Optimal. Leaf size=138 \[ \frac {3 i \text {Li}_4\left (1-\frac {2}{i a x+1}\right )}{4 a^2 c}-\frac {3 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)^2}{2 a^2 c}-\frac {3 \text {Li}_3\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)}{2 a^2 c}-\frac {i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac {\log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^3}{a^2 c} \]
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Rubi [A] time = 0.22, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 4854
Rule 4884
Rule 4920
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*}
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Mathematica [A] time = 0.01, size = 149, normalized size = 1.08 \[ \frac {3 i \text {Li}_4\left (\frac {a x+i}{a x-i}\right )}{4 a^2 c}-\frac {3 i \text {Li}_2\left (\frac {a x+i}{a x-i}\right ) \tan ^{-1}(a x)^2}{2 a^2 c}-\frac {3 \text {Li}_3\left (\frac {a x+i}{a x-i}\right ) \tan ^{-1}(a x)}{2 a^2 c}-\frac {i \tan ^{-1}(a x)^4}{4 a^2 c}-\frac {\log \left (\frac {2 i}{-a x+i}\right ) \tan ^{-1}(a x)^3}{a^2 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c}, x\right ) \]
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 = 0.73, size = 936, normalized size = 6.78 \[ \text {result too large to display} \]
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 \[ \int \frac {x \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{c\,a^2\,x^2+c} \,d x \]
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 \[ \frac {\int \frac {x \operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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