Optimal. Leaf size=447 \[ \frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3-\frac {1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac {1}{60} a^3 c^3 x^3-\frac {7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {29}{60} a^2 c^3 x^2 \tan ^{-1}(a x)-\frac {34}{15} i c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )+\frac {3}{4} i c^3 \text {Li}_4\left (1-\frac {2}{i a x+1}\right )-\frac {3}{4} i c^3 \text {Li}_4\left (\frac {2}{i a x+1}-1\right )-\frac {3}{2} i c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)^2+\frac {3}{2} i c^3 \text {Li}_2\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)^2-\frac {3}{2} c^3 \text {Li}_3\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)+\frac {3}{2} c^3 \text {Li}_3\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)-\frac {13}{30} a c^3 x-\frac {11}{4} a c^3 x \tan ^{-1}(a x)^2+\frac {11}{12} c^3 \tan ^{-1}(a x)^3-\frac {34}{15} i c^3 \tan ^{-1}(a x)^2+\frac {13}{30} c^3 \tan ^{-1}(a x)-\frac {68}{15} c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
[Out]
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Rubi [A] time = 1.66, antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 69, number of rules used = 17, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {4948, 4850, 4988, 4884, 4994, 4998, 6610, 4852, 4916, 4846, 4920, 4854, 2402, 2315, 321, 203, 302} \[ -\frac {34}{15} i c^3 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \text {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \text {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \tan ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \tan ^{-1}(a x) \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )-\frac {1}{60} a^3 c^3 x^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3-\frac {1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac {7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {29}{60} a^2 c^3 x^2 \tan ^{-1}(a x)-\frac {13}{30} a c^3 x-\frac {11}{4} a c^3 x \tan ^{-1}(a x)^2+\frac {11}{12} c^3 \tan ^{-1}(a x)^3-\frac {34}{15} i c^3 \tan ^{-1}(a x)^2+\frac {13}{30} c^3 \tan ^{-1}(a x)-\frac {68}{15} c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 4846
Rule 4850
Rule 4852
Rule 4854
Rule 4884
Rule 4916
Rule 4920
Rule 4948
Rule 4988
Rule 4994
Rule 4998
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3}{x} \, dx &=\int \left (\frac {c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+3 a^4 c^3 x^3 \tan ^{-1}(a x)^3+a^6 c^3 x^5 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^3 \int \frac {\tan ^{-1}(a x)^3}{x} \, dx+\left (3 a^2 c^3\right ) \int x \tan ^{-1}(a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^3 \tan ^{-1}(a x)^3 \, dx+\left (a^6 c^3\right ) \int x^5 \tan ^{-1}(a x)^3 \, dx\\ &=\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\left (6 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a^3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (9 a^5 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^7 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a c^3\right ) \int \tan ^{-1}(a x)^2 \, dx+\frac {1}{2} \left (9 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (9 a^3 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx+\frac {1}{4} \left (9 a^3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^5 c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\frac {1}{2} \left (a^5 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac {9}{2} a c^3 x \tan ^{-1}(a x)^2-\frac {3}{4} a^3 c^3 x^3 \tan ^{-1}(a x)^2-\frac {1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{2} c^3 \tan ^{-1}(a x)^3+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+\left (3 i a c^3\right ) \int \frac {\tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 i a c^3\right ) \int \frac {\tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{4} \left (9 a c^3\right ) \int \tan ^{-1}(a x)^2 \, dx-\frac {1}{4} \left (9 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (9 a^2 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{2} \left (a^3 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac {1}{2} \left (3 a^4 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^6 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {9}{2} i c^3 \tan ^{-1}(a x)^2-\frac {9}{4} a c^3 x \tan ^{-1}(a x)^2-\frac {7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2-\frac {1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac {3}{4} c^3 \tan ^{-1}(a x)^3+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} \left (a c^3\right ) \int \tan ^{-1}(a x)^2 \, dx+\frac {1}{2} \left (a c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac {1}{2} \left (3 a c^3\right ) \int \frac {\text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a c^3\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (9 a c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx+\frac {1}{2} \left (3 a^2 c^3\right ) \int x \tan ^{-1}(a x) \, dx-\frac {1}{2} \left (3 a^2 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a^2 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^4 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac {1}{5} \left (a^4 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^4 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {3}{4} a^2 c^3 x^2 \tan ^{-1}(a x)+\frac {1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac {3}{2} i c^3 \tan ^{-1}(a x)^2-\frac {11}{4} a c^3 x \tan ^{-1}(a x)^2-\frac {7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2-\frac {1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac {11}{12} c^3 \tan ^{-1}(a x)^3+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-9 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx+\frac {1}{2} \left (9 a c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx+\left (9 a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{5} \left (a^2 c^3\right ) \int x \tan ^{-1}(a x) \, dx+\frac {1}{5} \left (a^2 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^2 c^3\right ) \int x \tan ^{-1}(a x) \, dx+\frac {1}{3} \left (a^2 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (a^2 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {1}{20} \left (a^5 c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx\\ &=-\frac {3}{4} a c^3 x+\frac {29}{60} a^2 c^3 x^2 \tan ^{-1}(a x)+\frac {1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac {34}{15} i c^3 \tan ^{-1}(a x)^2-\frac {11}{4} a c^3 x \tan ^{-1}(a x)^2-\frac {7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2-\frac {1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac {11}{12} c^3 \tan ^{-1}(a x)^3+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-3 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )-\left (9 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )-\frac {1}{5} \left (a c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx-\frac {1}{3} \left (a c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx+\frac {1}{4} \left (3 a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\left (a c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx-\frac {1}{2} \left (3 a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{10} \left (a^3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx+\frac {1}{6} \left (a^3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {1}{20} \left (a^5 c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac {13}{30} a c^3 x-\frac {1}{60} a^3 c^3 x^3+\frac {3}{4} c^3 \tan ^{-1}(a x)+\frac {29}{60} a^2 c^3 x^2 \tan ^{-1}(a x)+\frac {1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac {34}{15} i c^3 \tan ^{-1}(a x)^2-\frac {11}{4} a c^3 x \tan ^{-1}(a x)^2-\frac {7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2-\frac {1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac {11}{12} c^3 \tan ^{-1}(a x)^3+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {68}{15} c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {9}{2} i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )+\frac {1}{2} \left (9 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )-\frac {1}{20} \left (a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{10} \left (a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{6} \left (a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{3} \left (a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac {13}{30} a c^3 x-\frac {1}{60} a^3 c^3 x^3+\frac {13}{30} c^3 \tan ^{-1}(a x)+\frac {29}{60} a^2 c^3 x^2 \tan ^{-1}(a x)+\frac {1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac {34}{15} i c^3 \tan ^{-1}(a x)^2-\frac {11}{4} a c^3 x \tan ^{-1}(a x)^2-\frac {7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2-\frac {1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac {11}{12} c^3 \tan ^{-1}(a x)^3+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {68}{15} c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{5} \left (i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )-\frac {1}{3} \left (i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )-\left (i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )\\ &=-\frac {13}{30} a c^3 x-\frac {1}{60} a^3 c^3 x^3+\frac {13}{30} c^3 \tan ^{-1}(a x)+\frac {29}{60} a^2 c^3 x^2 \tan ^{-1}(a x)+\frac {1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac {34}{15} i c^3 \tan ^{-1}(a x)^2-\frac {11}{4} a c^3 x \tan ^{-1}(a x)^2-\frac {7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2-\frac {1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac {11}{12} c^3 \tan ^{-1}(a x)^3+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {68}{15} c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )-\frac {34}{15} i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \tan ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \tan ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \text {Li}_4\left (1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \text {Li}_4\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}
