Optimal. Leaf size=138 \[ -\frac{i b \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e}+\frac{i b \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}-\frac{\log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e} \]
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Rubi [A] time = 0.0745539, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4856, 2402, 2315, 2447} \[ -\frac{i b \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e}+\frac{i b \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}-\frac{\log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e} \]
Antiderivative was successfully verified.
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Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{d+e x} \, dx &=-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac{(b c) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{e}-\frac{(b c) \int \frac{\log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{e}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}-\frac{i b \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{e}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e}+\frac{i b \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 e}-\frac{i b \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}\\ \end{align*}
Mathematica [A] time = 0.0559885, size = 138, normalized size = 1. \[ \frac{i b \text{PolyLog}\left (2,\frac{e (1-i c x)}{e+i c d}\right )-i b \text{PolyLog}\left (2,-\frac{e (c x-i)}{c d+i e}\right )+2 a \log (d+e x)+i b \log (1-i c x) \log \left (\frac{c (d+e x)}{c d-i e}\right )-i b \log (1+i c x) \log \left (\frac{c (d+e x)}{c d+i e}\right )}{2 e} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.038, size = 168, normalized size = 1.2 \begin{align*}{\frac{a\ln \left ( ecx+dc \right ) }{e}}+{\frac{b\ln \left ( ecx+dc \right ) \arctan \left ( cx \right ) }{e}}+{\frac{{\frac{i}{2}}b\ln \left ( ecx+dc \right ) }{e}\ln \left ({\frac{ie-ecx}{dc+ie}} \right ) }-{\frac{{\frac{i}{2}}b\ln \left ( ecx+dc \right ) }{e}\ln \left ({\frac{ie+ecx}{ie-dc}} \right ) }+{\frac{{\frac{i}{2}}b}{e}{\it dilog} \left ({\frac{ie-ecx}{dc+ie}} \right ) }-{\frac{{\frac{i}{2}}b}{e}{\it dilog} \left ({\frac{ie+ecx}{ie-dc}} \right ) } \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*} 2 \, b \int \frac{\arctan \left (c x\right )}{2 \,{\left (e x + d\right )}}\,{d x} + \frac{a \log \left (e x + d\right )}{e} \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 \arctan \left (c x\right ) + a}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{atan}{\left (c x \right )}}{d + e x}\, dx \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}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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