Optimal. Leaf size=125 \[ -\frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2}{12 c^4}+\frac {a b x^3}{6 c^3}+\frac {1}{12} x^{12} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2+\frac {b x^9 \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{18 c}+\frac {b^2 x^3 \tanh ^{-1}\left (c x^3\right )}{6 c^3}+\frac {b^2 x^6}{36 c^2}+\frac {b^2 \log \left (1-c^2 x^6\right )}{9 c^4} \]
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Rubi [C] time = 1.55, antiderivative size = 636, normalized size of antiderivative = 5.09, number of steps used = 62, number of rules used = 19, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.187, Rules used = {6099, 2454, 2398, 2411, 43, 2334, 12, 14, 2301, 2395, 2439, 2416, 2389, 2295, 2394, 2393, 2391, 2410, 2390} \[ \frac {b^2 \text {PolyLog}\left (2,\frac {1}{2} \left (1-c x^3\right )\right )}{24 c^4}+\frac {b^2 \text {PolyLog}\left (2,\frac {1}{2} \left (c x^3+1\right )\right )}{24 c^4}+\frac {a b x^3}{12 c^3}-\frac {b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}-\frac {1}{288} b \left (-\frac {3 \left (1-c x^3\right )^4}{c^4}+\frac {16 \left (1-c x^3\right )^3}{c^4}-\frac {36 \left (1-c x^3\right )^2}{c^4}+\frac {48 \left (1-c x^3\right )}{c^4}-\frac {12 \log \left (1-c x^3\right )}{c^4}\right ) \left (2 a-b \log \left (1-c x^3\right )\right )-\frac {b \log \left (\frac {1}{2} \left (c x^3+1\right )\right ) \left (2 a-b \log \left (1-c x^3\right )\right )}{24 c^4}+\frac {1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {1}{24} b x^{12} \log \left (c x^3+1\right ) \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}+\frac {b^2 x^6}{192 c^2}+\frac {23 b^2 x^3}{288 c^3}+\frac {b^2 \left (1-c x^3\right )^4}{384 c^4}-\frac {b^2 \left (1-c x^3\right )^3}{54 c^4}+\frac {b^2 \left (1-c x^3\right )^2}{16 c^4}+\frac {b^2 \log ^2\left (1-c x^3\right )}{48 c^4}-\frac {b^2 \log ^2\left (c x^3+1\right )}{48 c^4}+\frac {b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{24 c^4}-\frac {5 b^2 \log \left (1-c x^3\right )}{288 c^4}+\frac {b^2 \left (c x^3+1\right ) \log \left (c x^3+1\right )}{12 c^4}+\frac {b^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (c x^3+1\right )}{24 c^4}+\frac {b^2 \log \left (c x^3+1\right )}{36 c^4}-\frac {7 b^2 x^9}{864 c}+\frac {1}{48} b^2 x^{12} \log ^2\left (c x^3+1\right )+\frac {b^2 x^9 \log \left (c x^3+1\right )}{36 c}-\frac {1}{384} b^2 x^{12} \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 14
Rule 43
Rule 2295
Rule 2301
Rule 2334
Rule 2389
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2398
Rule 2410
Rule 2411
Rule 2416
Rule 2439
Rule 2454
Rule 6099
Rubi steps
\begin {align*} \int x^{11} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^{11} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {1}{2} b x^{11} \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac {1}{4} b^2 x^{11} \log ^2\left (1+c x^3\right )\right ) \, dx\\ &=\frac {1}{4} \int x^{11} \left (2 a-b \log \left (1-c x^3\right )\right )^2 \, dx-\frac {1}{2} b \int x^{11} \left (-2 a+b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right ) \, dx+\frac {1}{4} b^2 \int x^{11} \log ^2\left (1+c x^3\right ) \, dx\\ &=\frac {1}{12} \operatorname {Subst}\left (\int x^3 (2 a-b \log (1-c x))^2 \, dx,x,x^3\right )-\frac {1}{6} b \operatorname {Subst}\left (\int x^3 (-2 a+b \log (1-c x)) \log (1+c x) \, dx,x,x^3\right )+\frac {1}{12} b^2 \operatorname {Subst}\left (\int x^3 \log ^2(1+c x) \, dx,x,x^3\right )\\ &=\frac {1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac {1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac {1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )-\frac {1}{24} (b c) \operatorname {Subst}\left (\int \frac {x^4 (2 a-b \log (1-c x))}{1-c x} \, dx,x,x^3\right )+\frac {1}{24} (b c) \operatorname {Subst}\left (\int \frac {x^4 (-2 a+b \log (1-c x))}{1+c x} \, dx,x,x^3\right )-\frac {1}{24} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^4 \log (1+c x)}{1-c x} \, dx,x,x^3\right )-\frac {1}{24} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^4 \log (1+c x)}{1+c x} \, dx,x,x^3\right )\\ &=\frac {1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2+\frac {1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac {1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+\frac {1}{24} b \operatorname {Subst}\left (\int \frac {\left (\frac {1}{c}-\frac {x}{c}\right )^4 (2 a-b \log (x))}{x} \, dx,x,1-c x^3\right )+\frac {1}{24} (b c) \operatorname {Subst}\left (\int \left (-\frac {-2 a+b \log (1-c x)}{c^4}+\frac {x (-2 a+b \log (1-c x))}{c^3}-\frac {x^2 (-2 a+b \log (1-c x))}{c^2}+\frac {x^3 (-2 a+b \log (1-c x))}{c}+\frac {-2 a+b \log (1-c x)}{c^4 (1+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{24} \left (b^2 c\right ) \operatorname {Subst}\left (\int \left (-\frac {\log (1+c x)}{c^4}-\frac {x \log (1+c x)}{c^3}-\frac {x^2 \log (1+c x)}{c^2}-\frac {x^3 \log (1+c x)}{c}-\frac {\log (1+c x)}{c^4 (-1+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{24} \left (b^2 c\right ) \operatorname {Subst}\left (\int \left (-\frac {\log (1+c x)}{c^4}+\frac {x \log (1+c x)}{c^3}-\frac {x^2 \log (1+c x)}{c^2}+\frac {x^3 \log (1+c x)}{c}+\frac {\log (1+c x)}{c^4 (1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac {1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {48 \left (1-c x^3\right )}{c^4}-\frac {36 \left (1-c x^3\right )^2}{c^4}+\frac {16 \left (1-c x^3\right )^3}{c^4}-\frac {3 \left (1-c x^3\right )^4}{c^4}-\frac {12 \log \left (1-c x^3\right )}{c^4}\right )+\frac {1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac {1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+\frac {1}{24} b \operatorname {Subst}\left (\int x^3 (-2 a+b \log (1-c x)) \, dx,x,x^3\right )+\frac {1}{24} b^2 \operatorname {Subst}\left (\int \frac {x \left (-48+36 x-16 x^2+3 x^3\right )+12 \log (x)}{12 c^4 x} \, dx,x,1-c x^3\right )-\frac {b \operatorname {Subst}\left (\int (-2 a+b \log (1-c x)) \, dx,x,x^3\right )}{24 c^3}+\frac {b \operatorname {Subst}\left (\int \frac {-2 a+b \log (1-c x)}{1+c x} \, dx,x,x^3\right )}{24 c^3}+2 \frac {b^2 \operatorname {Subst}\left (\int \log (1+c x) \, dx,x,x^3\right )}{24 c^3}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (1+c x)}{-1+c x} \, dx,x,x^3\right )}{24 c^3}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (1+c x)}{1+c x} \, dx,x,x^3\right )}{24 c^3}+\frac {b \operatorname {Subst}\left (\int x (-2 a+b \log (1-c x)) \, dx,x,x^3\right )}{24 c^2}-\frac {b \operatorname {Subst}\left (\int x^2 (-2 a+b \log (1-c x)) \, dx,x,x^3\right )}{24 c}+2 \frac {b^2 \operatorname {Subst}\left (\int x^2 \log (1+c x) \, dx,x,x^3\right )}{24 c}\\ &=\frac {a b x^3}{12 c^3}-\frac {b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}+\frac {b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}-\frac {1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {48 \left (1-c x^3\right )}{c^4}-\frac {36 \left (1-c x^3\right )^2}{c^4}+\frac {16 \left (1-c x^3\right )^3}{c^4}-\frac {3 \left (1-c x^3\right )^4}{c^4}-\frac {12 \log \left (1-c x^3\right )}{c^4}\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^4}+\frac {b^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^4}+\frac {1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )+\frac {1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )-\frac {1}{72} b^2 \operatorname {Subst}\left (\int \frac {x^3}{1-c x} \, dx,x,x^3\right )+2 \left (\frac {b^2 x^9 \log \left (1+c x^3\right )}{72 c}-\frac {1}{72} b^2 \operatorname {Subst}\left (\int \frac {x^3}{1+c x} \, dx,x,x^3\right )\right )+\frac {b^2 \operatorname {Subst}\left (\int \frac {x \left (-48+36 x-16 x^2+3 x^3\right )+12 \log (x)}{x} \, dx,x,1-c x^3\right )}{288 c^4}+2 \frac {b^2 \operatorname {Subst}\left (\int \log (x) \, dx,x,1+c x^3\right )}{24 c^4}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+c x^3\right )}{24 c^4}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1-c x)\right )}{1+c x} \, dx,x,x^3\right )}{24 c^3}-\frac {b^2 \operatorname {Subst}\left (\int \log (1-c x) \, dx,x,x^3\right )}{24 c^3}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} (1+c x)\right )}{1-c x} \, dx,x,x^3\right )}{24 c^3}+\frac {b^2 \operatorname {Subst}\left (\int \frac {x^2}{1-c x} \, dx,x,x^3\right )}{48 c}+\frac {1}{96} \left (b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^4}{1-c x} \, dx,x,x^3\right )\\ &=\frac {a b x^3}{12 c^3}-\frac {b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}+\frac {b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}-\frac {1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {48 \left (1-c x^3\right )}{c^4}-\frac {36 \left (1-c x^3\right )^2}{c^4}+\frac {16 \left (1-c x^3\right )^3}{c^4}-\frac {3 \left (1-c x^3\right )^4}{c^4}-\frac {12 \log \left (1-c x^3\right )}{c^4}\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^4}+\frac {b^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^4}+\frac {1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b^2 \log ^2\left (1+c x^3\right )}{48 c^4}+\frac {1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+2 \left (-\frac {b^2 x^3}{24 c^3}+\frac {b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{24 c^4}\right )-\frac {1}{72} b^2 \operatorname {Subst}\left (\int \left (-\frac {1}{c^3}-\frac {x}{c^2}-\frac {x^2}{c}-\frac {1}{c^3 (-1+c x)}\right ) \, dx,x,x^3\right )+2 \left (\frac {b^2 x^9 \log \left (1+c x^3\right )}{72 c}-\frac {1}{72} b^2 \operatorname {Subst}\left (\int \left (\frac {1}{c^3}-\frac {x}{c^2}+\frac {x^2}{c}-\frac {1}{c^3 (1+c x)}\right ) \, dx,x,x^3\right )\right )+\frac {b^2 \operatorname {Subst}\left (\int \left (-48+36 x-16 x^2+3 x^3+\frac {12 \log (x)}{x}\right ) \, dx,x,1-c x^3\right )}{288 c^4}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-c x^3\right )}{24 c^4}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+c x^3\right )}{24 c^4}+\frac {b^2 \operatorname {Subst}\left (\int \log (x) \, dx,x,1-c x^3\right )}{24 c^4}+\frac {b^2 \operatorname {Subst}\left (\int \left (-\frac {1}{c^2}-\frac {x}{c}-\frac {1}{c^2 (-1+c x)}\right ) \, dx,x,x^3\right )}{48 c}+\frac {1}{96} \left (b^2 c\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{c^4}-\frac {x}{c^3}-\frac {x^2}{c^2}-\frac {x^3}{c}-\frac {1}{c^4 (-1+c x)}\right ) \, dx,x,x^3\right )\\ &=\frac {a b x^3}{12 c^3}+\frac {55 b^2 x^3}{288 c^3}-\frac {5 b^2 x^6}{576 c^2}+\frac {b^2 x^9}{864 c}-\frac {b^2 x^{12}}{384}+\frac {b^2 \left (1-c x^3\right )^2}{16 c^4}-\frac {b^2 \left (1-c x^3\right )^3}{54 c^4}+\frac {b^2 \left (1-c x^3\right )^4}{384 c^4}-\frac {5 b^2 \log \left (1-c x^3\right )}{288 c^4}+\frac {b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{24 