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Mathematica [A] time = 1.07, size = 350, normalized size = 0.78 \[ \frac {1}{960} c^3 \left (160 a^6 x^6 \tan ^{-1}(a x)^3-96 a^5 x^5 \tan ^{-1}(a x)^2+720 a^4 x^4 \tan ^{-1}(a x)^3+48 a^4 x^4 \tan ^{-1}(a x)-16 a^3 x^3-560 a^3 x^3 \tan ^{-1}(a x)^2+1440 a^2 x^2 \tan ^{-1}(a x)^3+464 a^2 x^2 \tan ^{-1}(a x)+1440 i \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )+32 i \left (45 \tan ^{-1}(a x)^2+68\right ) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+1440 \tan ^{-1}(a x) \text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )-1440 \tan ^{-1}(a x) \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )-720 i \text {Li}_4\left (e^{-2 i \tan ^{-1}(a x)}\right )-720 i \text {Li}_4\left (-e^{2 i \tan ^{-1}(a x)}\right )-416 a x-2640 a x \tan ^{-1}(a x)^2+480 i \tan ^{-1}(a x)^4+880 \tan ^{-1}(a x)^3+2176 i \tan ^{-1}(a x)^2+416 \tan ^{-1}(a x)+960 \tan ^{-1}(a x)^3 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-960 \tan ^{-1}(a x)^3 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-4352 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-15 i \pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 17.05, size = 664, normalized size = 1.49 \[ \frac {c^{3} \left (i a x +55 \arctan \left (a x \right )^{3} a x -3 i \arctan \left (a x \right ) a^{2} x^{2}+35 \arctan \left (a x \right )^{3} a^{3} x^{3}-35 i \arctan \left (a x \right )^{3} a^{2} x^{2}+10 \arctan \left (a x \right )^{3} a^{5} x^{5}-136 \arctan \left (a x \right )^{2}-10 i \arctan \left (a x \right )^{3} a^{4} x^{4}-29 \arctan \left (a x \right )^{2} x^{2} a^{2}+6 i \arctan \left (a x \right )^{2} a^{3} x^{3}-6 \arctan \left (a x \right )^{2} x^{4} a^{4}+29 i \arctan \left (a x \right )^{2} a x +26 \arctan \left (a x \right ) x a -26 i \arctan \left (a x \right )+3 \arctan \left (a x \right ) x^{3} a^{3}-25-55 i \arctan \left (a x \right )^{3}-a^{2} x^{2}\right ) \left (a x +i\right )}{60}+\frac {68 i c^{3} \arctan \left (a x \right )^{2}}{15}-\frac {68 c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{15}+6 i c^{3} \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-3 i c^{3} \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 c^{3} \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-3 i c^{3} \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i c^{3} \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+6 c^{3} \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {34 i c^{3} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{15}+c^{3} \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c^{3} \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 c^{3} \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {3 i c^{3} \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4} \]
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 \[ \frac {1}{96} \, {\left (2 \, a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} + 18 \, a^{2} c^{3} x^{2}\right )} \arctan \left (a x\right )^{3} - \frac {1}{128} \, {\left (2 \, a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} + 18 \, a^{2} c^{3} x^{2}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac {112 \, {\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3} - 4 \, {\left (2 \, a^{7} c^{3} x^{7} + 9 \, a^{5} c^{3} x^{5} + 18 \, a^{3} c^{3} x^{3}\right )} \arctan \left (a x\right )^{2} + 4 \, {\left (2 \, a^{8} c^{3} x^{8} + 9 \, a^{6} c^{3} x^{6} + 18 \, a^{4} c^{3} x^{4}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) + {\left (2 \, a^{7} c^{3} x^{7} + 9 \, a^{5} c^{3} x^{5} + 18 \, a^{3} c^{3} x^{3} + 12 \, {\left (a^{8} c^{3} x^{8} + 4 \, a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} + 4 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{128 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ c^{3} \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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