Optimal. Leaf size=51 \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i x}\right )+\frac{1}{2} i \text{PolyLog}\left (2,e^{2 i x}\right )+x \log (a \tan (x))+2 x \tanh ^{-1}\left (e^{2 i x}\right ) \]
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Rubi [A] time = 0.044377, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {2548, 4419, 4183, 2279, 2391} \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i x}\right )+\frac{1}{2} i \text{PolyLog}\left (2,e^{2 i x}\right )+x \log (a \tan (x))+2 x \tanh ^{-1}\left (e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 4419
Rule 4183
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log (a \tan (x)) \, dx &=x \log (a \tan (x))-\int x \csc (x) \sec (x) \, dx\\ &=x \log (a \tan (x))-2 \int x \csc (2 x) \, dx\\ &=2 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log (a \tan (x))+\int \log \left (1-e^{2 i x}\right ) \, dx-\int \log \left (1+e^{2 i x}\right ) \, dx\\ &=2 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log (a \tan (x))-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=2 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log (a \tan (x))-\frac{1}{2} i \text{Li}_2\left (-e^{2 i x}\right )+\frac{1}{2} i \text{Li}_2\left (e^{2 i x}\right )\\ \end{align*}
Mathematica [A] time = 0.0108714, size = 75, normalized size = 1.47 \[ -\frac{1}{2} i \text{PolyLog}(2,-i \tan (x))+\frac{1}{2} i \text{PolyLog}(2,i \tan (x))-\frac{1}{2} i \log (-i (-\tan (x)+i)) \log (a \tan (x))+\frac{1}{2} i \log (-i (\tan (x)+i)) \log (a \tan (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 82, normalized size = 1.6 \begin{align*} -{\frac{i}{2}}\ln \left ( a\tan \left ( x \right ) \right ) \ln \left ({\frac{ia\tan \left ( x \right ) +a}{a}} \right ) +{\frac{i}{2}}\ln \left ( a\tan \left ( x \right ) \right ) \ln \left ( -{\frac{ia\tan \left ( x \right ) -a}{a}} \right ) -{\frac{i}{2}}{\it dilog} \left ({\frac{ia\tan \left ( x \right ) +a}{a}} \right ) +{\frac{i}{2}}{\it dilog} \left ( -{\frac{ia\tan \left ( x \right ) -a}{a}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62799, size = 57, normalized size = 1.12 \begin{align*} x \log \left (a \tan \left (x\right )\right ) + \frac{1}{4} \, \pi \log \left (\tan \left (x\right )^{2} + 1\right ) - x \log \left (\tan \left (x\right )\right ) + \frac{1}{2} i \,{\rm Li}_2\left (i \, \tan \left (x\right ) + 1\right ) - \frac{1}{2} i \,{\rm Li}_2\left (-i \, \tan \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01159, size = 587, normalized size = 11.51 \begin{align*} x \log \left (a \tan \left (x\right )\right ) - \frac{1}{2} \, x \log \left (\frac{2 \,{\left (\tan \left (x\right )^{2} + i \, \tan \left (x\right )\right )}}{\tan \left (x\right )^{2} + 1}\right ) - \frac{1}{2} \, x \log \left (\frac{2 \,{\left (\tan \left (x\right )^{2} - i \, \tan \left (x\right )\right )}}{\tan \left (x\right )^{2} + 1}\right ) + \frac{1}{2} \, x \log \left (-\frac{2 \,{\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) + \frac{1}{2} \, x \log \left (-\frac{2 \,{\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) - \frac{1}{4} i \,{\rm Li}_2\left (-\frac{2 \,{\left (\tan \left (x\right )^{2} + i \, \tan \left (x\right )\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) + \frac{1}{4} i \,{\rm Li}_2\left (-\frac{2 \,{\left (\tan \left (x\right )^{2} - i \, \tan \left (x\right )\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) + \frac{1}{4} i \,{\rm Li}_2\left (\frac{2 \,{\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) - \frac{1}{4} i \,{\rm Li}_2\left (\frac{2 \,{\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\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 \log{\left (a \tan{\left (x \right )} \right )}\, 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 \log \left (a \tan \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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