Optimal. Leaf size=221 \[ \frac{i b \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}+\frac{b^2 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )}{2 d^2}+\frac{b \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-c x+i)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{\log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}+\frac{2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{i b^2}{2 d^2 (-c x+i)}+\frac{i b^2 \tan ^{-1}(c x)}{2 d^2} \]
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Rubi [A] time = 0.616117, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {4876, 4850, 4988, 4884, 4994, 6610, 4864, 4862, 627, 44, 203, 4854} \[ \frac{i b \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}+\frac{b^2 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )}{2 d^2}+\frac{b \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-c x+i)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{\log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}+\frac{2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{i b^2}{2 d^2 (-c x+i)}+\frac{i b^2 \tan ^{-1}(c x)}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4850
Rule 4988
Rule 4884
Rule 4994
Rule 6610
Rule 4864
Rule 4862
Rule 627
Rule 44
Rule 203
Rule 4854
Rubi steps
\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x (d+i c d x)^2} \, dx &=\int \left (\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-i+c x)^2}-\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-i+c x)}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx}{d^2}+\frac{(i c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{d^2}-\frac{c \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{-i+c x} \, dx}{d^2}\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{(2 i b c) \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}{d^2}-\frac{(2 b c) \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}{d^2}-\frac{(4 b c) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{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 )}{d^2}+\frac{(b c) \int \frac{a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{d^2}-\frac{(b c) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{d^2}+\frac{(2 b c) \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}{d^2}-\frac{(2 b c) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}-\frac{\left (i b^2 c\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}\\ &=\frac{b \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{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 )}{d^2}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 d^2}+\frac{\left (i b^2 c\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}-\frac{\left (i b^2 c\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac{\left (b^2 c\right ) \int \frac{1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^2}\\ &=\frac{b \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{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 )}{d^2}+\frac{b^2 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d^2}+\frac{\left (b^2 c\right ) \int \frac{1}{(-i+c x)^2 (i+c x)} \, dx}{d^2}\\ &=\frac{b \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{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 )}{d^2}+\frac{b^2 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d^2}+\frac{\left (b^2 c\right ) \int \left (-\frac{i}{2 (-i+c x)^2}+\frac{i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}\\ &=-\frac{i b^2}{2 d^2 (i-c x)}+\frac{b \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{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 )}{d^2}+\frac{b^2 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d^2}+\frac{\left (i b^2 c\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 d^2}\\ &=-\frac{i b^2}{2 d^2 (i-c x)}+\frac{i b^2 \tan ^{-1}(c x)}{2 d^2}+\frac{b \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}+\frac{2 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{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 )}{d^2}+\frac{b^2 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{2 d^2}\\ \end{align*}
Mathematica [A] time = 0.948125, size = 299, normalized size = 1.35 \[ \frac{-12 a b \left (2 i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )+4 i \tan ^{-1}(c x)^2+\sin \left (2 \tan ^{-1}(c x)\right )+i \cos \left (2 \tan ^{-1}(c x)\right )-2 \tan ^{-1}(c x) \left (2 \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )-i \sin \left (2 \tan ^{-1}(c x)\right )+\cos \left (2 \tan ^{-1}(c x)\right )\right )\right )+b^2 \left (24 i \tan ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(c x)}\right )+12 \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(c x)}\right )+24 \tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )-12 i \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )-12 \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )+6 i \sin \left (2 \tan ^{-1}(c x)\right )+12 \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )-12 i \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )-6 \cos \left (2 \tan ^{-1}(c x)\right )-i \pi ^3\right )-12 a^2 \log \left (c^2 x^2+1\right )-\frac{24 i a^2}{c x-i}+24 a^2 \log (c x)-24 i a^2 \tan ^{-1}(c x)}{24 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.416, size = 1921, normalized size = 8.7 \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} \log \left (-\frac{c x + i}{c x - i}\right )^{2} - 4 i \, a b \log \left (-\frac{c x + i}{c x - i}\right ) - 4 \, a^{2}}{4 \, c^{2} d^{2} x^{3} - 8 i \, c d^{2} x^{2} - 4 \, d^{2} x}, 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}}{{\left (i \, c d x + d\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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