Integrand size = 7, antiderivative size = 45 \[ \int \log \left (a \sec ^2(x)\right ) \, dx=-i x^2+2 x \log \left (1+e^{2 i x}\right )+x \log \left (a \sec ^2(x)\right )-i \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2628, 12, 3800, 2221, 2317, 2438} \[ \int \log \left (a \sec ^2(x)\right ) \, dx=x \log \left (a \sec ^2(x)\right )-i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )-i x^2+2 x \log \left (1+e^{2 i x}\right ) \]
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3800
Rubi steps \begin{align*} \text {integral}& = x \log \left (a \sec ^2(x)\right )-\int 2 x \tan (x) \, dx \\ & = x \log \left (a \sec ^2(x)\right )-2 \int x \tan (x) \, dx \\ & = -i x^2+x \log \left (a \sec ^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 \sec ^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 \sec ^2(x)\right )+i \text {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 \sec ^2(x)\right )-i \text {Li}_2\left (-e^{2 i x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \log \left (a \sec ^2(x)\right ) \, dx=x \left (-i x+2 \log \left (1+e^{2 i x}\right )+\log \left (a \sec ^2(x)\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (39 ) = 78\).
Time = 1.31 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.60
method | result | size |
default | \(-i \left (\ln \left ({\mathrm e}^{i x}\right ) \ln \left (\frac {a \,{\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right )-\ln \left ({\mathrm e}^{i x}\right )^{2}+2 \ln \left ({\mathrm e}^{i x}\right ) \ln \left (1+i {\mathrm e}^{i x}\right )+2 \ln \left ({\mathrm e}^{i x}\right ) \ln \left (1-i {\mathrm e}^{i x}\right )+2 \operatorname {dilog}\left (1+i {\mathrm e}^{i x}\right )+2 \operatorname {dilog}\left (1-i {\mathrm e}^{i x}\right )+2 \ln \left (2\right ) \ln \left ({\mathrm e}^{i x}\right )\right )\) | \(117\) |
risch | \(2 x \ln \left ({\mathrm e}^{i x}\right )-i x^{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right ) \operatorname {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right ) x}{2}-i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )\right ) {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )^{2}\right )}^{2} 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 \left (1+{\mathrm e}^{2 i x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )^{2}\right ) x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right )^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right )^{2} x}{2}+2 x \ln \left (2\right )+\ln \left (a \right ) x +2 i \ln \left ({\mathrm e}^{i x}\right ) \ln \left (1+{\mathrm e}^{2 i x}\right )-\frac {i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right )^{3} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}-2 i \operatorname {dilog}\left (1+i {\mathrm e}^{i x}\right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right )^{2} x}{2}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{i x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right )^{2} x +\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 i x}\right )^{2}\right )}^{3} x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{i x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 i x}\right ) x}{2}-2 i \ln \left ({\mathrm e}^{i x}\right ) \ln \left (1+i {\mathrm e}^{i x}\right )-2 i \operatorname {dilog}\left (1-i {\mathrm e}^{i x}\right )-\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right )^{3} x}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 i x}}{\left (1+{\mathrm e}^{2 i x}\right )^{2}}\right )^{2} \operatorname {csgn}\left (i a \right ) x}{2}\) | \(549\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (34) = 68\).
Time = 0.36 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.27 \[ \int \log \left (a \sec ^2(x)\right ) \, dx=x \log \left (\frac {a}{\cos \left (x\right )^{2}}\right ) + x \log \left (i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + x \log \left (i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + x \log \left (-i \, \cos \left (x\right ) + \sin \left (x\right ) + 1\right ) + x \log \left (-i \, \cos \left (x\right ) - \sin \left (x\right ) + 1\right ) + i \, {\rm Li}_2\left (i \, \cos \left (x\right ) + \sin \left (x\right )\right ) - i \, {\rm Li}_2\left (i \, \cos \left (x\right ) - \sin \left (x\right )\right ) - i \, {\rm Li}_2\left (-i \, \cos \left (x\right ) + \sin \left (x\right )\right ) + i \, {\rm Li}_2\left (-i \, \cos \left (x\right ) - \sin \left (x\right )\right ) \]
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\[ \int \log \left (a \sec ^2(x)\right ) \, dx=\int \log {\left (a \sec ^{2}{\left (x \right )} \right )}\, dx \]
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none
Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.36 \[ \int \log \left (a \sec ^2(x)\right ) \, dx=-i \, x^{2} + 2 i \, x \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + x \log \left (a \sec \left (x\right )^{2}\right ) + x \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) - i \, {\rm Li}_2\left (-e^{\left (2 i \, x\right )}\right ) \]
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\[ \int \log \left (a \sec ^2(x)\right ) \, dx=\int { \log \left (a \sec \left (x\right )^{2}\right ) \,d x } \]
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Time = 1.49 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \log \left (a \sec ^2(x)\right ) \, dx=x\,\ln \left (\frac {a}{{\cos \left (x\right )}^2}\right )-\mathrm {polylog}\left (2,-{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}-x\,\left (x+\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}+1\right )\,2{}\mathrm {i}\right )\,1{}\mathrm {i} \]
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