c^4}-\frac {b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}+\frac {b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}-\frac {1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {48 \left (1-c x^3\right )}{c^4}-\frac {36 \left (1-c x^3\right )^2}{c^4}+\frac {16 \left (1-c x^3\right )^3}{c^4}-\frac {3 \left (1-c x^3\right )^4}{c^4}-\frac {12 \log \left (1-c x^3\right )}{c^4}\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^4}+\frac {b^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^4}+\frac {1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b^2 \log ^2\left (1+c x^3\right )}{48 c^4}+\frac {1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+2 \left (-\frac {b^2 x^3}{72 c^3}+\frac {b^2 x^6}{144 c^2}-\frac {b^2 x^9}{216 c}+\frac {b^2 \log \left (1+c x^3\right )}{72 c^4}+\frac {b^2 x^9 \log \left (1+c x^3\right )}{72 c}\right )+2 \left (-\frac {b^2 x^3}{24 c^3}+\frac {b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{24 c^4}\right )+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{24 c^4}+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^4}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-c x^3\right )}{24 c^4}\\ &=\frac {a b x^3}{12 c^3}+\frac {55 b^2 x^3}{288 c^3}-\frac {5 b^2 x^6}{576 c^2}+\frac {b^2 x^9}{864 c}-\frac {b^2 x^{12}}{384}+\frac {b^2 \left (1-c x^3\right )^2}{16 c^4}-\frac {b^2 \left (1-c x^3\right )^3}{54 c^4}+\frac {b^2 \left (1-c x^3\right )^4}{384 c^4}-\frac {5 b^2 \log \left (1-c x^3\right )}{288 c^4}+\frac {b^2 \left (1-c x^3\right ) \log \left (1-c x^3\right )}{24 c^4}+\frac {b^2 \log ^2\left (1-c x^3\right )}{48 c^4}-\frac {b x^6 \left (2 a-b \log \left (1-c x^3\right )\right )}{48 c^2}+\frac {b x^9 \left (2 a-b \log \left (1-c x^3\right )\right )}{72 c}-\frac {1}{96} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )+\frac {1}{48} x^{12} \left (2 a-b \log \left (1-c x^3\right )\right )^2-\frac {1}{288} b \left (2 a-b \log \left (1-c x^3\right )\right ) \left (\frac {48 \left (1-c x^3\right )}{c^4}-\frac {36 \left (1-c x^3\right )^2}{c^4}+\frac {16 \left (1-c x^3\right )^3}{c^4}-\frac {3 \left (1-c x^3\right )^4}{c^4}-\frac {12 \log \left (1-c x^3\right )}{c^4}\right )-\frac {b \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^4}+\frac {b^2 \log \left (\frac {1}{2} \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )}{24 c^4}+\frac {1}{24} b x^{12} \left (2 a-b \log \left (1-c x^3\right )\right ) \log \left (1+c x^3\right )-\frac {b^2 \log ^2\left (1+c x^3\right )}{48 c^4}+\frac {1}{48} b^2 x^{12} \log ^2\left (1+c x^3\right )+2 \left (-\frac {b^2 x^3}{72 c^3}+\frac {b^2 x^6}{144 c^2}-\frac {b^2 x^9}{216 c}+\frac {b^2 \log \left (1+c x^3\right )}{72 c^4}+\frac {b^2 x^9 \log \left (1+c x^3\right )}{72 c}\right )+2 \left (-\frac {b^2 x^3}{24 c^3}+\frac {b^2 \left (1+c x^3\right ) \log \left (1+c x^3\right )}{24 c^4}\right )+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1-c x^3\right )\right )}{24 c^4}+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1+c x^3\right )\right )}{24 c^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 146, normalized size = 1.17 \[ \frac {3 a^2 c^4 x^{12}+2 a b c^3 x^9+2 b c x^3 \tanh ^{-1}\left (c x^3\right ) \left (3 a c^3 x^9+b \left (c^2 x^6+3\right )\right )+6 a b c x^3+b (3 a+4 b) \log \left (1-c x^3\right )-3 a b \log \left (c x^3+1\right )+3 b^2 \left (c^4 x^{12}-1\right ) \tanh ^{-1}\left (c x^3\right )^2+b^2 c^2 x^6+4 b^2 \log \left (c x^3+1\right )}{36 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 