Optimal. Leaf size=122 \[ -\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (-c x+i)}+\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2}+\frac {i b \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{2 c^2 d^2}-\frac {b}{2 c^2 d^2 (-c x+i)}+\frac {b \tan ^{-1}(c x)}{2 c^2 d^2} \]
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Rubi [A] time = 0.15, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {4876, 4862, 627, 44, 203, 4854, 2402, 2315} \[ \frac {i b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (-c x+i)}+\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2}-\frac {b}{2 c^2 d^2 (-c x+i)}+\frac {b \tan ^{-1}(c x)}{2 c^2 d^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 627
Rule 2315
Rule 2402
Rule 4854
Rule 4862
Rule 4876
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{(d+i c d x)^2} \, dx &=\int \left (-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (-i+c x)^2}-\frac {a+b \tan ^{-1}(c x)}{c d^2 (-i+c x)}\right ) \, dx\\ &=-\frac {i \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{c d^2}-\frac {\int \frac {a+b \tan ^{-1}(c x)}{-i+c x} \, dx}{c d^2}\\ &=-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}-\frac {(i b) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c d^2}-\frac {b \int \frac {\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 )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^2 d^2}-\frac {(i b) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c d^2}\\ &=-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}-\frac {(i b) \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 {b}{2 c^2 d^2 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}+\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{2 c d^2}\\ &=-\frac {b}{2 c^2 d^2 (i-c x)}+\frac {b \tan ^{-1}(c x)}{2 c^2 d^2}-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (i-c x)}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2 d^2}+\frac {i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 128, normalized size = 1.05 \[ -\frac {i \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2 (-c x+i)}+\frac {\log \left (\frac {2 i}{-c x+i}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2}+\frac {i b \text {Li}_2\left (-\frac {c x+i}{i-c x}\right )}{2 c^2 d^2}-\frac {b \left (-\frac {\tan ^{-1}(c x)}{c}+\frac {1}{c (-c x+i)}\right )}{2 c d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {-i \, b x \log \left (-\frac {c x + i}{c x - i}\right ) - 2 \, a x}{2 \, {\left (c^{2} d^{2} x^{2} - 2 i \, c d^{2} x - d^{2}\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 293, normalized size = 2.40 \[ \frac {i a}{c^{2} d^{2} \left (c x -i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 c^{2} d^{2}}-\frac {i a \arctan \left (c x \right )}{c^{2} d^{2}}+\frac {i b \arctan \left (c x \right )}{c^{2} d^{2} \left (c x -i\right )}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{2} d^{2}}+\frac {i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{16 c^{2} d^{2}}-\frac {b \arctan \left (\frac {c x}{2}\right )}{8 c^{2} d^{2}}+\frac {b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{8 c^{2} d^{2}}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{4 c^{2} d^{2}}+\frac {b}{2 c^{2} d^{2} \left (c x -i\right )}-\frac {i b \ln \left (c^{2} x^{2}+1\right )}{8 c^{2} d^{2}}+\frac {b \arctan \left (c x \right )}{4 c^{2} d^{2}}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 c^{2} d^{2}}+\frac {i b \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{2} d^{2}}-\frac {i b \ln \left (c x -i\right )^{2}}{4 c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\left (b c x \log {\left (i c x + 1 \right )} - i b \log {\left (i c x + 1 \right )} - i b\right ) \log {\left (- i c x + 1 \right )}}{2 i c^{3} d^{2} x + 2 c^{2} d^{2}} - \frac {\int \frac {i b}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \frac {i b \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \frac {2 a c^{2} x^{2}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {b c x}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \frac {2 i a c x}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\, dx + \int \left (- \frac {b c x \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx + \int \left (- \frac {2 i b c^{2} x^{2} \log {\left (i c x + 1 \right )}}{c^{3} x^{3} - i c^{2} x^{2} + c x - i}\right )\, dx}{2 c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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