3} x^{4} + \frac {3}{4} \, {\left (x^{4} \arctan \left (c x^{2}\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\arctan \left (c x^{2}\right )}{c^{3}}\right )}\right )} a^{2} b - \frac {3}{4} \, {\left (2 \, c {\left (\frac {x^{2}}{c^{2}} - \frac {\arctan \left (c x^{2}\right )}{c^{3}}\right )} \arctan \left (c x^{2}\right ) + \frac {\arctan \left (c x^{2}\right )^{2} - \log \left (4 \, c^{5} x^{4} + 4 \, c^{3}\right )}{c^{2}}\right )} a b^{2} + \frac {1}{128} \, {\left (4 \, x^{4} \arctan \left (c x^{2}\right )^{3} - 3 \, x^{4} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right )^{2} + 128 \, \int \frac {12 \, c^{2} x^{7} \arctan \left (c x^{2}\right ) \log \left (c^{2} x^{4} + 1\right ) - 12 \, c x^{5} \arctan \left (c x^{2}\right )^{2} + 56 \, {\left (c^{2} x^{7} + x^{3}\right )} \arctan \left (c x^{2}\right )^{3} + 3 \, {\left (c x^{5} + 2 \, {\left (c^{2} x^{7} + x^{3}\right )} \arctan \left (c x^{2}\right )\right )} \log \left (c^{2} x^{4} + 1\right )^{2}}{64 \, {\left (c^{2} x^{4} + 1\right )}}\,{d x}\right )} b^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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