Optimal. Leaf size=270 \[ \frac{3 i \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )}{4 a^4 c^2}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}-\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}+\frac{3 x}{8 a^3 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac{3 x \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)}{4 a^4 c^2 \left (a^2 x^2+1\right )}-\frac{i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac{\tan ^{-1}(a x)^3}{4 a^4 c^2}+\frac{3 \tan ^{-1}(a x)}{8 a^4 c^2}-\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^3}{a^4 c^2} \]
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Rubi [A] time = 0.412732, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4964, 4920, 4854, 4884, 4994, 4998, 6610, 4930, 4892, 199, 205} \[ \frac{3 i \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )}{4 a^4 c^2}-\frac{3 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}-\frac{3 \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}+\frac{3 x}{8 a^3 c^2 \left (a^2 x^2+1\right )}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac{3 x \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (a^2 x^2+1\right )}-\frac{3 \tan ^{-1}(a x)}{4 a^4 c^2 \left (a^2 x^2+1\right )}-\frac{i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac{\tan ^{-1}(a x)^3}{4 a^4 c^2}+\frac{3 \tan ^{-1}(a x)}{8 a^4 c^2}-\frac{\log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^3}{a^4 c^2} \]
Antiderivative was successfully verified.
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Rule 4964
Rule 4920
Rule 4854
Rule 4884
Rule 4994
Rule 4998
Rule 6610
Rule 4930
Rule 4892
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx &=-\frac{\int \frac{x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2}+\frac{\int \frac{x \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx}{a^2 c}\\ &=\frac{\tan ^{-1}(a x)^3}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac{3 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^3}-\frac{\int \frac{\tan ^{-1}(a x)^3}{i-a x} \, dx}{a^3 c^2}\\ &=-\frac{3 x \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^3}{4 a^4 c^2}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac{\tan ^{-1}(a x)^3 \log \left (\frac{2}{1+i a x}\right )}{a^4 c^2}+\frac{3 \int \frac{x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2}+\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^3 c^2}\\ &=-\frac{3 \tan ^{-1}(a x)}{4 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{3 x \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^3}{4 a^4 c^2}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac{\tan ^{-1}(a x)^3 \log \left (\frac{2}{1+i a x}\right )}{a^4 c^2}-\frac{3 i \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}+\frac{3 \int \frac{1}{\left (c+a^2 c x^2\right )^2} \, dx}{4 a^3}+\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^3 c^2}\\ &=\frac{3 x}{8 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{3 \tan ^{-1}(a x)}{4 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{3 x \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^3}{4 a^4 c^2}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac{\tan ^{-1}(a x)^3 \log \left (\frac{2}{1+i a x}\right )}{a^4 c^2}-\frac{3 i \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}-\frac{3 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}+\frac{3 \int \frac{\text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3 c^2}+\frac{3 \int \frac{1}{c+a^2 c x^2} \, dx}{8 a^3 c}\\ &=\frac{3 x}{8 a^3 c^2 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)}{8 a^4 c^2}-\frac{3 \tan ^{-1}(a x)}{4 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{3 x \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (1+a^2 x^2\right )}-\frac{\tan ^{-1}(a x)^3}{4 a^4 c^2}+\frac{\tan ^{-1}(a x)^3}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac{i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac{\tan ^{-1}(a x)^3 \log \left (\frac{2}{1+i a x}\right )}{a^4 c^2}-\frac{3 i \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}-\frac{3 \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{2 a^4 c^2}+\frac{3 i \text{Li}_4\left (1-\frac{2}{1+i a x}\right )}{4 a^4 c^2}\\ \end{align*}
Mathematica [A] time = 0.188492, size = 156, normalized size = 0.58 \[ \frac{24 i \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-24 \tan ^{-1}(a x) \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-12 i \text{PolyLog}\left (4,-e^{2 i \tan ^{-1}(a x)}\right )+4 i \tan ^{-1}(a x)^4-16 \tan ^{-1}(a x)^3 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-6 \tan ^{-1}(a x)^2 \sin \left (2 \tan ^{-1}(a x)\right )+3 \sin \left (2 \tan ^{-1}(a x)\right )+4 \tan ^{-1}(a x)^3 \cos \left (2 \tan ^{-1}(a x)\right )-6 \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )}{16 a^4 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.636, size = 1227, normalized size = 4.5 \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*} \int \frac{x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{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^{3} \arctan \left (a x\right )^{3}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, 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^{3} \operatorname{atan}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \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^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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