Optimal. Leaf size=59 \[ \frac{i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c d}-\frac{b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{2 c d} \]
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Rubi [A] time = 0.047172, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4854, 2402, 2315} \[ \frac{i \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c d}-\frac{b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{2 c d} \]
Antiderivative was successfully verified.
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Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{d+i c d x} \, dx &=\frac{i \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c d}-\frac{(i b) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c d}-\frac{b \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{c d}\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c d}-\frac{b \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{2 c d}\\ \end{align*}
Mathematica [A] time = 0.0140577, size = 60, normalized size = 1.02 \[ \frac{2 i \log \left (\frac{2 d}{d+i c d x}\right ) \left (a+b \tan ^{-1}(c x)\right )-b \text{PolyLog}\left (2,\frac{c x+i}{c x-i}\right )}{2 c d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 142, normalized size = 2.4 \begin{align*}{\frac{-{\frac{i}{2}}a\ln \left ({c}^{2}{x}^{2}+1 \right ) }{dc}}+{\frac{a\arctan \left ( cx \right ) }{dc}}-{\frac{ib\ln \left ( 1+icx \right ) \arctan \left ( cx \right ) }{dc}}-{\frac{b\ln \left ( 1+icx \right ) }{2\,dc}\ln \left ({\frac{1}{2}}-{\frac{i}{2}}cx \right ) }+{\frac{b}{2\,dc}\ln \left ({\frac{1}{2}}-{\frac{i}{2}}cx \right ) \ln \left ({\frac{i}{2}}cx+{\frac{1}{2}} \right ) }+{\frac{b}{2\,dc}{\it dilog} \left ({\frac{i}{2}}cx+{\frac{1}{2}} \right ) }+{\frac{b \left ( \ln \left ( 1+icx \right ) \right ) ^{2}}{4\,dc}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{-4 \,{\left (-2 i \, c^{2} \int \frac{x \arctan \left (c x\right )}{c^{2} x^{2} + 1}\,{d x} + \arctan \left (c x\right )^{2}\right )} b}{8 \, c d} - \frac{i \, a \log \left (i \, c d x + d\right )}{c d} \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 \log \left (-\frac{c x + i}{c x - i}\right ) - 2 i \, a}{2 \, c d x - 2 i \, d}, 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{b \arctan \left (c x\right ) + a}{i \, c d x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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