Optimal. Leaf size=159 \[ -\frac {a x}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \text {ArcTan}(a x)}{32 c^3}+\frac {\text {ArcTan}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\text {ArcTan}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \text {ArcTan}(a x)^2}{2 c^3}+\frac {\text {ArcTan}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{2 c^3} \]
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Rubi [A]
time = 0.21, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5086, 5044,
4988, 2497, 5050, 205, 211} \begin {gather*} \frac {\text {ArcTan}(a x)}{2 c^3 \left (a^2 x^2+1\right )}+\frac {\text {ArcTan}(a x)}{4 c^3 \left (a^2 x^2+1\right )^2}-\frac {11 a x}{32 c^3 \left (a^2 x^2+1\right )}-\frac {a x}{16 c^3 \left (a^2 x^2+1\right )^2}-\frac {i \text {ArcTan}(a x)^2}{2 c^3}-\frac {11 \text {ArcTan}(a x)}{32 c^3}+\frac {\text {ArcTan}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i \text {Li}_2\left (\frac {2}{1-i a x}-1\right )}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 211
Rule 2497
Rule 4988
Rule 5044
Rule 5050
Rule 5086
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )^3} \, dx &=-\left (a^2 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^3} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=\frac {\tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac {1}{4} a \int \frac {1}{\left (c+a^2 c x^2\right )^3} \, dx+\frac {\int \frac {\tan ^{-1}(a x)}{x \left (c+a^2 c x^2\right )} \, dx}{c^2}-\frac {a^2 \int \frac {x \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{c}\\ &=-\frac {a x}{16 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^2}{2 c^3}+\frac {i \int \frac {\tan ^{-1}(a x)}{x (i+a x)} \, dx}{c^3}-\frac {(3 a) \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}-\frac {a \int \frac {1}{\left (c+a^2 c x^2\right )^2} \, dx}{2 c}\\ &=-\frac {a x}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x}{32 c^3 \left (1+a^2 x^2\right )}+\frac {\tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^2}{2 c^3}+\frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx}{c^3}-\frac {(3 a) \int \frac {1}{c+a^2 c x^2} \, dx}{32 c^2}-\frac {a \int \frac {1}{c+a^2 c x^2} \, dx}{4 c^2}\\ &=-\frac {a x}{16 c^3 \left (1+a^2 x^2\right )^2}-\frac {11 a x}{32 c^3 \left (1+a^2 x^2\right )}-\frac {11 \tan ^{-1}(a x)}{32 c^3}+\frac {\tan ^{-1}(a x)}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac {\tan ^{-1}(a x)}{2 c^3 \left (1+a^2 x^2\right )}-\frac {i \tan ^{-1}(a x)^2}{2 c^3}+\frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^3}-\frac {i \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 90, normalized size = 0.57 \begin {gather*} -\frac {64 i \text {ArcTan}(a x)^2-4 \text {ArcTan}(a x) \left (12 \cos (2 \text {ArcTan}(a x))+\cos (4 \text {ArcTan}(a x))+32 \log \left (1-e^{2 i \text {ArcTan}(a x)}\right )\right )+64 i \text {PolyLog}\left (2,e^{2 i \text {ArcTan}(a x)}\right )+24 \sin (2 \text {ArcTan}(a x))+\sin (4 \text {ArcTan}(a x))}{128 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 291 vs. \(2 (141 ) = 282\).
