Optimal. Leaf size=260 \[ -\frac {3 i \text {ArcTan}(a x)^2}{2 a^4 c}-\frac {3 x \text {ArcTan}(a x)^2}{2 a^3 c}+\frac {\text {ArcTan}(a x)^3}{2 a^4 c}+\frac {x^2 \text {ArcTan}(a x)^3}{2 a^2 c}+\frac {i \text {ArcTan}(a x)^4}{4 a^4 c}-\frac {3 \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\text {ArcTan}(a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}-\frac {3 i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 i \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c}+\frac {3 \text {ArcTan}(a x) \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c}-\frac {3 i \text {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )}{4 a^4 c} \]
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Rubi [A]
time = 0.33, 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 = {5036, 4946,
4930, 5040, 4964, 2449, 2352, 5004, 5114, 5118, 6745} \begin {gather*} \frac {3 i \text {ArcTan}(a x)^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{2 a^4 c}+\frac {3 \text {ArcTan}(a x) \text {Li}_3\left (1-\frac {2}{i a x+1}\right )}{2 a^4 c}+\frac {i \text {ArcTan}(a x)^4}{4 a^4 c}+\frac {\text {ArcTan}(a x)^3}{2 a^4 c}-\frac {3 i \text {ArcTan}(a x)^2}{2 a^4 c}+\frac {\text {ArcTan}(a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^4 c}-\frac {3 \text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}-\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 x \text {ArcTan}(a x)^2}{2 a^3 c}+\frac {x^2 \text {ArcTan}(a x)^3}{2 a^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 5004
Rule 5036
Rule 5040
Rule 5114
Rule 5118
Rule 6745
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) \text {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 = 0.22, size = 162, normalized size = 0.62 \begin {gather*} \frac {6 i \text {ArcTan}(a x)^2-6 a x \text {ArcTan}(a x)^2+2 \left (1+a^2 x^2\right ) \text {ArcTan}(a x)^3-i \text {ArcTan}(a x)^4-12 \text {ArcTan}(a x) \log \left (1+e^{2 i \text {ArcTan}(a x)}\right )+4 \text {ArcTan}(a x)^3 \log \left (1+e^{2 i \text {ArcTan}(a x)}\right )-6 i \left (-1+\text {ArcTan}(a x)^2\right ) \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(a x)}\right )+6 \text {ArcTan}(a x) \text {PolyLog}\left (3,-e^{2 i \text {ArcTan}(a x)}\right )+3 i \text {PolyLog}\left (4,-e^{2 i \text {ArcTan}(a x)}\right )}{4 a^4 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 73.76, size = 259, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {i \arctan \left (a x \right )^{4}}{4 c}+\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )+\arctan \left (a x \right ) a x -3\right ) \left (a x +i\right )}{2 c}+\frac {\arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}-\frac {3 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}+\frac {3 \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}+\frac {3 i \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4 c}+\frac {3 i \arctan \left (a x \right )^{2}}{c}-\frac {3 \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}+\frac {3 i \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}}{a^{4}}\) | \(259\) |
default | \(\frac {-\frac {i \arctan \left (a x \right )^{4}}{4 c}+\frac {\arctan \left (a x \right )^{2} \left (-i \arctan \left (a x \right )+\arctan \left (a x \right ) a x -3\right ) \left (a x +i\right )}{2 c}+\frac {\arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}-\frac {3 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}+\frac {3 \arctan \left (a x \right ) \polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}+\frac {3 i \polylog \left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4 c}+\frac {3 i \arctan \left (a x \right )^{2}}{c}-\frac {3 \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{c}+\frac {3 i \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 c}}{a^{4}}\) | \(259\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {\int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^3}{c\,a^2\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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