Optimal. Leaf size=138 \[ \frac{3 i \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )}{4 a^2 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^2 c}-\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^3}{a^2 c} \]
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Rubi [A] time = 0.217725, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4920, 4854, 4884, 4994, 4998, 6610} \[ \frac{3 i \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )}{4 a^2 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^2 c}-\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^3}{a^2 c} \]
Antiderivative was successfully verified.
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Rule 4920
Rule 4854
Rule 4884
Rule 4994
Rule 4998
Rule 6610
Rubi steps
\begin{align*} \int \frac{x \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx &=-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\int \frac{\tan ^{-1}(a x)^3}{i-a x} \, dx}{a c}\\ &=-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\tan ^{-1}(a x)^3 \log \left (\frac{2}{1+i a x}\right )}{a^2 c}+\frac{3 \int \frac{\tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a c}\\ &=-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\tan ^{-1}(a x)^3 \log \left (\frac{2}{1+i a x}\right )}{a^2 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^2 c}+\frac{(3 i) \int \frac{\tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a c}\\ &=-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\tan ^{-1}(a x)^3 \log \left (\frac{2}{1+i a x}\right )}{a^2 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^2 c}-\frac{3 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a^2 c}+\frac{3 \int \frac{\text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a c}\\ &=-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\tan ^{-1}(a x)^3 \log \left (\frac{2}{1+i a x}\right )}{a^2 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^2 c}-\frac{3 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a^2 c}+\frac{3 i \text{Li}_4\left (1-\frac{2}{1+i a x}\right )}{4 a^2 c}\\ \end{align*}
Mathematica [A] time = 0.0118383, size = 149, normalized size = 1.08 \[ \frac{3 i \text{PolyLog}\left (4,\frac{a x+i}{a x-i}\right )}{4 a^2 c}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{a x+i}{a x-i}\right )}{2 a^2 c}-\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left (3,\frac{a x+i}{a x-i}\right )}{2 a^2 c}-\frac{i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac{\log \left (\frac{2 i}{-a x+i}\right ) \tan ^{-1}(a x)^3}{a^2 c} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.132, size = 936, normalized size = 6.8 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c}\,{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{x \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c}, 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{x \operatorname{atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \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{x \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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