Optimal. Leaf size=447 \[ -\frac{34}{15} i c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{4} i c^3 \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{3}{4} i c^3 \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} i c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{2} i c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} c^3 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{3}{2} c^3 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )-\frac{1}{60} a^3 c^3 x^3+\frac{1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^3-\frac{1}{10} a^5 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^3+\frac{1}{20} a^4 c^3 x^4 \tan ^{-1}(a x)-\frac{7}{12} a^3 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^3+\frac{29}{60} a^2 c^3 x^2 \tan ^{-1}(a x)-\frac{13}{30} a c^3 x-\frac{11}{4} a c^3 x \tan ^{-1}(a x)^2+\frac{11}{12} c^3 \tan ^{-1}(a x)^3-\frac{34}{15} i c^3 \tan ^{-1}(a x)^2+\frac{13}{30} c^3 \tan ^{-1}(a x)-\frac{68}{15} c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right ) \]
[Out]
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Rubi [A] time = 1.65542, antiderivative size = 447, normalized size of antiderivative = 1., 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 4948
Rule 4850
Rule 4988
Rule 4884
Rule 4994
Rule 4998
Rule 6610
Rule 4852
Rule 4916
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 321
Rule 203
Rule 302
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*}
Mathematica [A] time = 1.00972, size = 350, normalized size = 0.78 \[ \frac{1}{960} c^3 \left (1440 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+32 i \left (45 \tan ^{-1}(a x)^2+68\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+1440 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-1440 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-720 i \text{PolyLog}\left (4,e^{-2 i \tan ^{-1}(a x)}\right )-720 i \text{PolyLog}\left (4,-e^{2 i \tan ^{-1}(a x)}\right )-16 a^3 x^3+160 a^6 x^6 \tan ^{-1}(a x)^3-96 a^5 x^5 \tan ^{-1}(a x)^2+720 a^4 x^4 \tan ^{-1}(a x)^3+48 a^4 x^4 \tan ^{-1}(a x)-560 a^3 x^3 \tan ^{-1}(a x)^2+1440 a^2 x^2 \tan ^{-1}(a x)^3+464 a^2 x^2 \tan ^{-1}(a x)-416 a x-2640 a x \tan ^{-1}(a x)^2+480 i \tan ^{-1}(a x)^4+880 \tan ^{-1}(a x)^3+2176 i \tan ^{-1}(a x)^2+416 \tan ^{-1}(a x)+960 \tan ^{-1}(a x)^3 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-960 \tan ^{-1}(a x)^3 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-4352 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-15 i \pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.26, size = 664, normalized size = 1.5 \begin{align*} \text{result too large to display} \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}{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} \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 (\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} 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 ) \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 \frac{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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