Optimal. Leaf size=139 \[ -\frac {b^2 \log \left (\frac {2}{1-c x^3}\right ) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{c^2}-\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3}{6 c^2}+\frac {b \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{2 c^2}+\frac {b x^3 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{2 c}+\frac {1}{6} x^6 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3-\frac {b^3 \text {Li}_2\left (1-\frac {2}{1-c x^3}\right )}{2 c^2} \]
[Out]
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Rubi [B] time = 4.20, antiderivative size = 479, normalized size of antiderivative = 3.45, number of steps used = 155, number of rules used = 30, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.875, Rules used = {6099, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2439, 2416, 2396, 2433, 2374, 6589, 2411, 43, 2334, 12, 14, 2301, 6742, 2430, 2394, 2393, 2391, 2395, 2375, 2317, 2425} \[ -\frac {b^3 \text {PolyLog}\left (2,\frac {1}{2} \left (1-c x^3\right )\right )}{4 c^2}+\frac {b^3 \text {PolyLog}\left (2,\frac {1}{2} \left (c x^3+1\right )\right )}{4 c^2}-\frac {b^2 \log ^2\left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{16 c^2}+\frac {b^2 \log \left (\frac {1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{4 c^2}+\frac {1}{16} b^2 x^6 \log ^2\left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {b^2 x^3 \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{4 c}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^2}-\frac {b \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 c^2}+\frac {1}{16} b x^6 \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac {b^3 \left (c x^3+1\right )^2 \log ^3\left (c x^3+1\right )}{48 c^2}-\frac {b^3 \left (c x^3+1\right ) \log ^3\left (c x^3+1\right )}{24 c^2}+\frac {b^3 \left (c x^3+1\right ) \log ^2\left (c x^3+1\right )}{8 c^2}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )}{4 c^2} \]
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 6099
Rule 6589
Rule 6742
Rubi steps
\begin {align*} \int x^5 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^3 \, dx &=\int \left (\frac {1}{8} x^5 \left (2 a-b \log \left (1-c x^3\right )\right )^3+\frac {3}{8} b x^5 \left (-2 a+b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )-\frac {3}{8} b^2 x^5 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {1}{8} b^3 x^5 \log ^3\left (1+c x^3\right )\right ) \, dx\\ &=\frac {1}{8} \int x^5 \left (2 a-b \log \left (1-c x^3\right )\right )^3 \, dx+\frac {1}{8} (3 b) \int x^5 \left (-2 a+b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right ) \, dx-\frac {1}{8} \left (3 b^2\right ) \int x^5 \left (-2 a+b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right ) \, dx+\frac {1}{8} b^3 \int x^5 \log ^3\left (1+c x^3\right ) \, dx\\ &=\frac {1}{24} \operatorname {Subst}\left (\int x (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )+\frac {1}{8} b \operatorname {Subst}\left (\int x (-2 a+b \log (1-c x))^2 \log (1+c x) \, dx,x,x^3\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int x (-2 a+b \log (1-c x)) \log ^2(1+c x) \, dx,x,x^3\right )+\frac {1}{24} b^3 \operatorname {Subst}\left (\int x \log ^3(1+c x) \, dx,x,x^3\right )\\ &=\frac {1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {1}{24} \operatorname {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x))^3}{c}-\frac {(1-c x) (2 a-b \log (1-c x))^3}{c}\right ) \, dx,x,x^3\right )+\frac {1}{24} b^3 \operatorname {Subst}\left (\int \left (-\frac {\log ^3(1+c x)}{c}+\frac {(1+c x) \log ^3(1+c x)}{c}\right ) \, dx,x,x^3\right )-\frac {1}{16} (b c) \operatorname {Subst}\left (\int \frac {x^2 (-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^3\right )+\frac {1}{8} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2 (-2 a+b \log (1-c x)) \log (1+c x)}{1-c x} \, dx,x,x^3\right )+\frac {1}{8} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2 (-2 a+b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^3\right )-\frac {1}{16} \left (b^3 c\right ) \operatorname {Subst}\left (\int \frac {x^2 \log ^2(1+c x)}{1-c x} \, dx,x,x^3\right )\\ &=\frac {1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )+\frac {\operatorname {Subst}\left (\int (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )}{24 c}-\frac {\operatorname {Subst}\left (\int (1-c x) (2 a-b \log (1-c x))^3 \, dx,x,x^3\right )}{24 c}-\frac {b^3 \operatorname {Subst}\left (\int \log ^3(1+c x) \, dx,x,x^3\right )}{24 c}+\frac {b^3 \operatorname {Subst}\left (\int (1+c x) \log ^3(1+c x) \, dx,x,x^3\right )}{24 c}-\frac {1}{16} (b c) \operatorname {Subst}\left (\int \left (-\frac {(-2 a+b \log (1-c x))^2}{c^2}+\frac {x (-2 a+b \log (1-c x))^2}{c}+\frac {(-2 a+b \log (1-c x))^2}{c^2 (1+c x)}\right ) \, dx,x,x^3\right )+\frac {1}{8} \left (b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^2}+\frac {x (2 a-b \log (1-c x)) \log (1+c x)}{c}+\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^2 (-1+c x)}\right ) \, dx,x,x^3\right )+\frac {1}{8} \left (b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^2}-\frac {x (2 a-b \log (1-c x)) \log (1+c x)}{c}-\frac {(2 a-b \log (1-c x)) \log (1+c x)}{c^2 (1+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{16} \left (b^3 c\right ) \operatorname {Subst}\left (\int \left (-\frac {\log ^2(1+c x)}{c^2}-\frac {x \log ^2(1+c x)}{c}-\frac {\log ^2(1+c x)}{c^2 (-1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )-\frac {1}{16} b \operatorname {Subst}\left (\int x (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )+\frac {1}{16} b^3 \operatorname {Subst}\left (\int x \log ^2(1+c x) \, dx,x,x^3\right )-\frac {\operatorname {Subst}\left (\int (2 a-b \log (x))^3 \, dx,x,1-c x^3\right )}{24 c^2}+\frac {\operatorname {Subst}\left (\int x (2 a-b \log (x))^3 \, dx,x,1-c x^3\right )}{24 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \log ^3(x) \, dx,x,1+c x^3\right )}{24 c^2}+\frac {b^3 \operatorname {Subst}\left (\int x \log ^3(x) \, dx,x,1+c x^3\right )}{24 c^2}+\frac {b \operatorname {Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{16 c}-\frac {b \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x))^2}{1+c x} \, dx,x,x^3\right )}{16 c}+2 \frac {b^2 \operatorname {Subst}\left (\int (2 a-b \log (1-c x)) \log (1+c x) \, dx,x,x^3\right )}{8 c}+\frac {b^2 \operatorname {Subst}\left (\int \frac {(2 a-b \log (1-c x)) \log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{8 c}-\frac {b^2 \operatorname {Subst}\left (\int \frac {(2 a-b \log (1-c x)) \log (1+c x)}{1+c x} \, dx,x,x^3\right )}{8 c}+\frac {b^3 \operatorname {Subst}\left (\int \log ^2(1+c x) \, dx,x,x^3\right )}{16 c}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\log ^2(1+c x)}{-1+c x} \, dx,x,x^3\right )}{16 c}\\ &=-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{16 c^2}+\frac {1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^2}+\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}-\frac {1}{16} b \operatorname {Subst}\left (\int \left (\frac {(-2 a+b \log (1-c x))^2}{c}-\frac {(1-c x) (-2 a+b \log (1-c x))^2}{c}\right ) \, dx,x,x^3\right )+\frac {1}{16} b^3 \operatorname {Subst}\left (\int \left (-\frac {\log ^2(1+c x)}{c}+\frac {(1+c x) \log ^2(1+c x)}{c}\right ) \, dx,x,x^3\right )+2 \left (\frac {b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \frac {x (2 a-b \log (1-c x))}{1+c x} \, dx,x,x^3\right )-\frac {1}{8} b^3 \operatorname {Subst}\left (\int \frac {x \log (1+c x)}{1-c x} \, dx,x,x^3\right )\right )+\frac {b \operatorname {Subst}\left (\int x (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{16 c^2}-\frac {b \operatorname {Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^3\right )}{16 c^2}-\frac {b \operatorname {Subst}\left (\int (2 a-b \log (x))^2 \, dx,x,1-c x^3\right )}{8 c^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {(2 a-b \log (2-x)) \log (x)}{x} \, dx,x,1+c x^3\right )}{8 c^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (2-x) (2 a-b \log (x))}{x} \, dx,x,1-c x^3\right )}{8 c^2}+\frac {b^3 \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{16 c^2}-\frac {b^3 \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac {b^3 \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{8 c^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {(-2 a+b \log (1-c x)) \log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )}{8 c}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right ) \log (1+c x)}{1+c x} \, dx,x,x^3\right )}{8 c}\\ &=-\frac {3 b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 c^2}+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{32 c^2}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{16 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {3 b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{16 c^2}-\frac {b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{32 c^2}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^2}+\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+2 \left (\frac {b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \left (\frac {2 a-b \log (1-c x)}{c}-\frac {2 a-b \log (1-c x)}{c (1+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{8} b^3 \operatorname {Subst}\left (\int \left (-\frac {\log (1+c x)}{c}-\frac {\log (1+c x)}{c (-1+c x)}\right ) \, dx,x,x^3\right )\right )-\frac {b \operatorname {Subst}\left (\int \frac {(2 a-b \log (x))^2}{2-x} \, dx,x,1-c x^3\right )}{16 c^2}+\frac {b^2 \operatorname {Subst}\left (\int x (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{16 c^2}+\frac {b^2 \operatorname {Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^3\right )}{8 c^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {2-x}{2}\right ) (-2 a+b \log (x))}{x} \, dx,x,1-c x^3\right )}{8 c^2}-\frac {b^2 \operatorname {Subst}\left (\int (2 a-b \log (x)) \, dx,x,1-c x^3\right )}{4 c^2}+\frac {b^3 \operatorname {Subst}\left (\int x \log (x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{2-x} \, dx,x,1+c x^3\right )}{16 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{8 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (\frac {2-x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^3\right )}{8 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{4 c^2}-\frac {b \operatorname {Subst}\left (\int (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{16 c}+\frac {b \operatorname {Subst}\left (\int (1-c x) (-2 a+b \log (1-c x))^2 \, dx,x,x^3\right )}{16 c}-\frac {b^3 \operatorname {Subst}\left (\int \log ^2(1+c x) \, dx,x,x^3\right )}{16 c}+\frac {b^3 \operatorname {Subst}\left (\int (1+c x) \log ^2(1+c x) \, dx,x,x^3\right )}{16 c}\\ &=\frac {3 a b^2 x^3}{4 c}+\frac {3 b^3 x^3}{8 c}+\frac {b^3 \left (1-c x^3\right )^2}{64 c^2}-\frac {b^3 \left (1+c x^3\right )^2}{64 c^2}+\frac {b^2 \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{32 c^2}-\frac {3 b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{16 c^2}+\frac {b \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^2}{32 c^2}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac {3 b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac {b^3 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{32 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {3 b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{16 c^2}-\frac {b^3 \left (1+c x^3\right )^2 \log ^2\left (1+c x^3\right )}{32 c^2}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^2}+\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{8 c^2}+\frac {b^3 \log \left (1+c x^3\right ) \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{8 c^2}+\frac {b \operatorname {Subst}\left (\int (-2 a+b \log (x))^2 \, dx,x,1-c x^3\right )}{16 c^2}-\frac {b \operatorname {Subst}\left (\int x (-2 a+b \log (x))^2 \, dx,x,1-c x^3\right )}{16 c^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right ) (2 a-b \log (x))}{x} \, dx,x,1-c x^3\right )}{8 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac {b^3 \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac {b^3 \operatorname {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{8 c^2}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right ) \log (x)}{x} \, dx,x,1+c x^3\right )}{8 c^2}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{8 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{8 c^2}+\frac {b^3 \operatorname {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{4 c^2}+2 \left (\frac {b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}-\frac {b^2 \operatorname {Subst}\left (\int (2 a-b \log (1-c x)) \, dx,x,x^3\right )}{8 c}+\frac {b^2 \operatorname {Subst}\left (\int \frac {2 a-b \log (1-c x)}{1+c