Optimal. Leaf size=240 \[ -\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x^3}\right ) \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{3 c^3}-\frac{b^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x^3}\right )}{6 c^3}+\frac{a b^2 x^3}{3 c^2}-\frac{i \left (a+b \tan ^{-1}\left (c x^3\right )\right )^3}{9 c^3}-\frac{b \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{6 c^3}-\frac{b \log \left (\frac{2}{1+i c x^3}\right ) \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{3 c^3}+\frac{1}{9} x^9 \left (a+b \tan ^{-1}\left (c x^3\right )\right )^3-\frac{b x^6 \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{6 c}-\frac{b^3 \log \left (c^2 x^6+1\right )}{6 c^3}+\frac{b^3 x^3 \tan ^{-1}\left (c x^3\right )}{3 c^2} \]
[Out]
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Rubi [B] time = 7.0337, antiderivative size = 1867, normalized size of antiderivative = 7.78, number of steps used = 239, number of rules used = 32, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 2., Rules used = {5035, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2398, 2411, 43, 2334, 12, 14, 2301, 2439, 2416, 2396, 2433, 2374, 6589, 6742, 2430, 2394, 2393, 2391, 2395, 2375, 2317, 2410, 2425} \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 5035
Rule 2454
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rule 2398
Rule 2411
Rule 43
Rule 2334
Rule 12
Rule 14
Rule 2301
Rule 2439
Rule 2416
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 6742
Rule 2430
Rule 2394
Rule 2393
Rule 2391
Rule 2395
Rule 2375
Rule 2317
Rule 2410
Rule 2425
Rubi steps
\begin{align*} \int x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )^3 \, dx &=\int \left (\frac{1}{8} x^8 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3+\frac{3}{8} i b x^8 \left (-2 i a+b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )-\frac{3}{8} i b^2 x^8 \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )+\frac{1}{8} i b^3 x^8 \log ^3\left (1+i c x^3\right )\right ) \, dx\\ &=\frac{1}{8} \int x^8 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3 \, dx+\frac{1}{8} (3 i b) \int x^8 \left (-2 i a+b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right ) \, dx-\frac{1}{8} \left (3 i b^2\right ) \int x^8 \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right ) \, dx+\frac{1}{8} \left (i b^3\right ) \int x^8 \log ^3\left (1+i c x^3\right ) \, dx\\ &=\frac{1}{24} \operatorname{Subst}\left (\int x^2 (2 a+i b \log (1-i c x))^3 \, dx,x,x^3\right )+\frac{1}{8} (i b) \operatorname{Subst}\left (\int x^2 (-2 i a+b \log (1-i c x))^2 \log (1+i c x) \, dx,x,x^3\right )-\frac{1}{8} \left (i b^2\right ) \operatorname{Subst}\left (\int x^2 (-2 i a+b \log (1-i c x)) \log ^2(1+i c x) \, dx,x,x^3\right )+\frac{1}{24} \left (i b^3\right ) \operatorname{Subst}\left (\int x^2 \log ^3(1+i c x) \, dx,x,x^3\right )\\ &=\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )+\frac{1}{24} \operatorname{Subst}\left (\int \left (-\frac{(2 a+i b \log (1-i c x))^3}{c^2}+\frac{2 (1-i c x) (2 a+i b \log (1-i c x))^3}{c^2}-\frac{(1-i c x)^2 (2 a+i b \log (1-i c x))^3}{c^2}\right ) \, dx,x,x^3\right )+\frac{1}{24} \left (i b^3\right ) \operatorname{Subst}\left (\int \left (-\frac{\log ^3(1+i c x)}{c^2}+\frac{2 (1+i c x) \log ^3(1+i c x)}{c^2}-\frac{(1+i c x)^2 \log ^3(1+i c x)}{c^2}\right ) \, dx,x,x^3\right )+\frac{1}{24} (b c) \operatorname{Subst}\left (\int \frac{x^3 (-2 i a+b \log (1-i c x))^2}{1+i c x} \, dx,x,x^3\right )-\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3 (-2 i a+b \log (1-i c x)) \log (1+i c x)}{1-i c x} \, dx,x,x^3\right )-\frac{1}{12} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3 (-2 i a+b \log (1-i c x)) \log (1+i c x)}{1+i c x} \, dx,x,x^3\right )+\frac{1}{24} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{x^3 \log ^2(1+i c x)}{1-i c x} \, dx,x,x^3\right )\\ &=\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )-\frac{\operatorname{Subst}\left (\int (2 a+i b \log (1-i c x))^3 \, dx,x,x^3\right )}{24 c^2}-\frac{\operatorname{Subst}\left (\int (1-i c x)^2 (2 a+i b \log (1-i c x))^3 \, dx,x,x^3\right )}{24 c^2}+\frac{\operatorname{Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^3 \, dx,x,x^3\right )}{12 c^2}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \log ^3(1+i c x) \, dx,x,x^3\right )}{24 c^2}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int (1+i c x)^2 \log ^3(1+i