Optimal. Leaf size=203 \[ -\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}+\frac {b}{2 c^4 d^2 (i-c x)}-\frac {b \text {ArcTan}(c x)}{c^4 d^2}-\frac {2 i b x \text {ArcTan}(c x)}{c^3 d^2}-\frac {x^2 (a+b \text {ArcTan}(c x))}{2 c^2 d^2}+\frac {i (a+b \text {ArcTan}(c x))}{c^4 d^2 (i-c x)}-\frac {3 (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {i b \log \left (1+c^2 x^2\right )}{c^4 d^2}-\frac {3 i b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d^2} \]
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Rubi [A]
time = 0.17, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4996, 4930,
266, 4946, 327, 209, 4972, 641, 46, 4964, 2449, 2352} \begin {gather*} \frac {i (a+b \text {ArcTan}(c x))}{c^4 d^2 (-c x+i)}-\frac {3 \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))}{c^4 d^2}-\frac {x^2 (a+b \text {ArcTan}(c x))}{2 c^2 d^2}-\frac {2 i a x}{c^3 d^2}-\frac {b \text {ArcTan}(c x)}{c^4 d^2}-\frac {2 i b x \text {ArcTan}(c x)}{c^3 d^2}-\frac {3 i b \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{2 c^4 d^2}+\frac {b}{2 c^4 d^2 (-c x+i)}+\frac {b x}{2 c^3 d^2}+\frac {i b \log \left (c^2 x^2+1\right )}{c^4 d^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 209
Rule 266
Rule 327
Rule 641
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 4972
Rule 4996
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{(d+i c d x)^2} \, dx &=\int \left (-\frac {2 i \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^2}-\frac {x \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^2 (-i+c x)^2}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^2 (-i+c x)}\right ) \, dx\\ &=\frac {i \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{c^3 d^2}-\frac {(2 i) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^3 d^2}+\frac {3 \int \frac {a+b \tan ^{-1}(c x)}{-i+c x} \, dx}{c^3 d^2}-\frac {\int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^2 d^2}\\ &=-\frac {2 i a x}{c^3 d^2}-\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^2 (i-c x)}-\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {(i b) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^3 d^2}-\frac {(2 i b) \int \tan ^{-1}(c x) \, dx}{c^3 d^2}+\frac {(3 b) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d^2}+\frac {b \int \frac {x^2}{1+c^2 x^2} \, dx}{2 c d^2}\\ &=-\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}-\frac {2 i b x \tan ^{-1}(c x)}{c^3 d^2}-\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^2 (i-c x)}-\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}-\frac {(3 i b) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^4 d^2}+\frac {(i b) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^3 d^2}-\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{2 c^3 d^2}+\frac {(2 i b) \int \frac {x}{1+c^2 x^2} \, dx}{c^2 d^2}\\ &=-\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}-\frac {b \tan ^{-1}(c x)}{2 c^4 d^2}-\frac {2 i b x \tan ^{-1}(c x)}{c^3 d^2}-\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^2 (i-c x)}-\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {i b \log \left (1+c^2 x^2\right )}{c^4 d^2}-\frac {3 i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^4 d^2}+\frac {(i b) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^3 d^2}\\ &=-\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}+\frac {b}{2 c^4 d^2 (i-c x)}-\frac {b \tan ^{-1}(c x)}{2 c^4 d^2}-\frac {2 i b x \tan ^{-1}(c x)}{c^3 d^2}-\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^2 (i-c x)}-\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {i b \log \left (1+c^2 x^2\right )}{c^4 d^2}-\frac {3 i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^4 d^2}-\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{2 c^3 d^2}\\ &=-\frac {2 i a x}{c^3 d^2}+\frac {b x}{2 c^3 d^2}+\frac {b}{2 c^4 d^2 (i-c x)}-\frac {b \tan ^{-1}(c x)}{c^4 d^2}-\frac {2 i b x \tan ^{-1}(c x)}{c^3 d^2}-\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^2 (i-c x)}-\frac {3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^2}+\frac {i b \log \left (1+c^2 x^2\right )}{c^4 d^2}-\frac {3 i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^4 d^2}\\ \end {align*}
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Mathematica [A]
