Optimal. Leaf size=130 \[ \frac {3 \text {Li}_3\left (1-\frac {2}{i a x+1}\right )}{2 a^3 c}+\frac {3 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)}{a^3 c}-\frac {\tan ^{-1}(a x)^4}{4 a^3 c}+\frac {i \tan ^{-1}(a x)^3}{a^3 c}+\frac {3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{a^3 c}+\frac {x \tan ^{-1}(a x)^3}{a^2 c} \]
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Rubi [A] time = 0.25, antiderivative size = 130, normalized size of antiderivative = 1.00, 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 4846
Rule 4854
Rule 4884
Rule 4916
Rule 4920
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*}
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Mathematica [A] time = 0.22, size = 93, normalized size = 0.72 \[ \frac {-3 i \tan ^{-1}(a x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+\frac {3}{2} \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )-\frac {1}{4} \tan ^{-1}(a x)^2 \left (\tan ^{-1}(a x)^2+(-4 a x+4 i) \tan ^{-1}(a x)-12 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{a^3 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.54, size = 925, normalized size = 7.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^3}{c\,a^2\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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