Optimal. Leaf size=270 \[ \frac {3 i \text {Li}_4\left (1-\frac {2}{i a x+1}\right )}{4 a^4 c^2}-\frac {3 i \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)^2}{2 a^4 c^2}-\frac {3 \text {Li}_3\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)}{2 a^4 c^2}-\frac {i \tan ^{-1}(a x)^4}{4 a^4 c^2}-\frac {\tan ^{-1}(a x)^3}{4 a^4 c^2}+\frac {3 \tan ^{-1}(a x)}{8 a^4 c^2}-\frac {\log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^3}{a^4 c^2}+\frac {\tan ^{-1}(a x)^3}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac {3 \tan ^{-1}(a x)}{4 a^4 c^2 \left (a^2 x^2+1\right )}+\frac {3 x}{8 a^3 c^2 \left (a^2 x^2+1\right )}-\frac {3 x \tan ^{-1}(a x)^2}{4 a^3 c^2 \left (a^2 x^2+1\right )} \]
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Rubi [A] time = 0.41, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 199
Rule 205
Rule 4854
Rule 4884
Rule 4892
Rule 4920
Rule 4930
Rule 4964
Rule 4994
Rule 4998
Rule 6610
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*}
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Mathematica [A] time = 0.20, size = 156, normalized size = 0.58 \[ \frac {24 i \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )-24 \tan ^{-1}(a x) \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )-12 i \text {Li}_4\left (-e^{2 i \tan ^{-1}(a x)}\right )+4 i \tan ^{-1}(a x)^4-16 \tan ^{-1}(a x)^3 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-6 \tan ^{-1}(a x)^2 \sin \left (2 \tan ^{-1}(a x)\right )+3 \sin \left (2 \tan ^{-1}(a x)\right )+4 \tan ^{-1}(a x)^3 \cos \left (2 \tan ^{-1}(a x)\right )-6 \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )}{16 a^4 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \arctan \left (a x\right )^{3}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
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 = 1.01, size = 1227, normalized size = 4.54 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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 \[ \frac {\int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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