x} \, dx,x,x^3\right )}{8 c}+\frac {b^3 \operatorname {Subst}\left (\int \log (1+c x) \, dx,x,x^3\right )}{8 c}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{8 c}\right )\\ &=\frac {3 a b^2 x^3}{4 c}+\frac {3 b^3 x^3}{4 c}+\frac {b^3 \left (1-c x^3\right )^2}{64 c^2}-\frac {b^3 \left (1+c x^3\right )^2}{64 c^2}+\frac {3 b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{8 c^2}+\frac {b^2 \left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )}{32 c^2}-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^2}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac {3 b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac {b^3 \left (1+c x^3\right )^2 \log \left (1+c x^3\right )}{32 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{8 c^2}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^2}+\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1-c x^3\right )\right )}{8 c^2}-\frac {b^3 \text {Li}_3\left (\frac {1}{2} \left (1+c x^3\right )\right )}{8 c^2}+\frac {b^2 \operatorname {Subst}\left (\int x (-2 a+b \log (x)) \, dx,x,1-c x^3\right )}{16 c^2}-\frac {b^2 \operatorname {Subst}\left (\int (-2 a+b \log (x)) \, dx,x,1-c x^3\right )}{8 c^2}-\frac {b^3 \operatorname {Subst}\left (\int x \log (x) \, dx,x,1+c x^3\right )}{16 c^2}+\frac {b^3 \operatorname {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{8 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{8 c^2}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{8 c^2}+2 \left (-\frac {a b^2 x^3}{4 c}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{8 c^2}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac {b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}+\frac {b^3 \operatorname {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{8 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )}{8 c}+\frac {b^3 \operatorname {Subst}\left (\int \log (1-c x) \, dx,x,x^3\right )}{8 c}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )}{8 c}\right )\\ &=\frac {a b^2 x^3}{2 c}+\frac {5 b^3 x^3}{8 c}+\frac {3 b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{8 c^2}-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^2}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac {b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{4 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{8 c^2}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^2}+\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+2 \left (-\frac {a b^2 x^3}{4 c}-\frac {b^3 x^3}{8 c}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{8 c^2}+\frac {b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac {b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}+\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{8 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{8 c^2}-\frac {b^3 \operatorname {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{8 c^2}\right )-\frac {b^3 \operatorname {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{8 c^2}\\ &=\frac {a b^2 x^3}{2 c}+\frac {b^3 x^3}{2 c}+\frac {b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{4 c^2}-\frac {b \left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^2}{8 c^2}-\frac {\left (1-c x^3\right ) \left (2 a-b \log \left (1-c x^3\right )\right )^3}{24 c^2}+\frac {\left (1-c x^3\right )^2 \left (2 a-b \log \left (1-c x^3\right )\right )^3}{48 c^2}-\frac {b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{4 c^2}-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )^2 \log \left (1+c x^3\right )+\frac {b^3 \left (1+c x^3\right ) \log ^2\left (1+c x^3\right )}{8 c^2}-\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )}{16 c^2}+\frac {1}{16} b^2 x^6 \left (2 a-b \log \left (1-c x^3\right )\right ) \log ^2\left (1+c x^3\right )-\frac {b^3 \left (1+c x^3\right ) \log ^3\left (1+c x^3\right )}{24 c^2}+\frac {b^3 \left (1+c x^3\right )^2 \log ^3\left (1+c x^3\right )}{48 c^2}+2 \left (-\frac {a b^2 x^3}{4 c}-\frac {b^3 x^3}{4 c}-\frac {b^3 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{8 c^2}+\frac {b^2 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{8 c^2}+\frac {b^3 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac {b^3 