c x) \, dx,x,x^3\right )}{24 c^2}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int (1+i c x) \log ^3(1+i c x) \, dx,x,x^3\right )}{12 c^2}+\frac{1}{24} (b c) \operatorname{Subst}\left (\int \left (\frac{i (-2 i a+b \log (1-i c x))^2}{c^3}+\frac{x (-2 i a+b \log (1-i c x))^2}{c^2}-\frac{i x^2 (-2 i a+b \log (1-i c x))^2}{c}-\frac{(-2 i a+b \log (1-i c x))^2}{c^3 (-i+c x)}\right ) \, dx,x,x^3\right )-\frac{1}{12} \left (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^3}-\frac{x^2 (2 a+i b \log (1-i c x)) \log (1+i c x)}{c}+\frac{i (2 a+i b \log (1-i c x)) \log (1+i c x)}{c^3 (-i+c x)}+\frac{x (-2 i a+b \log (1-i c x)) \log (1+i c x)}{c^2}\right ) \, dx,x,x^3\right )-\frac{1}{12} \left (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^3}+\frac{x^2 (2 a+i b \log (1-i c x)) \log (1+i c x)}{c}+\frac{i (2 a+i b \log (1-i c x)) \log (1+i c x)}{c^3 (i+c x)}+\frac{x (-2 i a+b \log (1-i c x)) \log (1+i c x)}{c^2}\right ) \, dx,x,x^3\right )+\frac{1}{24} \left (b^3 c\right ) \operatorname{Subst}\left (\int \left (-\frac{i \log ^2(1+i c x)}{c^3}+\frac{x \log ^2(1+i c x)}{c^2}+\frac{i x^2 \log ^2(1+i c x)}{c}-\frac{\log ^2(1+i c x)}{c^3 (i+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )-\frac{1}{24} (i b) \operatorname{Subst}\left (\int x^2 (-2 i a+b \log (1-i c x))^2 \, dx,x,x^3\right )+\frac{1}{24} \left (i b^3\right ) \operatorname{Subst}\left (\int x^2 \log ^2(1+i c x) \, dx,x,x^3\right )-\frac{i \operatorname{Subst}\left (\int (2 a+i b \log (x))^3 \, dx,x,1-i c x^3\right )}{24 c^3}-\frac{i \operatorname{Subst}\left (\int x^2 (2 a+i b \log (x))^3 \, dx,x,1-i c x^3\right )}{24 c^3}+\frac{i \operatorname{Subst}\left (\int x (2 a+i b \log (x))^3 \, dx,x,1-i c x^3\right )}{12 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \log ^3(x) \, dx,x,1+i c x^3\right )}{24 c^3}-\frac{b^3 \operatorname{Subst}\left (\int x^2 \log ^3(x) \, dx,x,1+i c x^3\right )}{24 c^3}+\frac{b^3 \operatorname{Subst}\left (\int x \log ^3(x) \, dx,x,1+i c x^3\right )}{12 c^3}+\frac{(i b) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x))^2 \, dx,x,x^3\right )}{24 c^2}-\frac{b \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x))^2}{-i+c x} \, dx,x,x^3\right )}{24 c^2}-\frac{\left (i 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^3\right )}{12 c^2}-\frac{\left (i 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^3\right )}{12 c^2}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^3\right )}{24 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{i+c x} \, dx,x,x^3\right )}{24 c^2}+\frac{b \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x))^2 \, dx,x,x^3\right )}{24 c}-2 \frac{b^2 \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^3\right )}{12 c}+\frac{b^3 \operatorname{Subst}\left (\int x \log ^2(1+i c x) \, dx,x,x^3\right )}{24 c}\\ &=-\frac{1}{72} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2-\frac{i \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}+\frac{i \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}-\frac{i \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{72 c^3}-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )+\frac{1}{72} i b^3 x^9 \log ^2\left (1+i c x^3\right )-\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{24 c^3}+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )-\frac{b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log ^3\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log ^3\left (1+i c x^3\right )}{72 c^3}-2 \left (-\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c}-\frac{1}{24} \left (i b^2\right ) \operatorname{Subst}\left (\int \frac{x^2 (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^3\right )+\frac{1}{24} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{x^2 \log (1+i c x)}{1-i c x} \, dx,x,x^3\right )\right )-\frac{b \operatorname{Subst}\left (\int x^2 (2 a+i b \log (x))^2 \, dx,x,1-i c x^3\right )}{24 c^3}-\frac{b \operatorname{Subst}\left (\int (-2 i a+b \log (x))^2 \, dx,x,1-i c x^3\right )}{24 c^3}-\frac{b \operatorname{Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-i c x^3\right )}{8 c^3}+\frac{b \operatorname{Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-i c x^3\right )}{8 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (2-x)) \log (x)}{x} \, dx,x,1+i c x^3\right )}{12 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2-x) (2 a+i b \log (x))}{x} \, dx,x,1-i