Optimal. Leaf size=503 \[ -\frac{3}{2} i a^2 c^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{7}{2} i a^2 c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{9}{4} i a^2 c^3 \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{9}{4} i a^2 c^3 \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{1}{4} a^3 c^3 x-\frac{15}{4} a^3 c^3 x \tan ^{-1}(a x)^2+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^3-5 i a^2 c^3 \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c^3 \tan ^{-1}(a x)-7 a^2 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+3 a^2 c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x} \]
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Rubi [A] time = 1.19198, antiderivative size = 503, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 20, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.909, Rules used = {4948, 4852, 4918, 4924, 4868, 2447, 4884, 4850, 4988, 4994, 4998, 6610, 4916, 4846, 4920, 4854, 2402, 2315, 321, 203} \[ -\frac{3}{2} i a^2 c^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{7}{2} i a^2 c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{9}{4} i a^2 c^3 \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{9}{4} i a^2 c^3 \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-\frac{1}{4} a^3 c^3 x-\frac{15}{4} a^3 c^3 x \tan ^{-1}(a x)^2+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^3-5 i a^2 c^3 \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c^3 \tan ^{-1}(a x)-7 a^2 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)+3 a^2 c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x} \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4852
Rule 4918
Rule 4924
Rule 4868
Rule 2447
Rule 4884
Rule 4850
Rule 4988
Rule 4994
Rule 4998
Rule 6610
Rule 4916
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3}{x^3} \, dx &=\int \left (\frac{c^3 \tan ^{-1}(a x)^3}{x^3}+\frac{3 a^2 c^3 \tan ^{-1}(a x)^3}{x}+3 a^4 c^3 x \tan ^{-1}(a x)^3+a^6 c^3 x^3 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)^3}{x^3} \, dx+\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^3}{x} \, dx+\left (3 a^4 c^3\right ) \int x \tan ^{-1}(a x)^3 \, dx+\left (a^6 c^3\right ) \int x^3 \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} \left (3 a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (18 a^3 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^5 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{4} \left (3 a^7 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} \left (3 a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx-\frac{1}{2} \left (3 a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{2} \left (9 a^3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx+\frac{1}{2} \left (9 a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (9 a^3 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 (9 a^3 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}{4} \left (3 a^5 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx+\frac{1}{4} \left (3 a^5 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac{9}{2} a^3 c^3 x \tan ^{-1}(a x)^2-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+a^2 c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )+\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx+\left (9 i a^3 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 (9 i a^3 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 (3 a^3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx-\frac{1}{4} \left (3 a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (9 a^4 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{2} \left (a^6 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-6 i a^2 c^3 \tan ^{-1}(a x)^2-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac{15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\left (3 i a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx+\frac{1}{2} \left (9 a^3 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 (9 a^3 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^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx+\frac{1}{2} \left (a^4 c^3\right ) \int x \tan ^{-1}(a x) \, dx-\frac{1}{2} \left (a^4 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{2} \left (3 a^4 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-5 i a^2 c^3 \tan ^{-1}(a x)^2-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac{15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-9 a^2 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+3 a^2 c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{9}{4} i a^2 c^3 \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{9}{4} i a^2 c^3 \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )+\frac{1}{2} \left (a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx+\frac{1}{2} \left (3 a^3 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx-\left (3 a^3 c^3\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx+\left (9 a^3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{4} \left (a^5 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx\\ &=-\frac{1}{4} a^3 c^3 x+\frac{1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-5 i a^2 c^3 \tan ^{-1}(a x)^2-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac{15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-7 a^2 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+3 a^2 c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{9}{4} i a^2 c^3 \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{9}{4} i a^2 c^3 \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )-\left (9 i a^2 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}{4} \left (a^3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx-\frac{1}{2} \left (a^3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{2} \left (3 a^3 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{1}{4} a^3 c^3 x+\frac{1}{4} a^2 c^3 \tan ^{-1}(a x)+\frac{1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-5 i a^2 c^3 \tan ^{-1}(a x)^2-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac{15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-7 a^2 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+3 a^2 c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-\frac{9}{2} i a^2 c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{9}{4} i a^2 c^3 \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{9}{4} i a^2 c^3 \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )+\frac{1}{2} \left (i a^2 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 (3 i a^2 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}{4} a^3 c^3 x+\frac{1}{4} a^2 c^3 \tan ^{-1}(a x)+\frac{1}{4} a^4 c^3 x^2 \tan ^{-1}(a x)-5 i a^2 c^3 \tan ^{-1}(a x)^2-\frac{3 a c^3 \tan ^{-1}(a x)^2}{2 x}-\frac{15}{4} a^3 c^3 x \tan ^{-1}(a x)^2-\frac{1}{4} a^5 c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{4} a^2 c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{2 x^2}+\frac{3}{2} a^4 c^3 x^2 \tan ^{-1}(a x)^3+\frac{1}{4} a^6 c^3 x^4 \tan ^{-1}(a x)^3+6 a^2 c^3 \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-7 a^2 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )+3 a^2 c^3 \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-\frac{7}{2} i a^2 c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )-\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} i a^2 c^3 \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{9}{2} a^2 c^3 \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{9}{4} i a^2 c^3 \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{9}{4} i a^2 c^3 \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.749681, size = 464, normalized size = 0.92 \[ \frac{c^3 \left (288 i a^2 x^2 \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )+32 i a^2 x^2 \left (9 \tan ^{-1}(a x)^2+7\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-96 i a^2 x^2 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )+288 a^2 x^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )-288 a^2 x^2 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-144 i a^2 x^2 \text{PolyLog}\left (4,e^{-2 i \tan ^{-1}(a x)}\right )-144 i a^2 x^2 \text{PolyLog}\left (4,-e^{2 i \tan ^{-1}(a x)}\right )-16 a^3 x^3-3 i \pi ^4 a^2 x^2+16 a^6 x^6 \tan ^{-1}(a x)^3-16 a^5 x^5 \tan ^{-1}(a x)^2+96 a^4 x^4 \tan ^{-1}(a x)^3+16 a^4 x^4 \tan ^{-1}(a x)-240 a^3 x^3 \tan ^{-1}(a x)^2+96 i a^2 x^2 \tan ^{-1}(a x)^4+48 a^2 x^2 \tan ^{-1}(a x)^3+128 i a^2 x^2 \tan ^{-1}(a x)^2+16 a^2 x^2 \tan ^{-1}(a x)+192 a^2 x^2 \tan ^{-1}(a x)^3 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+192 a^2 x^2 \tan ^{-1}(a x) \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )-192 a^2 x^2 \tan ^{-1}(a x)^3 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-448 a^2 x^2 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-96 a x \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^3\right )}{64 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 6.651, size = 790, normalized size = 1.6 \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{4 \,{\left (a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} - 2 \, c^{3}\right )} \arctan \left (a x\right )^{3} - 3 \,{\left (a^{6} c^{3} x^{6} + 6 \, a^{4} c^{3} x^{4} - 2 \, c^{3}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + x^{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} - 12 \,{\left (a^{7} c^{3} x^{7} + 6 \, a^{5} c^{3} x^{5} - 2 \, a c^{3} x\right )} \arctan \left (a x\right )^{2} + 12 \,{\left (a^{8} c^{3} x^{8} + 6 \, a^{6} c^{3} x^{6} - 2 \, a^{2} c^{3} x^{2}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) + 3 \,{\left (a^{7} c^{3} x^{7} + 6 \, a^{5} c^{3} x^{5} - 2 \, a c^{3} x + 4 \,{\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}}{a^{2} x^{5} + x^{3}}\,{d x}}{128 \, x^{2}} \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^{3}}, 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^{3}}\, dx + \int \frac{3 a^{2} \operatorname{atan}^{3}{\left (a x \right )}}{x}\, dx + \int 3 a^{4} x \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{3} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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