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c^2}+\frac {b^2 x^3 \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{8 c}-\frac {b^3 \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{8 c^2}+\frac {b^3 \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{8 c^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.32, size = 185, normalized size = 1.33 \[ \frac {a \left (2 a^2 c^2 x^6+6 a b c x^3+3 a b \log \left (1-c x^3\right )-3 a b \log \left (c x^3+1\right )+6 b^2 \log \left (1-c^2 x^6\right )\right )+6 b^2 \left (c x^3-1\right ) \tanh ^{-1}\left (c x^3\right )^2 \left (a c x^3+a+b\right )+6 b \tanh ^{-1}\left (c x^3\right ) \left (a c x^3 \left (a c x^3+2 b\right )-2 b^2 \log \left (e^{-2 \tanh ^{-1}\left (c x^3\right )}+1\right )\right )+2 b^3 \left (c^2 x^6-1\right ) \tanh ^{-1}\left (c x^3\right )^3+6 b^3 \text {Li}_2\left (-e^{-2 \tanh ^{-1}\left (c x^3\right )}\right )}{12 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{5} \operatorname {artanh}\left (c x^{3}\right )^{3} + 3 \, a b^{2} x^{5} \operatorname {artanh}\left (c x^{3}\right )^{2} + 3 \, a^{2} b x^{5} \operatorname {artanh}\left (c x^{3}\right ) + a^{3} x^{5}, x\right ) \]
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 \[ \int {\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{3} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.44, size = 750, normalized size = 5.40 \[ \text {result too large to display} \]
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 {1}{2} \, a b^{2} x^{6} \operatorname {artanh}\left (c x^{3}\right )^{2} + \frac {1}{6} \, a^{3} x^{6} + \frac {1}{4} \, {\left (2 \, x^{6} \operatorname {artanh}\left (c x^{3}\right ) + c {\left (\frac {2 \, x^{3}}{c^{2}} - \frac {\log \left (c x^{3} + 1\right )}{c^{3}} + \frac {\log \left (c x^{3} - 1\right )}{c^{3}}\right )}\right )} a^{2} b + \frac {1}{8} \, {\left (4 \, c {\left (\frac {2 \, x^{3}}{c^{2}} - \frac {\log \left (c x^{3} + 1\right )}{c^{3}} + \frac {\log \left (c x^{3} - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c x^{3}\right ) - \frac {2 \, {\left (\log \left (c x^{3} - 1\right ) - 2\right )} \log \left (c x^{3} + 1\right ) - \log \left (c x^{3} + 1\right )^{2} - \log \left (c x^{3} - 1\right )^{2} - 4 \, \log \left (c x^{3} - 1\right )}{c^{2}}\right )} a b^{2} - \frac {1}{192} \, {\left (4 \, x^{6} \log \left (-c x^{3} + 1\right )^{3} + 3 \, {\left (\frac {x^{6}}{c^{3}} + \frac {\log \left (c^{2} x^{6} - 1\right )}{c^{5}}\right )} c^{3} - 6 \, c {\left (\frac {c x^{6} + 2 \, x^{3}}{c^{2}} + \frac {2 \, \log \left (c x^{3} - 1\right )}{c^{3}}\right )} \log \left (-c x^{3} + 1\right )^{2} + 21 \, c^{2} {\left (\frac {2 \, x^{3}}{c^{3}} - \frac {\log \left (c x^{3} + 1\right )}{c^{4}} + \frac {\log \left (c x^{3} - 1\right )}{c^{4}}\right )} + c {\left (\frac {6 \, {\left (c^{2} x^{6} + 6 \, c x^{3} + 2 \, \log \left (c x^{3} - 1\right )^{2} + 6 \, \log \left (c x^{3} - 1\right )\right )} \log \left (-c x^{3} + 1\right )}{c^{3}} - \frac {3 \, c^{2} x^{6} + 42 \, c x^{3} + 4 \, \log \left (c x^{3} - 1\right )^{3} + 18 \, \log \left (c x^{3} - 1\right )^{2} + 42 \, \log \left (c x^{3} - 1\right )}{c^{3}}\right )} - 1728 \, c \int \frac {x^{5} \log \left (c x^{3} + 1\right )}{4 \, {\left (c^{3} x^{6} - c\right )}}\,{d x} - \frac {2 \, {\left (12 \, c x^{3} \log \left (c x^{3} + 1\right )^{2} + 2 \, {\left (c^{2} x^{6} - 1\right )} \log \left (c x^{3} + 1\right )^{3} - 3 \, {\left (c^{2} x^{6} - 2 \, c x^{3} - 2 \, {\left (c^{2} x^{6} - 1\right )} \log \left (c x^{3} + 1\right ) + 1\right )} \log \left (-c x^{3} + 1\right )^{2} + 3 \, {\left (c^{2} x^{6} + 6 \, c x^{3} - 2 \, {\left (c^{2} x^{6} - 1\right )} \log \left (c x^{3} + 1\right )^{2} - 8 \, {\left (c x^{3} + 1\right )} \log \left (c x^{3} + 1\right )\right )} \log \left (-c x^{3} + 1\right )\right )}}{c^{2}} + \frac {18 \, \log \left (4 \, c^{3} x^{6} - 4 \, c\right )}{c^{2}} - 576 \, \int \frac {x^{2} \log \left (c x^{3} + 1\right )}{4 \, {\left (c^{3} x^{6} - c\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 \[ \int x^5\,{\left (a+b\,\mathrm {atanh}\left (c\,x^3\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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