c x^3\right )}{12 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^3\right )}{24 c^3}+\frac{b^3 \operatorname{Subst}\left (\int x^2 \log ^2(x) \, dx,x,1+i c x^3\right )}{24 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^3\right )}{8 c^3}-\frac{b^3 \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+i c x^3\right )}{8 c^3}-\frac{\left (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^3\right )}{12 c^2}+\frac{\left (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^3\right )}{12 c^2}+\frac{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^3\right )}{24 c}+\frac{b^3 \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^3\right )}{24 c}+\frac{1}{36} \left (b^2 c\right ) \operatorname{Subst}\left (\int \frac{x^3 (-2 i a+b \log (1-i c x))}{1-i c x} \, dx,x,x^3\right )+\frac{1}{36} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{x^3 \log (1+i c x)}{1+i c x} \, dx,x,x^3\right )\\ &=-\frac{1}{72} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2-\frac{b \left (1-i c x^3\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )^2}{24 c^3}-\frac{b \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{8 c^3}+\frac{b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 c^3}-\frac{b \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{72 c^3}-\frac{i \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}+\frac{i \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}-\frac{i \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{72 c^3}-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )-\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )}{24 c^3}+\frac{1}{72} i b^3 x^9 \log ^2\left (1+i c x^3\right )+\frac{b^3 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 c^3}-\frac{b^3 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{16 c^3}+\frac{b^3 \left (1+i c x^3\right )^3 \log ^2\left (1+i c x^3\right )}{72 c^3}-\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{24 c^3}+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log ^3\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log ^3\left (1+i c x^3\right )}{72 c^3}+\frac{1}{36} \left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i}{c}+\frac{i x}{c}\right )^3 (-2 i a+b \log (x))}{x} \, dx,x,1-i c x^3\right )-2 \left (-\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c}-\frac{1}{24} \left (i b^2\right ) \operatorname{Subst}\left (\int \left (\frac{-2 i a+b \log (1-i c x)}{c^2}-\frac{i x (-2 i a+b \log (1-i c x))}{c}+\frac{i (-2 i a+b \log (1-i c x))}{c^2 (-i+c x)}\right ) \, dx,x,x^3\right )+\frac{1}{24} \left (i b^3\right ) \operatorname{Subst}\left (\int \left (\frac{\log (1+i c x)}{c^2}+\frac{i x \log (1+i c x)}{c}-\frac{i \log (1+i c x)}{c^2 (i+c x)}\right ) \, dx,x,x^3\right )\right )-\frac{b \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{2-x} \, dx,x,1-i c x^3\right )}{24 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int x^2 (2 a+i b \log (x)) \, dx,x,1-i c x^3\right )}{36 c^3}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-i c x^3\right )}{8 c^3}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-i c x^3\right )}{4 c^3}+\frac{b^2 \operatorname{Subst}\left (\int (-2 i a+b \log (x)) \, dx,x,1-i c x^3\right )}{12 c^3}+\frac{b^2 \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^3\right )}{12 c^3}-\frac{b^3 \operatorname{Subst}\left (\int x^2 \log (x) \, dx,x,1+i c x^3\right )}{36 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log ^2(x)}{2-x} \, dx,x,1+i c x^3\right )}{24 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{12 c^3}+\frac{b^3 \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^3\right )}{12 c^3}+\frac{b^3 \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+i c x^3\right )}{8 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{4 c^3}-\frac{(i b) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x))^2 \, dx,x,x^3\right )}{24 c^2}+\frac{(i b) \operatorname{Subst}\left (\int (1-i c x) (-2 i a+b \log (1-i c x))^2 \, dx,x,x^3\right )}{24 c^2}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^3\right )}{24 c^2}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,x^3\right )}{24 c^2}+\frac{1}{36} \left (b^3 c\right ) \operatorname{Subst}\left (\int \left (\frac{i \log (1+i c x)}{c^3}+\frac{x \log (1+i c x)}{c^2}-\frac{i x^2 \log (1+i c x)}{c}-\frac{\log (1+i c x)}{c^3 (-i+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{a