Optimal. Leaf size=216 \[ \frac{i b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 c^2 d^2}-\frac{b \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (-c x+i)}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2 (-c x+i)}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2 d^2}+\frac{\log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2}+\frac{i b^2}{2 c^2 d^2 (-c x+i)}-\frac{i b^2 \tan ^{-1}(c x)}{2 c^2 d^2} \]
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Rubi [A] time = 0.342402, antiderivative size = 216, normalized size of antiderivative = 1., 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 4876
Rule 4864
Rule 4862
Rule 627
Rule 44
Rule 203
Rule 4884
Rule 4854
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*}
Mathematica [A] time = 0.760157, size = 300, normalized size = 1.39 \[ \frac{-6 i a b \left (2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+4 \tan ^{-1}(c x)^2+i \sin \left (2 \tan ^{-1}(c x)\right )-\cos \left (2 \tan ^{-1}(c x)\right )-2 i \tan ^{-1}(c x) \left (-2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-i \sin \left (2 \tan ^{-1}(c x)\right )+\cos \left (2 \tan ^{-1}(c x)\right )\right )\right )+b^2 \left (-12 i \tan ^{-1}(c x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+6 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )-8 i \tan ^{-1}(c x)^3+12 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+6 i \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )+6 \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )-3 i \sin \left (2 \tan ^{-1}(c x)\right )-6 \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )+6 i \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )+3 \cos \left (2 \tan ^{-1}(c x)\right )\right )-6 a^2 \log \left (c^2 x^2+1\right )+\frac{12 i a^2}{c x-i}-12 i a^2 \tan ^{-1}(c x)}{12 c^2 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.316, size = 1059, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 \, c^{2} d^{2} x^{2} - 8 i \, c d^{2} x - 4 \, d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )}^{2} x}{{\left (i \, c d x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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