Optimal. Leaf size=46 \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i x}\right )+x \log (a \sec (x))-\frac{i x^2}{2}+x \log \left (1+e^{2 i x}\right ) \]
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Rubi [A] time = 0.0518107, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {2548, 3719, 2190, 2279, 2391} \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i x}\right )+x \log (a \sec (x))-\frac{i x^2}{2}+x \log \left (1+e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Rule 2548
Rule 3719
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \log (a \sec (x)) \, dx &=x \log (a \sec (x))-\int x \tan (x) \, dx\\ &=-\frac{i x^2}{2}+x \log (a \sec (x))+2 i \int \frac{e^{2 i x} x}{1+e^{2 i x}} \, dx\\ &=-\frac{i x^2}{2}+x \log \left (1+e^{2 i x}\right )+x \log (a \sec (x))-\int \log \left (1+e^{2 i x}\right ) \, dx\\ &=-\frac{i x^2}{2}+x \log \left (1+e^{2 i x}\right )+x \log (a \sec (x))+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=-\frac{i x^2}{2}+x \log \left (1+e^{2 i x}\right )+x \log (a \sec (x))-\frac{1}{2} i \text{Li}_2\left (-e^{2 i x}\right )\\ \end{align*}
Mathematica [A] time = 0.0064459, size = 46, normalized size = 1. \[ -\frac{1}{2} i \text{PolyLog}\left (2,-e^{2 i x}\right )+x \log (a \sec (x))-\frac{i x^2}{2}+x \log \left (1+e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 118, normalized size = 2.6 \begin{align*} -i\ln \left ( 2 \right ) \ln \left ({{\rm e}^{ix}} \right ) -i\ln \left ({{\rm e}^{ix}} \right ) \ln \left ({\frac{a{{\rm e}^{ix}}}{1+{{\rm e}^{2\,ix}}}} \right ) +{\frac{i}{2}} \left ( \ln \left ({{\rm e}^{ix}} \right ) \right ) ^{2}-i\ln \left ({{\rm e}^{ix}} \right ) \ln \left ( 1+i{{\rm e}^{ix}} \right ) -i\ln \left ({{\rm e}^{ix}} \right ) \ln \left ( 1-i{{\rm e}^{ix}} \right ) -i{\it dilog} \left ( 1+i{{\rm e}^{ix}} \right ) -i{\it dilog} \left ( 1-i{{\rm e}^{ix}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.0649, size = 81, normalized size = 1.76 \begin{align*} -\frac{1}{2} i \, x^{2} + i \, x \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + x \log \left (a \sec \left (x\right )\right ) - \frac{1}{2} i \,{\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20141, size = 396, normalized size = 8.61 \begin{align*} x \log \left (\frac{a}{\cos \left (x\right )}\right ) + \frac{1}{2} \, x \log \left (i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (-i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, x \log \left (-i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + \frac{1}{2} i \,{\rm Li}_2\left (i \, \cos \left (x\right ) + \sin \left (x\right )\right ) - \frac{1}{2} i \,{\rm Li}_2\left (i \, \cos \left (x\right ) - \sin \left (x\right )\right ) - \frac{1}{2} i \,{\rm Li}_2\left (-i \, \cos \left (x\right ) + \sin \left (x\right )\right ) + \frac{1}{2} i \,{\rm Li}_2\left (-i \, \cos \left (x\right ) - \sin \left (x\right )\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 \sec{\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 \sec \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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