Optimal. Leaf size=260 \[ -\frac {3 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{2 a^4 c}-\frac {3 i \text {Li}_4\left (1-\frac {2}{i a x+1}\right )}{4 a^4 c}+\frac {3 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)^2}{2 a^4 c}+\frac {3 \text {Li}_3\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)}{2 a^4 c}+\frac {i \tan ^{-1}(a x)^4}{4 a^4 c}+\frac {\tan ^{-1}(a x)^3}{2 a^4 c}-\frac {3 i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac {\log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^3}{a^4 c}-\frac {3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^4 c}-\frac {3 x \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {x^2 \tan ^{-1}(a x)^3}{2 a^2 c} \]
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Rubi [A] time = 0.45, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4916, 4852, 4846, 4920, 4854, 2402, 2315, 4884, 4994, 4998, 6610} \[ -\frac {3 i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c}-\frac {3 i \text {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )}{4 a^4 c}+\frac {3 i \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 \tan ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^4}{4 a^4 c}+\frac {\tan ^{-1}(a x)^3}{2 a^4 c}-\frac {3 x \tan ^{-1}(a x)^2}{2 a^3 c}-\frac {3 i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac {\log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^3}{a^4 c}-\frac {3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^4 c} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4884
Rule 4916
Rule 4920
Rule 4994
Rule 4998
Rule 6610
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx &=-\frac {\int \frac {x \tan ^{-1}(a x)^3}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x \tan ^{-1}(a x)^3 \, dx}{a^2 c}\\ &=\frac {x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^4}{4 a^4 c}+\frac {\int \frac {\tan ^{-1}(a x)^3}{i-a x} \, dx}{a^3 c}-\frac {3 \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a c}\\ &=\frac {x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^4}{4 a^4 c}+\frac {\tan ^{-1}(a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}-\frac {3 \int \tan ^{-1}(a x)^2 \, dx}{2 a^3 c}+\frac {3 \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a^3 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^3 c}\\ &=-\frac {3 x \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {\tan ^{-1}(a x)^3}{2 a^4 c}+\frac {x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^4}{4 a^4 c}+\frac {\tan ^{-1}(a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {3 i \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{2 a^4 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^3 c}+\frac {3 \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2 c}\\ &=-\frac {3 i \tan ^{-1}(a x)^2}{2 a^4 c}-\frac {3 x \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {\tan ^{-1}(a x)^3}{2 a^4 c}+\frac {x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^4}{4 a^4 c}+\frac {\tan ^{-1}(a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {3 i \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{2 a^4 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^3 c}-\frac {3 \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{a^3 c}\\ &=-\frac {3 i \tan ^{-1}(a x)^2}{2 a^4 c}-\frac {3 x \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {\tan ^{-1}(a x)^3}{2 a^4 c}+\frac {x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^4}{4 a^4 c}-\frac {3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\tan ^{-1}(a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {3 i \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c}-\frac {3 i \text {Li}_4\left (1-\frac {2}{1+i a x}\right )}{4 a^4 c}+\frac {3 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c}\\ &=-\frac {3 i \tan ^{-1}(a x)^2}{2 a^4 c}-\frac {3 x \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {\tan ^{-1}(a x)^3}{2 a^4 c}+\frac {x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^4}{4 a^4 c}-\frac {3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\tan ^{-1}(a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {3 i \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c}-\frac {3 i \text {Li}_4\left (1-\frac {2}{1+i a x}\right )}{4 a^4 c}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^4 c}\\ &=-\frac {3 i \tan ^{-1}(a x)^2}{2 a^4 c}-\frac {3 x \tan ^{-1}(a x)^2}{2 a^3 c}+\frac {\tan ^{-1}(a x)^3}{2 a^4 c}+\frac {x^2 \tan ^{-1}(a x)^3}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^4}{4 a^4 c}-\frac {3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\tan ^{-1}(a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}-\frac {3 i \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 i \tan ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 \tan ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c}-\frac {3 i \text {Li}_4\left (1-\frac {2}{1+i a x}\right )}{4 a^4 c}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 162, normalized size = 0.62 \[ \frac {2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3+6 \tan ^{-1}(a x) \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )-6 i \left (\tan ^{-1}(a x)^2-1\right ) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+3 i \text {Li}_4\left (-e^{2 i \tan ^{-1}(a x)}\right )-i \tan ^{-1}(a x)^4-6 a x \tan ^{-1}(a x)^2+6 i \tan ^{-1}(a x)^2+4 \tan ^{-1}(a x)^3 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-12 \tan ^{-1}(a x) \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )}{4 a^4 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c}, x\right ) \]
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 [A] time = 8.53, size = 292, normalized size = 1.12 \[ -\frac {i \arctan \left (a x \right )^{4}}{4 a^{4} c}+\frac {x^{2} \arctan \left (a x \right )^{3}}{2 a^{2} c}+\frac {\arctan \left (a x \right )^{3}}{2 a^{4} c}-\frac {3 x \arctan \left (a x \right )^{2}}{2 a^{3} c}+\frac {3 i \arctan \left (a x \right )^{2}}{2 a^{4} c}+\frac {\arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{a^{4} c}-\frac {3 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a^{4} c}+\frac {3 \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a^{4} c}+\frac {3 i \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4 a^{4} c}-\frac {3 \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{a^{4} c}+\frac {3 i \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a^{4} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^3}{c\,a^2\,x^2+c} \,d x \]
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 \[ \frac {\int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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