time = 0.08, size = 292, normalized size = 1.84
method | result | size |
derivativedivides | \(\frac {\arctan \left (a x \right )}{2 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{3}}+\frac {\arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c^{3}}-\frac {\frac {\frac {11}{8} a^{3} x^{3}+\frac {13}{8} a x}{\left (a^{2} x^{2}+1\right )^{2}}+\frac {11 \arctan \left (a x \right )}{8}-2 i \ln \left (a x \right ) \ln \left (i a x +1\right )+2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-2 i \dilog \left (i a x +1\right )+2 i \dilog \left (-i a x +1\right )+i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {i \ln \left (a x -i\right )^{2}}{2}-i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )-i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )+\frac {i \ln \left (a x +i\right )^{2}}{2}+i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )+i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{4 c^{3}}\) | \(292\) |
default | \(\frac {\arctan \left (a x \right )}{2 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2 c^{3}}+\frac {\arctan \left (a x \right )}{4 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right ) \ln \left (a x \right )}{c^{3}}-\frac {\frac {\frac {11}{8} a^{3} x^{3}+\frac {13}{8} a x}{\left (a^{2} x^{2}+1\right )^{2}}+\frac {11 \arctan \left (a x \right )}{8}-2 i \ln \left (a x \right ) \ln \left (i a x +1\right )+2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )-2 i \dilog \left (i a x +1\right )+2 i \dilog \left (-i a x +1\right )+i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {i \ln \left (a x -i\right )^{2}}{2}-i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )-i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )+\frac {i \ln \left (a x +i\right )^{2}}{2}+i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )+i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{4 c^{3}}\) | \(292\) |
risch | \(-\frac {11 \arctan \left (a x \right )}{64 c^{3}}+\frac {i \ln \left (-i a x +1\right ) x^{2} a^{2}}{128 c^{3} \left (-i a x -1\right )^{2}}+\frac {i \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{4 c^{3}}-\frac {i \ln \left (i a x +1\right )}{32 c^{3} \left (i a x +1\right )^{2}}-\frac {3 i \ln \left (i a x +1\right )}{128 c^{3} \left (i a x -1\right )^{2}}+\frac {5 i \ln \left (i a x +1\right )}{64 c^{3} \left (i a x -1\right )}+\frac {5 i \ln \left (-i a x +1\right )}{32 c^{3} \left (-i a x +1\right )}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{4 c^{3}}+\frac {i \ln \left (-i a x +1\right )}{32 c^{3} \left (-i a x +1\right )^{2}}+\frac {3 i \ln \left (-i a x +1\right )}{128 c^{3} \left (-i a x -1\right )^{2}}-\frac {5 i \ln \left (-i a x +1\right )}{64 c^{3} \left (-i a x -1\right )}-\frac {5 i \ln \left (i a x +1\right )}{32 c^{3} \left (i a x +1\right )}+\frac {\ln \left (-i a x +1\right ) a x}{64 c^{3} \left (-i a x -1\right )^{2}}-\frac {5 \ln \left (-i a x +1\right ) a x}{64 c^{3} \left (-i a x -1\right )}-\frac {5 \ln \left (i a x +1\right ) a x}{64 c^{3} \left (i a x -1\right )}+\frac {\ln \left (i a x +1\right ) a x}{64 c^{3} \left (i a x -1\right )^{2}}-\frac {5 i}{32 c^{3} \left (i a x +1\right )}-\frac {i}{64 c^{3} \left (i a x +1\right )^{2}}-\frac {i}{64 c^{3} \left (i a x -1\right )}+\frac {i \ln \left (i a x +1\right )^{2}}{8 c^{3}}+\frac {i}{64 c^{3} \left (-i a x +1\right )^{2}}+\frac {i}{64 c^{3} \left (-i a x -1\right )}-\frac {i \ln \left (-i a x +1\right )^{2}}{8 c^{3}}+\frac {5 i}{32 c^{3} \left (-i a x +1\right )}-\frac {i \dilog \left (\frac {1}{2}+\frac {i a x}{2}\right )}{4 c^{3}}+\frac {i \dilog \left (i a x +1\right )}{2 c^{3}}+\frac {i \dilog \left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c^{3}}-\frac {i \dilog \left (-i a x +1\right )}{2 c^{3}}-\frac {i \ln \left (i a x +1\right ) x^{2} a^{2}}{128 c^{3} \left (i a x -1\right )^{2}}\) | \(571\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RecursionError} \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 {\mathrm {atan}\left (a\,x\right )}{x\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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