b^2 x^3}{3 c^2}+\frac{i b^3 x^3}{6 c^2}-\frac{b^3 \left (1-i c x^3\right )^2}{32 c^3}-\frac{b^3 \left (1+i c x^3\right )^2}{32 c^3}+\frac{b^3 \left (1+i c x^3\right )^3}{324 c^3}+\frac{i b^3 \left (i+c x^3\right )^3}{324 c^3}-\frac{1}{72} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2-\frac{b \left (1-i c x^3\right ) \left (2 i a-b \log \left (1-i c x^3\right )\right )^2}{24 c^3}-\frac{i b^2 \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{16 c^3}+\frac{i b^2 \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{108 c^3}-\frac{b \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{8 c^3}+\frac{b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 c^3}-\frac{b \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{72 c^3}-\frac{i \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}+\frac{i \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}-\frac{i \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{72 c^3}+\frac{1}{216} i b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{6 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{16 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log \left (1+i c x^3\right )}{108 c^3}+\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )-\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )}{24 c^3}+\frac{1}{72} i b^3 x^9 \log ^2\left (1+i c x^3\right )+\frac{b^3 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 c^3}-\frac{b^3 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{16 c^3}+\frac{b^3 \left (1+i c x^3\right )^3 \log ^2\left (1+i c x^3\right )}{72 c^3}-\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{12 c^3}+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log ^3\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log ^3\left (1+i c x^3\right )}{72 c^3}+\frac{b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}-\frac{b^3 \log \left (1+i c x^3\right ) \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{12 c^3}-\frac{1}{36} \left (i b^3\right ) \operatorname{Subst}\left (\int -\frac{i \left (x \left (18-9 x+2 x^2\right )-6 \log (x)\right )}{6 c^3 x} \, dx,x,1-i c x^3\right )-\frac{1}{36} \left (i b^3\right ) \operatorname{Subst}\left (\int x^2 \log (1+i c x) \, dx,x,x^3\right )+\frac{b \operatorname{Subst}\left (\int (-2 i a+b \log (x))^2 \, dx,x,1-i c x^3\right )}{24 c^3}-\frac{b \operatorname{Subst}\left (\int x (-2 i a+b \log (x))^2 \, dx,x,1-i c x^3\right )}{24 c^3}-\frac{\left (i 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^3\right )}{12 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^3\right )}{24 c^3}-\frac{b^3 \operatorname{Subst}\left (\int x \log ^2(x) \, dx,x,1+i c x^3\right )}{24 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{12 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right ) \log (x)}{x} \, dx,x,1+i c x^3\right )}{12 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-i c x^3\right )}{12 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+i c x^3\right )}{12 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{4 c^3}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \log (1+i c x) \, dx,x,x^3\right )}{36 c^2}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{-i+c x} \, dx,x,x^3\right )}{36 c^2}+\frac{b^3 \operatorname{Subst}\left (\int x \log (1+i c x) \, dx,x,x^3\right )}{36 c}-2 \left (-\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )}{24 c^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^3\right )}{24 c^2}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \log (1+i c x) \, dx,x,x^3\right )}{24 c^2}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log (1+i c x)}{i+c x} \, dx,x,x^3\right )}{24 c^2}-\frac{b^2 \operatorname{Subst}\left (\int x (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )}{24 c}-\frac{b^3 \operatorname{Subst}\left (\int x \log (1+i c x) \, dx,x,x^3\right )}{24 c}\right )\\ &=\frac{a b^2 x^3}{3 c^2}-\frac{b^3 \left (1-i c x^3\right )^2}{32 c^3}-\frac{b^3 \left (1+i c x^3\right )^2}{32 c^3}+\frac{b^3 \left (1+i c x^3\right )^3}{324 c^3}+\frac{i b^3 \left (i+c x^3\right )^3}{324 c^3}-\frac{b^3 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{6 c^3}-\frac{1}{72} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2-\frac{b \left (1-i c x^3\right )^2 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2}{48 c^3}-\frac{i b^2 \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{16 c^3}+\frac{i b^2 \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{108 c^3}-\frac{b \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{8 c^3}+\frac{b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 c^3}-\frac{b \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{72 c^3}-\frac{i \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}+\frac{i \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}-\frac{i \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{72 c^3}+\frac{1}{216} i b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{b^3 x^6 \log \left (1+i c x^3\right )}{72 c}-\frac{1}{108} i b^3 x^9 \log \left (1+i c x^3\right )-\frac{b^3 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{6 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{16 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log \left (1+i c x^3\right )}{108 c^3}+\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )-\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )}{24 c^3}+\frac{1}{72} i b^3 x^9 \log ^2\left (1+i c x^3\right )+\frac{b^3 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{8 c^3}-\frac{b^3 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{12 c^3}+\frac{b^3 \left (1+i c x^3\right )^3 \log ^2\left (1+i c x^3\right )}{72 c^3}-\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{12 c^3}+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log ^3\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log ^3\left (1+i c x^3\right )}{72 c^3}+\frac{b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}+\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}-\frac{b^3 \log \left (1+i c x^3\right ) \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{6 c^3}+\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}+\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{12 c^3}-\frac{1}{72} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+i c x} \, dx,x,x^3\right )+\frac{b^2 \operatorname{Subst}\left (\int x (-2 i a+b \log (x)) \, dx,x,1-i c x^3\right )}{24 c^3}-\frac{b^2 \operatorname{Subst}\left (\int (-2 i a+b \log (x)) \, dx,x,1-i c x^3\right )}{12 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{x \left (18-9 x+2 x^2\right )-6 \log (x)}{x} \, dx,x,1-i c x^3\right )}{216 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{36 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+i c x^3\right )}{36 c^3}+\frac{b^3 \operatorname{Subst}\left (\int x \log (x) \, dx,x,1+i c x^3\right )}{24 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{12 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1-i c x^3\right )}{12 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{2}\right )}{x} \, dx,x,1+i c x^3\right )}{12 c^3}-2 \left (-\frac{a b^2 x^3}{12 c^2}+\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c}-\frac{b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}-\frac{b^3 x^6 \log \left (1+i c x^3\right )}{48 c}+\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c^3}-\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c}-\frac{1}{48} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-i c x} \, dx,x,x^3\right )+\frac{1}{48} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+i c x} \, dx,x,x^3\right )+\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{24 c^3}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \log (1-i c x) \, dx,x,x^3\right )}{24 c^2}+\frac{\left (i 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^3\right )}{24 c^2}-\frac{\left (i 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^3\right )}{24 c^2}\right )-\frac{1}{108} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{x^3}{1+i c x} \, dx,x,x^3\right )\\ &=\frac{a b^2 x^3}{2 c^2}+\frac{i b^3 x^3}{18 c^2}-\frac{b^3 \left (1-i c x^3\right )^2}{24 c^3}-\frac{b^3 \left (1+i c x^3\right )^2}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^3}{324 c^3}+\frac{i b^3 \left (i+c x^3\right )^3}{324 c^3}-\frac{b^3 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{6 c^3}-\frac{b^2 \left (1-i c x^3\right )^2 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c^3}-\frac{1}{72} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2-\frac{b \left (1-i c x^3\right )^2 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2}{48 c^3}-\frac{i b^2 \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{16 c^3}+\frac{i b^2 \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{108 c^3}-\frac{b \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{8 c^3}+\frac{b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 c^3}-\frac{b \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{72 c^3}-\frac{i \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}+\frac{i \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}-\frac{i \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{72 c^3}+\frac{1}{216} i b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{b^3 x^6 \log \left (1+i c x^3\right )}{72 c}-\frac{1}{108} i b^3 x^9 \log \left (1+i c x^3\right )-\frac{2 b^3 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{9 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{12 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log \left (1+i c x^3\right )}{108 c^3}+\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )-\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \log ^2\left (1+i c x^3\right )}{72 c^3}+\frac{1}{72} i b^3 x^9 \log ^2\left (1+i c x^3\right )+\frac{b^3 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{8 c^3}-\frac{b^3 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{12 c^3}+\frac{b^3 \left (1+i c x^3\right )^3 \log ^2\left (1+i c x^3\right )}{72 c^3}-\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{12 c^3}+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log ^3\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log ^3\left (1+i c x^3\right )}{72 c^3}+\frac{b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}+\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}-\frac{b^3 \log \left (1+i c x^3\right ) \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{6 c^3}+\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{6 c^3}+\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{6 c^3}-\frac{1}{72} \left (i b^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{i x}{c}+\frac{i}{c^2 (-i+c x)}\right ) \, dx,x,x^3\right )-\frac{b^3 \operatorname{Subst}\left (\int \left (18-9 x+2 x^2-\frac{6 \log (x)}{x}\right ) \, dx,x,1-i c x^3\right )}{216 c^3}-2 \left (-\frac{a b^2 x^3}{12 c^2}-\frac{i b^3 x^3}{24 c^2}+\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c}-\frac{b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}-\frac{b^3 x^6 \log \left (1+i c x^3\right )}{48 c}+\frac{b^3 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c^3}-\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c}+\frac{1}{48} \left (i b^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{i x}{c}+\frac{i}{c^2 (-i+c x)}\right ) \, dx,x,x^3\right )-\frac{1}{48} \left (i b^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{i x}{c}-\frac{i}{c^2 (i+c x)}\right ) \, dx,x,x^3\right )-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1-i c x^3\right )}{24 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2}\right )}{x} \, dx,x,1+i c x^3\right )}{24 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{24 c^3}\right )-\frac{b^3 \operatorname{Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{12 c^3}-\frac{1}{108} \left (b^3 c\right ) \operatorname{Subst}\left (\int \left (\frac{i}{c^3}+\frac{x}{c^2}-\frac{i x^2}{c}-\frac{1}{c^3 (-i+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac{a b^2 x^3}{2 c^2}+\frac{7 i b^3 x^3}{216 c^2}-\frac{5 b^3 x^6}{432 c}+\frac{1}{324} i b^3 x^9-\frac{b^3 \left (1-i c x^3\right )^2}{48 c^3}-\frac{b^3 \left (1-i c x^3\right )^3}{324 c^3}-\frac{b^3 \left (1+i c x^3\right )^2}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^3}{324 c^3}+\frac{i b^3 \left (i+c x^3\right )^3}{324 c^3}+\frac{5 b^3 \log \left (i-c x^3\right )}{216 c^3}-\frac{b^3 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{4 c^3}-\frac{b^2 \left (1-i c x^3\right )^2 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c^3}-\frac{1}{72} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2-\frac{b \left (1-i c x^3\right )^2 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2}{48 c^3}-\frac{i b^2 \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{16 c^3}+\frac{i b^2 \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{108 c^3}-\frac{b \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{8 c^3}+\frac{b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 c^3}-\frac{b \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{72 c^3}-\frac{i \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}+\frac{i \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}-\frac{i \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{72 c^3}+\frac{1}{216} i b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{b^3 x^6 \log \left (1+i c x^3\right )}{72 c}-\frac{1}{108} i b^3 x^9 \log \left (1+i c x^3\right )-\frac{2 b^3 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{9 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{12 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log \left (1+i c x^3\right )}{108 c^3}+\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )-\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \log ^2\left (1+i c x^3\right )}{72 c^3}+\frac{1}{72} i b^3 x^9 \log ^2\left (1+i c x^3\right )+\frac{b^3 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{8 c^3}-\frac{b^3 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{12 c^3}+\frac{b^3 \left (1+i c x^3\right )^3 \log ^2\left (1+i c x^3\right )}{72 c^3}-\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{12 c^3}+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log ^3\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log ^3\left (1+i c x^3\right )}{72 c^3}+\frac{b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}+\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}-\frac{b^3 \log \left (1+i c x^3\right ) \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{6 c^3}-2 \left (-\frac{a b^2 x^3}{12 c^2}+\frac{b^3 x^6}{48 c}-\frac{b^3 \log \left (i-c x^3\right )}{48 c^3}+\frac{b^3 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{24 c^3}+\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c}-\frac{b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}-\frac{b^3 x^6 \log \left (1+i c x^3\right )}{48 c}+\frac{b^3 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c^3}-\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c}-\frac{b^3 \log \left (i+c x^3\right )}{48 c^3}+\frac{b^3 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{24 c^3}+\frac{b^3 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}\right )+\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{6 c^3}+\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{6 c^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1-i c x^3\right )}{36 c^3}\\ &=\frac{a b^2 x^3}{2 c^2}+\frac{7 i b^3 x^3}{216 c^2}-\frac{5 b^3 x^6}{432 c}+\frac{1}{324} i b^3 x^9-\frac{b^3 \left (1-i c x^3\right )^2}{48 c^3}-\frac{b^3 \left (1-i c x^3\right )^3}{324 c^3}-\frac{b^3 \left (1+i c x^3\right )^2}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^3}{324 c^3}+\frac{i b^3 \left (i+c x^3\right )^3}{324 c^3}+\frac{5 b^3 \log \left (i-c x^3\right )}{216 c^3}-\frac{b^3 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{4 c^3}+\frac{b^3 \log ^2\left (1-i c x^3\right )}{72 c^3}-\frac{b^2 \left (1-i c x^3\right )^2 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c^3}-\frac{1}{72} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2-\frac{b \left (1-i c x^3\right )^2 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2}{48 c^3}-\frac{i b^2 \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{16 c^3}+\frac{i b^2 \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{108 c^3}-\frac{b \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{8 c^3}+\frac{b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 c^3}-\frac{b \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{72 c^3}-\frac{i \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}+\frac{i \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 c^3}-\frac{i \left (1-i c x^3\right )^3 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{72 c^3}+\frac{1}{216} i b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \left (\frac{18 i \left (1-i c x^3\right )}{c^3}-\frac{9 i \left (1-i c x^3\right )^2}{c^3}-\frac{2 \left (i+c x^3\right )^3}{c^3}-\frac{6 i \log \left (1-i c x^3\right )}{c^3}\right )-\frac{b \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}+\frac{b^3 x^6 \log \left (1+i c x^3\right )}{72 c}-\frac{1}{108} i b^3 x^9 \log \left (1+i c x^3\right )-\frac{2 b^3 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{9 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{12 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log \left (1+i c x^3\right )}{108 c^3}+\frac{1}{24} i b x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )-\frac{b \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \log ^2\left (1+i c x^3\right )}{72 