Integrand size = 5, antiderivative size = 46 \[ \int \log (a \sec (x)) \, dx=-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right )+x \log (a \sec (x))-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2628, 3800, 2221, 2317, 2438} \[ \int \log (a \sec (x)) \, dx=x \log (a \sec (x))-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right ) \]
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Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3800
Rubi steps \begin{align*} \text {integral}& = 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 \text {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*}
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \log (a \sec (x)) \, dx=-\frac {i x^2}{2}+x \log \left (1+e^{2 i x}\right )+x \log (a \sec (x))-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (36 ) = 72\).
Time = 1.32 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.35
method | result | size |
default | \(-i \left (\ln \left (2\right ) \ln \left ({\mathrm e}^{i x}\right )+\ln \left ({\mathrm e}^{i x}\right ) \ln \left (\frac {a \,{\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right )+\ln \left ({\mathrm e}^{i x}\right ) \ln \left (1+i {\mathrm e}^{i x}\right )+\ln \left ({\mathrm e}^{i x}\right ) \ln \left (1-i {\mathrm e}^{i x}\right )+\operatorname {dilog}\left (1+i {\mathrm e}^{i x}\right )+\operatorname {dilog}\left (1-i {\mathrm e}^{i x}\right )-\frac {\ln \left ({\mathrm e}^{i x}\right )^{2}}{2}\right )\) | \(108\) |
risch | \(x \ln \left ({\mathrm e}^{i x}\right )-\frac {i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right )^{3} x}{2}-\frac {i x^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{i x}\right ) \operatorname {csgn}\left (\frac {i}{1+{\mathrm e}^{2 i x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right ) x}{2}-i \ln \left ({\mathrm e}^{i x}\right ) \ln \left (1-i {\mathrm e}^{i x}\right )+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{i x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}+x \ln \left (2\right )+\ln \left (a \right ) x -i \operatorname {dilog}\left (1+i {\mathrm e}^{i x}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{1+{\mathrm e}^{2 i x}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right )^{2} x}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right )^{2} \operatorname {csgn}\left (i a \right ) x}{2}+i \ln \left ({\mathrm e}^{i x}\right ) \ln \left (1+{\mathrm e}^{2 i x}\right )-i \ln \left ({\mathrm e}^{i x}\right ) \ln \left (1+i {\mathrm e}^{i x}\right )-i \operatorname {dilog}\left (1-i {\mathrm e}^{i x}\right )-\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{1+{\mathrm e}^{2 i x}}\right )^{2} x}{2}\) | \(399\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (31) = 62\).
Time = 0.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.30 \[ \int \log (a \sec (x)) \, dx=x \log \left (\frac {a}{\cos \left (x\right )}\right ) + \frac {1}{2} \, x \log \left (i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (i \, \cos \left (x\right ) + \sin \left (x\right )\right ) - \frac {1}{2} i \, {\rm Li}_2\left (i \, \cos \left (x\right ) - \sin \left (x\right )\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-i \, \cos \left (x\right ) + \sin \left (x\right )\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-i \, \cos \left (x\right ) - \sin \left (x\right )\right ) \]
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\[ \int \log (a \sec (x)) \, dx=\int \log {\left (a \sec {\left (x \right )} \right )}\, dx \]
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none
Time = 0.40 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \log (a \sec (x)) \, dx=-\frac {1}{2} i \, x^{2} + i \, x \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) + x \log \left (a \sec \left (x\right )\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right ) \]
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\[ \int \log (a \sec (x)) \, dx=\int { \log \left (a \sec \left (x\right )\right ) \,d x } \]
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Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \log (a \sec (x)) \, dx=x\,\ln \left (\frac {a}{\cos \left (x\right )}\right )-\frac {\mathrm {polylog}\left (2,-{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}-\frac {x\,\left (x+\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \]
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