Optimal. Leaf size=221 \[ \frac {i b \text {Li}_2\left (\frac {2}{i c x+1}-1\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}+\frac {b \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{d^2 (-c x+i)}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d^2}+\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}+\frac {2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d^2}+\frac {b^2 \text {Li}_3\left (\frac {2}{i c x+1}-1\right )}{2 d^2}-\frac {i b^2}{2 d^2 (-c x+i)}+\frac {i b^2 \tan ^{-1}(c x)}{2 d^2} \]
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Rubi [A] time = 0.62, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, 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 44
Rule 203
Rule 627
Rule 4850
Rule 4854
Rule 4862
Rule 4864
Rule 4876
Rule 4884
Rule 4988
Rule 4994
Rule 6610
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*}
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Mathematica [A] time = 1.04, size = 299, normalized size = 1.35 \[ \frac {-12 a^2 \log \left (c^2 x^2+1\right )-\frac {24 i a^2}{c x-i}+24 a^2 \log (c x)-24 i a^2 \tan ^{-1}(c x)-12 a b \left (2 i \text {Li}_2\left (e^{2 i \tan ^{-1}(c x)}\right )+4 i \tan ^{-1}(c x)^2+\sin \left (2 \tan ^{-1}(c x)\right )+i \cos \left (2 \tan ^{-1}(c x)\right )-2 \tan ^{-1}(c x) \left (2 \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )-i \sin \left (2 \tan ^{-1}(c x)\right )+\cos \left (2 \tan ^{-1}(c x)\right )\right )\right )+b^2 \left (24 i \tan ^{-1}(c x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(c x)}\right )+12 \text {Li}_3\left (e^{-2 i \tan ^{-1}(c x)}\right )+24 \tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )-12 i \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )-12 \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )+6 i \sin \left (2 \tan ^{-1}(c x)\right )+12 \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )-12 i \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )-6 \cos \left (2 \tan ^{-1}(c x)\right )-i \pi ^3\right )}{24 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 4 i \, a b \log \left (-\frac {c x + i}{c x - i}\right ) - 4 \, a^{2}}{4 \, {\left (c^{2} d^{2} x^{3} - 2 i \, c d^{2} x^{2} - d^{2} x\right )}}, x\right ) \]
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 = 0.54, size = 1921, normalized size = 8.69 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
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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