Optimal. Leaf size=113 \[ -\frac {x}{2 a^3 c}+\frac {\text {ArcTan}(a x)}{2 a^4 c}+\frac {x^2 \text {ArcTan}(a x)}{2 a^2 c}+\frac {i \text {ArcTan}(a x)^2}{2 a^4 c}+\frac {\text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^4 c} \]
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Rubi [A]
time = 0.10, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5036, 4946,
327, 209, 5040, 4964, 2449, 2352} \begin {gather*} \frac {i \text {ArcTan}(a x)^2}{2 a^4 c}+\frac {\text {ArcTan}(a x)}{2 a^4 c}+\frac {\text {ArcTan}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{2 a^4 c}-\frac {x}{2 a^3 c}+\frac {x^2 \text {ArcTan}(a x)}{2 a^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 4946
Rule 4964
Rule 5036
Rule 5040
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx &=-\frac {\int \frac {x \tan ^{-1}(a x)}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int x \tan ^{-1}(a x) \, dx}{a^2 c}\\ &=\frac {x^2 \tan ^{-1}(a x)}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac {\int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{a^3 c}-\frac {\int \frac {x^2}{1+a^2 x^2} \, dx}{2 a c}\\ &=-\frac {x}{2 a^3 c}+\frac {x^2 \tan ^{-1}(a x)}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {\int \frac {1}{1+a^2 x^2} \, dx}{2 a^3 c}-\frac {\int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c}\\ &=-\frac {x}{2 a^3 c}+\frac {\tan ^{-1}(a x)}{2 a^4 c}+\frac {x^2 \tan ^{-1}(a x)}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {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 {x}{2 a^3 c}+\frac {\tan ^{-1}(a x)}{2 a^4 c}+\frac {x^2 \tan ^{-1}(a x)}{2 a^2 c}+\frac {i \tan ^{-1}(a x)^2}{2 a^4 c}+\frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4 c}+\frac {i \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{2 a^4 c}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 120, normalized size = 1.06 \begin {gather*} -\frac {x}{2 a^3 c}+\frac {\text {ArcTan}(a x)}{2 a^4 c}+\frac {x^2 \text {ArcTan}(a x)}{2 a^2 c}+\frac {i \text {ArcTan}(a x)^2}{2 a^4 c}+\frac {\text {ArcTan}(a x) \log \left (\frac {2 i}{i-a x}\right )}{a^4 c}+\frac {i \text {PolyLog}\left (2,-\frac {i+a x}{i-a x}\right )}{2 a^4 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 185, normalized size = 1.64
method | result | size |
derivativedivides | \(\frac {\frac {\arctan \left (a x \right ) a^{2} x^{2}}{2 c}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {a x -\arctan \left (a x \right )+\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {i \ln \left (a x -i\right )^{2}}{4}-\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}-\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}-\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i \ln \left (a x +i\right )^{2}}{4}+\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{2}+\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{2}}{2 c}}{a^{4}}\) | \(185\) |
default | \(\frac {\frac {\arctan \left (a x \right ) a^{2} x^{2}}{2 c}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c}-\frac {a x -\arctan \left (a x \right )+\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {i \ln \left (a x -i\right )^{2}}{4}-\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}-\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{2}-\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {i \ln \left (a x +i\right )^{2}}{4}+\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{2}+\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{2}}{2 c}}{a^{4}}\) | \(185\) |
risch | \(\frac {i \ln \left (-i a x +1\right ) x^{2}}{4 c \,a^{2}}+\frac {\arctan \left (a x \right )}{2 a^{4} c}-\frac {x}{2 a^{3} c}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c \,a^{4}}+\frac {i \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c \,a^{4}}-\frac {i \ln \left (-i a x +1\right )^{2}}{8 c \,a^{4}}-\frac {i \ln \left (i a x +1\right ) x^{2}}{4 c \,a^{2}}+\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c \,a^{4}}-\frac {i \dilog \left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c \,a^{4}}+\frac {i \ln \left (i a x +1\right )^{2}}{8 c \,a^{4}}\) | \(185\) |
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}{\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 = -0.01 \begin {gather*} \int \frac {x^3\,\mathrm {atan}\left (a\,x\right )}{c\,a^2\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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