Optimal. Leaf size=130 \[ \frac{3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a^3 c}+\frac{3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a^3 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 c}+\frac{x \tan ^{-1}(a x)^3}{a^2 c}+\frac{i \tan ^{-1}(a x)^3}{a^3 c}+\frac{3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{a^3 c} \]
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Rubi [A] time = 0.245269, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {4916, 4846, 4920, 4854, 4884, 4994, 6610} \[ \frac{3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a^3 c}+\frac{3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{a^3 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 c}+\frac{x \tan ^{-1}(a x)^3}{a^2 c}+\frac{i \tan ^{-1}(a x)^3}{a^3 c}+\frac{3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{a^3 c} \]
Antiderivative was successfully verified.
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Rule 4916
Rule 4846
Rule 4920
Rule 4854
Rule 4884
Rule 4994
Rule 6610
Rubi steps
\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx &=-\frac{\int \frac{\tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx}{a^2}+\frac{\int \tan ^{-1}(a x)^3 \, dx}{a^2 c}\\ &=\frac{x \tan ^{-1}(a x)^3}{a^2 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 c}-\frac{3 \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a c}\\ &=\frac{i \tan ^{-1}(a x)^3}{a^3 c}+\frac{x \tan ^{-1}(a x)^3}{a^2 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 c}+\frac{3 \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx}{a^2 c}\\ &=\frac{i \tan ^{-1}(a x)^3}{a^3 c}+\frac{x \tan ^{-1}(a x)^3}{a^2 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 c}+\frac{3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a^3 c}-\frac{6 \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2 c}\\ &=\frac{i \tan ^{-1}(a x)^3}{a^3 c}+\frac{x \tan ^{-1}(a x)^3}{a^2 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 c}+\frac{3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a^3 c}+\frac{3 i \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^3 c}-\frac{(3 i) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2 c}\\ &=\frac{i \tan ^{-1}(a x)^3}{a^3 c}+\frac{x \tan ^{-1}(a x)^3}{a^2 c}-\frac{\tan ^{-1}(a x)^4}{4 a^3 c}+\frac{3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{a^3 c}+\frac{3 i \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^3 c}+\frac{3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a^3 c}\\ \end{align*}
Mathematica [A] time = 0.209472, size = 93, normalized size = 0.72 \[ \frac{-3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-\frac{1}{4} \tan ^{-1}(a x)^2 \left (\tan ^{-1}(a x)^2+(-4 a x+4 i) \tan ^{-1}(a x)-12 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{a^3 c} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.033, size = 925, normalized size = 7.1 \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{x^{2} \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^{2} \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^{2} \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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