Optimal. Leaf size=110 \[ -\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d}-\frac {i a x}{c d}-\frac {i b \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{2 c^2 d}+\frac {i b \log \left (c^2 x^2+1\right )}{2 c^2 d}-\frac {i b x \tan ^{-1}(c x)}{c d} \]
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Rubi [A] time = 0.10, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4866, 4846, 260, 4854, 2402, 2315} \[ -\frac {i b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2 d}-\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d}-\frac {i a x}{c d}+\frac {i b \log \left (c^2 x^2+1\right )}{2 c^2 d}-\frac {i b x \tan ^{-1}(c x)}{c d} \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2402
Rule 4846
Rule 4854
Rule 4866
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{d+i c d x} \, dx &=\frac {i \int \frac {a+b \tan ^{-1}(c x)}{d+i c d x} \, dx}{c}-\frac {i \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c d}\\ &=-\frac {i a x}{c d}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2 d}-\frac {(i b) \int \tan ^{-1}(c x) \, dx}{c d}+\frac {b \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c d}\\ &=-\frac {i a x}{c d}-\frac {i b x \tan ^{-1}(c x)}{c d}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2 d}+\frac {(i b) \int \frac {x}{1+c^2 x^2} \, dx}{d}-\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}\\ &=-\frac {i a x}{c d}-\frac {i b x \tan ^{-1}(c x)}{c d}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2 d}+\frac {i b \log \left (1+c^2 x^2\right )}{2 c^2 d}-\frac {i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^2 d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 108, normalized size = 0.98 \[ \frac {2 i \tan ^{-1}(c x) \left (a-b c x+i b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+a \log \left (c^2 x^2+1\right )-2 i a c x+i b \log \left (c^2 x^2+1\right )+i b \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+2 i b \tan ^{-1}(c x)^2}{2 c^2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \log \left (-\frac {c x + i}{c x - i}\right ) - 2 i \, a x}{2 \, c d x - 2 i \, d}, x\right ) \]
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 = 0.07, size = 244, normalized size = 2.22 \[ -\frac {i a x}{d c}+\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 c^{2} d}+\frac {i a \arctan \left (c x \right )}{c^{2} d}-\frac {i b x \arctan \left (c x \right )}{d c}+\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{2} d}-\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 c^{2} d}-\frac {i b \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{2} d}+\frac {i b \ln \left (c x -i\right )^{2}}{4 c^{2} d}+\frac {i b \ln \left (c^{8} x^{8}+12 c^{6} x^{6}+30 c^{4} x^{4}+28 c^{2} x^{2}+9\right )}{8 c^{2} d}-\frac {b \arctan \left (\frac {1}{12} c^{3} x^{3}+\frac {13}{12} c x \right )}{4 c^{2} d}-\frac {b \arctan \left (\frac {c x}{4}\right )}{4 c^{2} d}+\frac {b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{2 c^{2} d} \]
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 \[ a {\left (-\frac {i \, x}{c d} + \frac {\log \left (i \, c x + 1\right )}{c^{2} d}\right )} - \frac {{\left (2 i \, {\left (2 \, {\left (\frac {x}{c^{3} d} - \frac {\arctan \left (c x\right )}{c^{4} d}\right )} \arctan \left (c x\right ) + \frac {\arctan \left (c x\right )^{2} - \log \left (c^{2} x^{2} + 1\right )}{c^{4} d}\right )} c^{4} d + 2 \, c^{4} d \int \frac {x^{2} \log \left (c^{2} x^{2} + 1\right )}{c^{3} d x^{2} + c d}\,{d x} - 8 \, c^{3} d \int \frac {x \arctan \left (c x\right )}{c^{3} d x^{2} + c d}\,{d x} + 2 \, c^{2} d \int \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{3} d x^{2} + c d}\,{d x} - 2 \, c x \log \left (c^{2} x^{2} + 1\right ) + 4 \, c x - 4 \, {\left (-i \, c x + 1\right )} \arctan \left (c x\right ) - 2 i \, \arctan \left (c x\right )^{2} - 2 i \, \log \left (2 \, c^{3} d x^{2} + 2 \, c d\right )\right )} b}{8 \, c^{2} d} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]
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 \[ - \frac {i \left (\int \left (- \frac {i b \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\right )\, dx + \int \frac {2 a c^{2} x^{2}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {b c x}{c^{2} x^{2} + 1}\right )\, dx + \int \frac {2 i a c x}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {i b c^{2} x^{2}}{c^{2} x^{2} + 1}\right )\, dx + \int \frac {2 b c x \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {i b c^{2} x^{2} \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\right )\, dx\right )}{2 c d} + \frac {\left (b c x + i b \log {\left (i c x + 1 \right )}\right ) \log {\left (- i c x + 1 \right )}}{2 c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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