Optimal. Leaf size=216 \[ \frac {i b \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2}-\frac {b \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (-c x+i)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (-c x+i)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{i c x+1}\right )}{2 c^2 d^2}+\frac {i b^2}{2 c^2 d^2 (-c x+i)}-\frac {i b^2 \tan ^{-1}(c x)}{2 c^2 d^2} \]
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Rubi [A] time = 0.34, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4876, 4864, 4862, 627, 44, 203, 4884, 4854, 4994, 6610} \[ \frac {i b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2}+\frac {b^2 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {b \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (-c x+i)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (-c x+i)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2}+\frac {i b^2}{2 c^2 d^2 (-c x+i)}-\frac {i b^2 \tan ^{-1}(c x)}{2 c^2 d^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 627
Rule 4854
Rule 4862
Rule 4864
Rule 4876
Rule 4884
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{(d+i c d x)^2} \, dx &=\int \left (-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c d^2 (-i+c x)^2}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{c d^2 (-i+c x)}\right ) \, dx\\ &=-\frac {i \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{c d^2}-\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{-i+c x} \, dx}{c d^2}\\ &=-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}-\frac {(2 i b) \int \left (-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c d^2}-\frac {(2 b) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c d^2}\\ &=-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^2 d^2}-\frac {b \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{c d^2}+\frac {b \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c d^2}-\frac {\left (i b^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c d^2}\\ &=-\frac {b \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {b^2 \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c d^2}\\ &=-\frac {b \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {b^2 \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c d^2}\\ &=-\frac {b \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {b^2 \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c d^2}\\ &=\frac {i b^2}{2 c^2 d^2 (i-c x)}-\frac {b \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {\left (i b^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 c d^2}\\ &=\frac {i b^2}{2 c^2 d^2 (i-c x)}-\frac {i b^2 \tan ^{-1}(c x)}{2 c^2 d^2}-\frac {b \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.85, size = 300, normalized size = 1.39 \[ \frac {-6 a^2 \log \left (c^2 x^2+1\right )+\frac {12 i a^2}{c x-i}-12 i a^2 \tan ^{-1}(c x)-6 i a b \left (2 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+4 \tan ^{-1}(c x)^2+i \sin \left (2 \tan ^{-1}(c x)\right )-\cos \left (2 \tan ^{-1}(c x)\right )-2 i \tan ^{-1}(c x) \left (-2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-i \sin \left (2 \tan ^{-1}(c x)\right )+\cos \left (2 \tan ^{-1}(c x)\right )\right )\right )+b^2 \left (-12 i \tan ^{-1}(c x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+6 \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )-8 i \tan ^{-1}(c x)^3+12 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+6 i \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )+6 \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )-3 i \sin \left (2 \tan ^{-1}(c x)\right )-6 \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )+6 i \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )+3 \cos \left (2 \tan ^{-1}(c x)\right )\right )}{12 c^2 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} x \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 4 i \, a b x \log \left (-\frac {c x + i}{c x - i}\right ) - 4 \, a^{2} x}{4 \, {\left (c^{2} d^{2} x^{2} - 2 i \, c d^{2} x - d^{2}\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 1059, normalized size = 4.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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