time = 0.70, size = 186, normalized size = 0.92 \begin {gather*} -\frac {8 i a c x+2 a c^2 x^2+\frac {4 i a}{-i+c x}-12 i a \text {ArcTan}(c x)-6 a \log \left (1+c^2 x^2\right )+b \left (-2 c x-12 i \text {ArcTan}(c x)^2+i \cos (2 \text {ArcTan}(c x))-4 i \log \left (1+c^2 x^2\right )-6 i \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )+2 \text {ArcTan}(c x) \left (1+4 i c x+c^2 x^2-\cos (2 \text {ArcTan}(c x))+6 \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )+i \sin (2 \text {ArcTan}(c x))\right )+\sin (2 \text {ArcTan}(c x))\right )}{4 c^4 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 317, normalized size = 1.56
method | result | size |
derivativedivides | \(\frac {-\frac {2 i b \arctan \left (c x \right ) c x}{d^{2}}-\frac {a \,c^{2} x^{2}}{2 d^{2}}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a}{d^{2} \left (c x -i\right )}+\frac {3 i b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}+\frac {3 i a \arctan \left (c x \right )}{d^{2}}-\frac {b \arctan \left (c x \right ) c^{2} x^{2}}{2 d^{2}}+\frac {3 b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}-\frac {i b}{2 d^{2}}+\frac {i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}-\frac {i b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}-\frac {3 i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d^{2}}+\frac {b c x}{2 d^{2}}-\frac {2 i a c x}{d^{2}}-\frac {b}{2 d^{2} \left (c x -i\right )}-\frac {3 i b \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{2}}-\frac {3 b \arctan \left (c x \right )}{2 d^{2}}+\frac {3 i b \ln \left (c x -i\right )^{2}}{4 d^{2}}-\frac {b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}}{c^{4}}\) | \(317\) |
default | \(\frac {-\frac {2 i b \arctan \left (c x \right ) c x}{d^{2}}-\frac {a \,c^{2} x^{2}}{2 d^{2}}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}-\frac {i a}{d^{2} \left (c x -i\right )}+\frac {3 i b \ln \left (c^{2} x^{2}+1\right )}{4 d^{2}}+\frac {3 i a \arctan \left (c x \right )}{d^{2}}-\frac {b \arctan \left (c x \right ) c^{2} x^{2}}{2 d^{2}}+\frac {3 b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}-\frac {i b}{2 d^{2}}+\frac {i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{8 d^{2}}-\frac {i b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}-\frac {3 i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d^{2}}+\frac {b c x}{2 d^{2}}-\frac {2 i a c x}{d^{2}}-\frac {b}{2 d^{2} \left (c x -i\right )}-\frac {3 i b \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{2}}-\frac {3 b \arctan \left (c x \right )}{2 d^{2}}+\frac {3 i b \ln \left (c x -i\right )^{2}}{4 d^{2}}-\frac {b \arctan \left (\frac {c x}{2}\right )}{4 d^{2}}+\frac {b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{4 d^{2}}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 d^{2}}}{c^{4}}\) | \(317\) |
risch | \(-\frac {i b \ln \left (-i c x +1\right )}{4 d^{2} c^{4} \left (-i c x -1\right )}+\left (\frac {i b \left (\frac {1}{2} c \,x^{2}+2 i x \right )}{2 c^{3} d^{2}}-\frac {b}{2 c^{4} d^{2} \left (c x -i\right )}\right ) \ln \left (i c x +1\right )+\frac {9 i b \ln \left (c^{2} x^{2}+1\right )}{8 c^{4} d^{2}}+\frac {\ln \left (-i c x +1\right ) x b}{d^{2} c^{3}}-\frac {2 i a x}{c^{3} d^{2}}-\frac {3 b \arctan \left (c x \right )}{4 c^{4} d^{2}}+\frac {b x}{2 c^{3} d^{2}}+\frac {3 i a \arctan \left (c x \right )}{d^{2} c^{4}}-\frac {3 i b \ln \left (i c x +1\right )^{2}}{4 c^{4} d^{2}}-\frac {5 i b}{8 c^{4} d^{2}}+\frac {3 i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{2 d^{2} c^{4}}-\frac {3 i b \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2} c^{4}}-\frac {b \ln \left (-i c x +1\right ) x}{4 d^{2} c^{3} \left (-i c x -1\right )}-\frac {i \ln \left (-i c x +1\right ) x^{2} b}{4 d^{2} c^{2}}-\frac {x^{2} a}{2 d^{2} c^{2}}-\frac {3 i b \dilog \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2 d^{2} c^{4}}+\frac {3 a}{2 d^{2} c^{4}}+\frac {3 a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2} c^{4}}-\frac {a}{d^{2} c^{4} \left (-i c x -1\right )}+\frac {b}{2 c^{4} d^{2} \left (-c x +i\right )}\) | \(386\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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