Optimal. Leaf size=45 \[ x \log \left (a \cos ^2(x)\right )+i \text {Li}_2\left (-e^{2 i x}\right )+i x^2-2 x \log \left (1+e^{2 i x}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2548, 12, 3719, 2190, 2279, 2391} \[ i \text {PolyLog}\left (2,-e^{2 i x}\right )+x \log \left (a \cos ^2(x)\right )+i x^2-2 x \log \left (1+e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2190
Rule 2279
Rule 2391
Rule 2548
Rule 3719
Rubi steps
\begin {align*} \int \log \left (a \cos ^2(x)\right ) \, dx &=x \log \left (a \cos ^2(x)\right )-\int -2 x \tan (x) \, dx\\ &=x \log \left (a \cos ^2(x)\right )+2 \int x \tan (x) \, dx\\ &=i x^2+x \log \left (a \cos ^2(x)\right )-4 i \int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx\\ &=i x^2-2 x \log \left (1+e^{2 i x}\right )+x \log \left (a \cos ^2(x)\right )+2 \int \log \left (1+e^{2 i x}\right ) \, dx\\ &=i x^2-2 x \log \left (1+e^{2 i x}\right )+x \log \left (a \cos ^2(x)\right )-i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=i x^2-2 x \log \left (1+e^{2 i x}\right )+x \log \left (a \cos ^2(x)\right )+i \text {Li}_2\left (-e^{2 i x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.96 \[ x \left (\log \left (a \cos ^2(x)\right )+i x-2 \log \left (1+e^{2 i x}\right )\right )+i \text {Li}_2\left (-e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 106, normalized size = 2.36 \[ x \log \left (a \cos \relax (x)^{2}\right ) - x \log \left (i \, \cos \relax (x) + \sin \relax (x) + 1\right ) - x \log \left (i \, \cos \relax (x) - \sin \relax (x) + 1\right ) - x \log \left (-i \, \cos \relax (x) + \sin \relax (x) + 1\right ) - x \log \left (-i \, \cos \relax (x) - \sin \relax (x) + 1\right ) - i \, {\rm Li}_2\left (i \, \cos \relax (x) + \sin \relax (x)\right ) + i \, {\rm Li}_2\left (i \, \cos \relax (x) - \sin \relax (x)\right ) + i \, {\rm Li}_2\left (-i \, \cos \relax (x) + \sin \relax (x)\right ) - i \, {\rm Li}_2\left (-i \, \cos \relax (x) - \sin \relax (x)\right ) \]
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 \[ \int \log \left (a \cos \relax (x)^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 118, normalized size = 2.62 \[ -i \ln \left (\left ({\mathrm e}^{2 i x}+1\right )^{2} a \,{\mathrm e}^{-2 i x}\right ) \ln \left ({\mathrm e}^{i x}\right )+2 i \ln \left (-i {\mathrm e}^{i x}+1\right ) \ln \left ({\mathrm e}^{i x}\right )+2 i \ln \left (i {\mathrm e}^{i x}+1\right ) \ln \left ({\mathrm e}^{i x}\right )-i \ln \left ({\mathrm e}^{i x}\right )^{2}+2 i \dilog \left (-i {\mathrm e}^{i x}+1\right )+2 i \dilog \left (i {\mathrm e}^{i x}+1\right )+2 i \ln \relax (2) \ln \left ({\mathrm e}^{i x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.55, size = 62, normalized size = 1.38 \[ i \, x^{2} - 2 i \, x \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + x \log \left (a \cos \relax (x)^{2}\right ) - x \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + i \, {\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 39, normalized size = 0.87 \[ x\,\ln \left (a\,{\cos \relax (x)}^2\right )+\mathrm {polylog}\left (2,-{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}+x\,\left (x+\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )\,2{}\mathrm {i}\right )\,1{}\mathrm {i} \]
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 \[ \int \log {\left (a \cos ^{2}{\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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