Optimal. Leaf size=432 \[ -\frac{5 a^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{10 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^3}-\frac{141 a^3}{128 c^3 \left (a^2 x^2+1\right )}-\frac{3 a^3}{128 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^3 \log \left (a^2 x^2+1\right )}{2 c^3}+\frac{11 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (a^2 x^2+1\right )}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{141 a^4 x \tan ^{-1}(a x)}{64 c^3 \left (a^2 x^2+1\right )}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{33 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac{a^3 \log (x)}{c^3}+\frac{35 a^3 \tan ^{-1}(a x)^4}{32 c^3}+\frac{10 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{205 a^3 \tan ^{-1}(a x)^2}{128 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{10 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3} \]
[Out]
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Rubi [A] time = 2.15952, antiderivative size = 432, normalized size of antiderivative = 1., number of steps used = 57, number of rules used = 17, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.773, Rules used = {4966, 4918, 4852, 266, 36, 29, 31, 4884, 4924, 4868, 4992, 6610, 4892, 4930, 261, 4900, 4896} \[ -\frac{5 a^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{10 i a^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )}{c^3}-\frac{141 a^3}{128 c^3 \left (a^2 x^2+1\right )}-\frac{3 a^3}{128 c^3 \left (a^2 x^2+1\right )^2}-\frac{a^3 \log \left (a^2 x^2+1\right )}{2 c^3}+\frac{11 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (a^2 x^2+1\right )}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac{141 a^4 x \tan ^{-1}(a x)}{64 c^3 \left (a^2 x^2+1\right )}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (a^2 x^2+1\right )^2}+\frac{33 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (a^2 x^2+1\right )^2}+\frac{a^3 \log (x)}{c^3}+\frac{35 a^3 \tan ^{-1}(a x)^4}{32 c^3}+\frac{10 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{205 a^3 \tan ^{-1}(a x)^2}{128 c^3}+\frac{3 a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{10 a^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2}{c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4966
Rule 4918
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4884
Rule 4924
Rule 4868
Rule 4992
Rule 6610
Rule 4892
Rule 4930
Rule 261
Rule 4900
Rule 4896
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^3}{x^4 \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )^3} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^4 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=a^4 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^4 \left (c+a^2 c x^2\right )} \, dx}{c^2}-2 \frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{1}{8} \left (3 a^4\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^3}{x^4} \, dx}{c^3}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}+\frac{\left (3 a^4\right ) \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}-2 \left (\frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac{a^4 \int \frac{\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\right )\\ &=-\frac{3 a^3}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^3 \tan ^{-1}(a x)^4}{32 c^3}+\frac{a \int \frac{\tan ^{-1}(a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{x^2} \, dx}{c^3}+\frac{a^4 \int \frac{\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx}{c^2}-\frac{\left (9 a^4\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{32 c}-\frac{\left (9 a^5\right ) \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-2 \left (-\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{a^3 \tan ^{-1}(a x)^4}{8 c^3}+\frac{a^2 \int \frac{\tan ^{-1}(a x)^3}{x^2} \, dx}{c^3}-\frac{a^4 \int \frac{\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx}{c^2}+\frac{\left (3 a^5\right ) \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )\\ &=-\frac{3 a^3}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{9 a^4 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{9 a^3 \tan ^{-1}(a x)^2}{128 c^3}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^4}{32 c^3}+\frac{a \int \frac{\tan ^{-1}(a x)^2}{x^3} \, dx}{c^3}-\frac{a^3 \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac{\left (3 a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac{\left (9 a^4\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{8 c}-2 \left (-\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)^4}{8 c^3}+\frac{\left (3 a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx}{c^3}+\frac{\left (3 a^4\right ) \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\right )+\frac{\left (9 a^5\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{64 c}\\ &=-\frac{3 a^3}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac{9 a^3}{128 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^4 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{45 a^3 \tan ^{-1}(a x)^2}{128 c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}+\frac{4 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^4}{32 c^3}+\frac{a^2 \int \frac{\tan ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac{\left (i a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^3}-\frac{\left (3 i a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^3}+\frac{\left (9 a^5\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-2 \left (\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{8 c^3}-\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{i a^3 \tan ^{-1}(a x)^3}{c^3}-\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)^4}{8 c^3}+\frac{\left (3 i a^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx}{c^3}-\frac{\left (3 a^5\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\right )\\ &=-\frac{3 a^3}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^3}{128 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^4 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{45 a^3 \tan ^{-1}(a x)^2}{128 c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}+\frac{4 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^4}{32 c^3}-\frac{4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{a^2 \int \frac{\tan ^{-1}(a x)}{x^2} \, dx}{c^3}-\frac{a^4 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{c^3}+\frac{\left (2 a^4\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-2 \left (\frac{3 a^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{8 c^3}-\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{i a^3 \tan ^{-1}(a x)^3}{c^3}-\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)^4}{8 c^3}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{\left (6 a^4\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\right )+\frac{\left (6 a^4\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\\ &=-\frac{3 a^3}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^3}{128 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^4 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{109 a^3 \tan ^{-1}(a x)^2}{128 c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}+\frac{4 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^4}{32 c^3}-\frac{4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{4 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{a^3 \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx}{c^3}-\frac{\left (i a^4\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-\frac{\left (3 i a^4\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-2 \left (\frac{3 a^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{8 c^3}-\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{i a^3 \tan ^{-1}(a x)^3}{c^3}-\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)^4}{8 c^3}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{3 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{\left (3 i a^4\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}\right )\\ &=-\frac{3 a^3}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^3}{128 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^4 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{109 a^3 \tan ^{-1}(a x)^2}{128 c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}+\frac{4 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^4}{32 c^3}-\frac{4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{4 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}-\frac{2 a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{c^3}-2 \left (\frac{3 a^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{8 c^3}-\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{i a^3 \tan ^{-1}(a x)^3}{c^3}-\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)^4}{8 c^3}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{3 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{3 a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}\right )+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )}{2 c^3}\\ &=-\frac{3 a^3}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^3}{128 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^4 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{109 a^3 \tan ^{-1}(a x)^2}{128 c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}+\frac{4 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^4}{32 c^3}-\frac{4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{4 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}-\frac{2 a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{c^3}-2 \left (\frac{3 a^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{8 c^3}-\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{i a^3 \tan ^{-1}(a x)^3}{c^3}-\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)^4}{8 c^3}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{3 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{3 a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}\right )+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 c^3}-\frac{a^5 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )}{2 c^3}\\ &=-\frac{3 a^3}{128 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^3}{128 c^3 \left (1+a^2 x^2\right )}-\frac{a^2 \tan ^{-1}(a x)}{c^3 x}-\frac{3 a^4 x \tan ^{-1}(a x)}{32 c^3 \left (1+a^2 x^2\right )^2}-\frac{45 a^4 x \tan ^{-1}(a x)}{64 c^3 \left (1+a^2 x^2\right )}-\frac{109 a^3 \tan ^{-1}(a x)^2}{128 c^3}-\frac{a \tan ^{-1}(a x)^2}{2 c^3 x^2}+\frac{3 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac{9 a^3 \tan ^{-1}(a x)^2}{16 c^3 \left (1+a^2 x^2\right )}+\frac{4 i a^3 \tan ^{-1}(a x)^3}{3 c^3}-\frac{\tan ^{-1}(a x)^3}{3 c^3 x^3}+\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}+\frac{a^4 x \tan ^{-1}(a x)^3}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 a^4 x \tan ^{-1}(a x)^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{11 a^3 \tan ^{-1}(a x)^4}{32 c^3}+\frac{a^3 \log (x)}{c^3}-\frac{a^3 \log \left (1+a^2 x^2\right )}{2 c^3}-\frac{4 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}+\frac{4 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}-\frac{2 a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{c^3}-2 \left (\frac{3 a^3}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^4 x \tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )}+\frac{3 a^3 \tan ^{-1}(a x)^2}{8 c^3}-\frac{3 a^3 \tan ^{-1}(a x)^2}{4 c^3 \left (1+a^2 x^2\right )}-\frac{i a^3 \tan ^{-1}(a x)^3}{c^3}-\frac{a^2 \tan ^{-1}(a x)^3}{c^3 x}-\frac{a^4 x \tan ^{-1}(a x)^3}{2 c^3 \left (1+a^2 x^2\right )}-\frac{3 a^3 \tan ^{-1}(a x)^4}{8 c^3}+\frac{3 a^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )}{c^3}-\frac{3 i a^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{c^3}+\frac{3 a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )}{2 c^3}\right )\\ \end{align*}
Mathematica [A] time = 1.16333, size = 301, normalized size = 0.7 \[ \frac{a^3 \left (-10 i \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )-5 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )+\log \left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-\frac{\tan ^{-1}(a x)^3}{3 a^3 x^3}-\frac{\tan ^{-1}(a x)^2}{2 a^2 x^2}+\frac{35}{32} \tan ^{-1}(a x)^4+\frac{3 \tan ^{-1}(a x)^3}{a x}-\frac{10}{3} i \tan ^{-1}(a x)^3-\frac{1}{2} \tan ^{-1}(a x)^2-\frac{\tan ^{-1}(a x)}{a x}-10 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+\frac{3}{4} \tan ^{-1}(a x)^3 \sin \left (2 \tan ^{-1}(a x)\right )+\frac{1}{32} \tan ^{-1}(a x)^3 \sin \left (4 \tan ^{-1}(a x)\right )-\frac{9}{8} \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )-\frac{3}{256} \tan ^{-1}(a x) \sin \left (4 \tan ^{-1}(a x)\right )+\frac{9}{8} \tan ^{-1}(a x)^2 \cos \left (2 \tan ^{-1}(a x)\right )+\frac{3}{128} \tan ^{-1}(a x)^2 \cos \left (4 \tan ^{-1}(a x)\right )-\frac{9}{16} \cos \left (2 \tan ^{-1}(a x)\right )-\frac{3 \cos \left (4 \tan ^{-1}(a x)\right )}{1024}+\frac{5 i \pi ^3}{12}\right )}{c^3} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 11.602, size = 2523, normalized size = 5.8 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\arctan \left (a x\right )^{3}}{a^{6} c^{3} x^{10} + 3 \, a^{4} c^{3} x^{8} + 3 \, a^{2} c^{3} x^{6} + c^{3} x^{4}}, 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*} \frac{\int \frac{\operatorname{atan}^{3}{\left (a x \right )}}{a^{6} x^{10} + 3 a^{4} x^{8} + 3 a^{2} x^{6} + x^{4}}\, dx}{c^{3}} \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{\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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