176, normalized size = 1.41 \[ \frac {12 \, a^{2} c^{4} x^{12} + 8 \, a b c^{3} x^{9} + 4 \, b^{2} c^{2} x^{6} + 24 \, a b c x^{3} + 3 \, {\left (b^{2} c^{4} x^{12} - b^{2}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )^{2} - 4 \, {\left (3 \, a b - 4 \, b^{2}\right )} \log \left (c x^{3} + 1\right ) + 4 \, {\left (3 \, a b + 4 \, b^{2}\right )} \log \left (c x^{3} - 1\right ) + 4 \, {\left (3 \, a b c^{4} x^{12} + b^{2} c^{3} x^{9} + 3 \, b^{2} c x^{3}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{144 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 175, normalized size = 1.40 \[ \frac {1}{12} \, a^{2} x^{12} + \frac {a b x^{9}}{18 \, c} + \frac {b^{2} x^{6}}{36 \, c^{2}} + \frac {1}{48} \, {\left (b^{2} x^{12} - \frac {b^{2}}{c^{4}}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )^{2} + \frac {a b x^{3}}{6 \, c^{3}} + \frac {1}{36} \, {\left (3 \, a b x^{12} + \frac {b^{2} x^{9}}{c} + \frac {3 \, b^{2} x^{3}}{c^{3}}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) - \frac {{\left (3 \, a b - 4 \, b^{2}\right )} \log \left (c x^{3} + 1\right )}{36 \, c^{4}} + \frac {{\left (3 \, a b + 4 \, b^{2}\right )} \log \left (c x^{3} - 1\right )}{36 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{11} \left (a +b \arctanh \left (c \,x^{3}\right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 217, normalized size = 1.74 \[ \frac {1}{12} \, b^{2} x^{12} \operatorname {artanh}\left (c x^{3}\right )^{2} + \frac {1}{12} \, a^{2} x^{12} + \frac {1}{36} \, {\left (6 \, x^{12} \operatorname {artanh}\left (c x^{3}\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{9} + 3 \, x^{3}\right )}}{c^{4}} - \frac {3 \, \log \left (c x^{3} + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x^{3} - 1\right )}{c^{5}}\right )}\right )} a b + \frac {1}{144} \, {\left (4 \, c {\left (\frac {2 \, {\left (c^{2} x^{9} + 3 \, x^{3}\right )}}{c^{4}} - \frac {3 \, \log \left (c x^{3} + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x^{3} - 1\right )}{c^{5}}\right )} \operatorname {artanh}\left (c x^{3}\right ) + \frac {4 \, c^{2} x^{6} - 2 \, {\left (3 \, \log \left (c x^{3} - 1\right ) - 8\right )} \log \left (c x^{3} + 1\right ) + 3 \, \log \left (c x^{3} + 1\right )^{2} + 3 \, \log \left (c x^{3} - 1\right )^{2} + 16 \, \log \left (c x^{3} - 1\right )}{c^{4}}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.60, size = 335, normalized size = 2.68 \[ \frac {a^2\,x^{12}}{12}+\frac {b^2\,\ln \left (c\,x^3-1\right )}{9\,c^4}+\frac {b^2\,\ln \left (c\,x^3+1\right )}{9\,c^4}-\frac {b^2\,{\ln \left (c\,x^3+1\right )}^2}{48\,c^4}-\frac {b^2\,{\ln \left (1-c\,x^3\right )}^2}{48\,c^4}+\frac {b^2\,x^6}{36\,c^2}+\frac {b^2\,x^{12}\,{\ln \left (c\,x^3+1\right )}^2}{48}+\frac {b^2\,x^{12}\,{\ln \left (1-c\,x^3\right )}^2}{48}+\frac {b^2\,x^3\,\ln \left (c\,x^3+1\right )}{12\,c^3}-\frac {b^2\,x^3\,\ln \left (1-c\,x^3\right )}{12\,c^3}+\frac {b^2\,x^9\,\ln \left (c\,x^3+1\right )}{36\,c}-\frac {b^2\,x^9\,\ln \left (1-c\,x^3\right )}{36\,c}+\frac {a\,b\,\ln \left (c\,x^3-1\right )}{12\,c^4}-\frac {a\,b\,\ln \left (c\,x^3+1\right )}{12\,c^4}+\frac {a\,b\,x^{12}\,\ln \left (c\,x^3+1\right )}{12}-\frac {a\,b\,x^{12}\,\ln \left (1-c\,x^3\right )}{12}+\frac {b^2\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{24\,c^4}+\frac {a\,b\,x^3}{6\,c^3}+\frac {a\,b\,x^9}{18\,c}-\frac {b^2\,x^{12}\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{24} \]
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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