c^3}+\frac{1}{72} i b^3 x^9 \log ^2\left (1+i c x^3\right )+\frac{b^3 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{8 c^3}-\frac{b^3 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{12 c^3}+\frac{b^3 \left (1+i c x^3\right )^3 \log ^2\left (1+i c x^3\right )}{72 c^3}-\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{12 c^3}+\frac{1}{24} i b^2 x^9 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \left (1+i c x^3\right )^2 \log ^3\left (1+i c x^3\right )}{24 c^3}-\frac{b^3 \left (1+i c x^3\right )^3 \log ^3\left (1+i c x^3\right )}{72 c^3}+\frac{b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}+\frac{i b^2 \left (2 a+i b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{12 c^3}-\frac{b^3 \log \left (1+i c x^3\right ) \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{6 c^3}-2 \left (-\frac{a b^2 x^3}{12 c^2}+\frac{b^3 x^6}{48 c}-\frac{b^3 \log \left (i-c x^3\right )}{48 c^3}+\frac{b^3 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{24 c^3}+\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )}{48 c}-\frac{b^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}-\frac{b^3 x^6 \log \left (1+i c x^3\right )}{48 c}+\frac{b^3 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{24 c^3}+\frac{b^3 \log \left (\frac{1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c^3}-\frac{b^2 x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{24 c}-\frac{b^3 \log \left (i+c x^3\right )}{48 c^3}+\frac{b^3 \text{Li}_2\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{24 c^3}+\frac{b^3 \text{Li}_2\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{24 c^3}\right )+\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1-i c x^3\right )\right )}{6 c^3}+\frac{b^3 \text{Li}_3\left (\frac{1}{2} \left (1+i c x^3\right )\right )}{6 c^3}\\ \end{align*}
Mathematica [A] time = 0.504979, size = 346, normalized size = 1.44 \[ \frac{6 i b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}\left (c x^3\right )}\right ) \left (a+b \tan ^{-1}\left (c x^3\right )\right )-3 b^3 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}\left (c x^3\right )}\right )-3 a^2 b c^2 x^6+3 a^2 b \log \left (c^2 x^6+1\right )+6 a^2 b c^3 x^9 \tan ^{-1}\left (c x^3\right )+2 a^3 c^3 x^9+6 a b^2 c^3 x^9 \tan ^{-1}\left (c x^3\right )^2-6 a b^2 c^2 x^6 \tan ^{-1}\left (c x^3\right )+6 a b^2 c x^3+6 i a b^2 \tan ^{-1}\left (c x^3\right )^2-6 a b^2 \tan ^{-1}\left (c x^3\right )-12 a b^2 \tan ^{-1}\left (c x^3\right ) \log \left (1+e^{2 i \tan ^{-1}\left (c x^3\right )}\right )-3 b^3 \log \left (c^2 x^6+1\right )+2 b^3 c^3 x^9 \tan ^{-1}\left (c x^3\right )^3-3 b^3 c^2 x^6 \tan ^{-1}\left (c x^3\right )^2+6 b^3 c x^3 \tan ^{-1}\left (c x^3\right )+2 i b^3 \tan ^{-1}\left (c x^3\right )^3-3 b^3 \tan ^{-1}\left (c x^3\right )^2-6 b^3 \tan ^{-1}\left (c x^3\right )^2 \log \left (1+e^{2 i \tan ^{-1}\left (c x^3\right )}\right )}{18 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.279, size = 0, normalized size = 0. \begin{align*} \int{x}^{8} \left ( a+b\arctan \left ( c{x}^{3} \right ) \right ) ^{3}\, dx \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}{72} \, b^{3} x^{9} \arctan \left (c x^{3}\right )^{3} - \frac{1}{96} \, b^{3} x^{9} \arctan \left (c x^{3}\right ) \log \left (c^{2} x^{6} + 1\right )^{2} + \frac{1}{9} \, a^{3} x^{9} + \frac{1}{6} \,{\left (2 \, x^{9} \arctan \left (c x^{3}\right ) -{\left (\frac{x^{6}}{c^{2}} - \frac{\log \left (c^{2} x^{6} + 1\right )}{c^{4}}\right )} c\right )} a^{2} b + \int \frac{4 \, b^{3} c^{2} x^{14} \arctan \left (c x^{3}\right ) \log \left (c^{2} x^{6} + 1\right ) + 28 \,{\left (b^{3} c^{2} x^{14} + b^{3} x^{8}\right )} \arctan \left (c x^{3}\right )^{3} + 4 \,{\left (24 \, a b^{2} c^{2} x^{14} - b^{3} c x^{11} + 24 \, a b^{2} x^{8}\right )} \arctan \left (c x^{3}\right )^{2} +{\left (b^{3} c x^{11} + 3 \,{\left (b^{3} c^{2} x^{14} + b^{3} x^{8}\right )} \arctan \left (c x^{3}\right )\right )} \log \left (c^{2} x^{6} + 1\right )^{2}}{32 \,{\left (c^{2} x^{6} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{8} \arctan \left (c x^{3}\right )^{3} + 3 \, a b^{2} x^{8} \arctan \left (c x^{3}\right )^{2} + 3 \, a^{2} b x^{8} \arctan \left (c x^{3}\right ) + a^{3} x^{8}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{3}\right ) + a\right )}^{3} x^{8}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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