Optimal. Leaf size=306 \[ \frac{2 b c \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{i b^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-c x+i)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{2 b c \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{2 i c \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{4 i c \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{b^2 c}{2 d^2 (-c x+i)}+\frac{b^2 c \tan ^{-1}(c x)}{2 d^2} \]
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Rubi [A] time = 0.767048, antiderivative size = 306, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {4876, 4852, 4924, 4868, 2447, 4850, 4988, 4884, 4994, 6610, 4864, 4862, 627, 44, 203, 4854} \[ \frac{2 b c \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{i b^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-c x+i)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac{2 b c \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{2 i c \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{4 i c \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{b^2 c}{2 d^2 (-c x+i)}+\frac{b^2 c \tan ^{-1}(c x)}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4852
Rule 4924
Rule 4868
Rule 2447
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^2 (d+i c d x)^2} \, dx &=\int \left (\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-i+c x)^2}+\frac{2 i c^2 \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^2} \, dx}{d^2}-\frac{(2 i c) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx}{d^2}+\frac{\left (2 i c^2\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{-i+c x} \, dx}{d^2}+\frac{c^2 \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{d^2}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{(2 b c) \int \frac{a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{d^2}+\frac{\left (4 i b c^2\right ) \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{\left (8 i b c^2\right ) \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{\left (2 b c^2\right ) \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{i c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{(2 i b c) \int \frac{a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx}{d^2}-\frac{\left (i b c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{d^2}+\frac{\left (i b c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{d^2}-\frac{\left (4 i b c^2\right ) \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{\left (4 i b c^2\right ) \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 (2 b^2 c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{\left (i b^2 c^2\right ) \int \frac{1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac{\left (2 b^2 c^2\right ) \int \frac{\log \left (2-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac{\left (2 b^2 c^2\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 (2 b^2 c^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{\left (i b^2 c^2\right ) \int \frac{1}{(-i+c x)^2 (i+c x)} \, dx}{d^2}\\ &=-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{\left (i b^2 c^2\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{b^2 c}{2 d^2 (i-c x)}-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (b^2 c^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 d^2}\\ &=-\frac{b^2 c}{2 d^2 (i-c x)}+\frac{b^2 c \tan ^{-1}(c x)}{2 d^2}-\frac{i b c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{i c \left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (i-c x)}-\frac{4 i c \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )}{d^2}+\frac{2 b c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{d^2}-\frac{i b^2 c \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 2.57128, size = 398, normalized size = 1.3 \[ -\frac{6 a b c \left (4 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )-4 \log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )+8 \tan ^{-1}(c x)^2-i \sin \left (2 \tan ^{-1}(c x)\right )+\cos \left (2 \tan ^{-1}(c x)\right )+\tan ^{-1}(c x) \left (\frac{4}{c x}+8 i \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )+2 \sin \left (2 \tan ^{-1}(c x)\right )+2 i \cos \left (2 \tan ^{-1}(c x)\right )\right )\right )+b^2 c \left (-24 \tan ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(c x)}\right )+12 i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )+12 i \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(c x)}\right )+\frac{12 \tan ^{-1}(c x)^2}{c x}+12 i \tan ^{-1}(c x)^2+24 i \tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )-24 \tan ^{-1}(c x) \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )+6 \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )-6 i \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )-3 \sin \left (2 \tan ^{-1}(c x)\right )+6 i \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )+6 \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )-3 i \cos \left (2 \tan ^{-1}(c x)\right )+\pi ^3\right )-12 i a^2 c \log \left (c^2 x^2+1\right )+\frac{12 a^2 c}{c x-i}+24 i a^2 c \log (c x)+24 a^2 c \tan ^{-1}(c x)+\frac{12 a^2}{x}}{12 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.996, size = 9420, normalized size = 30.8 \begin{align*} \text{output 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^{4} - 8 i \, c d^{2} x^{3} - 4 \, d^{2} x^{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}}{{\left (i \, c d x + d\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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