Integrand size = 7, antiderivative size = 35 \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=-x^2+2 x \log \left (1+e^{2 x}\right )+x \log \left (a \text {sech}^2(x)\right )+\operatorname {PolyLog}\left (2,-e^{2 x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 35, 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, 3799, 2221, 2317, 2438} \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=x \log \left (a \text {sech}^2(x)\right )+\operatorname {PolyLog}\left (2,-e^{2 x}\right )-x^2+2 x \log \left (e^{2 x}+1\right ) \]
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3799
Rubi steps \begin{align*} \text {integral}& = x \log \left (a \text {sech}^2(x)\right )-\int -2 x \tanh (x) \, dx \\ & = x \log \left (a \text {sech}^2(x)\right )+2 \int x \tanh (x) \, dx \\ & = -x^2+x \log \left (a \text {sech}^2(x)\right )+4 \int \frac {e^{2 x} x}{1+e^{2 x}} \, dx \\ & = -x^2+2 x \log \left (1+e^{2 x}\right )+x \log \left (a \text {sech}^2(x)\right )-2 \int \log \left (1+e^{2 x}\right ) \, dx \\ & = -x^2+2 x \log \left (1+e^{2 x}\right )+x \log \left (a \text {sech}^2(x)\right )-\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 x}\right ) \\ & = -x^2+2 x \log \left (1+e^{2 x}\right )+x \log \left (a \text {sech}^2(x)\right )+\text {Li}_2\left (-e^{2 x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=x \left (x+2 \log \left (1+e^{-2 x}\right )+\log \left (a \text {sech}^2(x)\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 x}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.15 (sec) , antiderivative size = 480, normalized size of antiderivative = 13.71
method | result | size |
risch | \(\ln \left (a \right ) x -\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{3} x}{2}+2 x \ln \left (2\right )-x^{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{3} x}{2}+2 \operatorname {dilog}\left (1+i {\mathrm e}^{x}\right )+2 \operatorname {dilog}\left (1-i {\mathrm e}^{x}\right )-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (i a \right ) x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{2} x}{2}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} x -i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )}^{2} x +\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right )}^{3} x}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3} x}{2}+\frac {i \pi {\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right )}^{2} \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )^{2}\right ) x}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right )^{2} \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}\right ) x}{2}+2 x \ln \left ({\mathrm e}^{x}\right )-2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )+2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1+i {\mathrm e}^{x}\right )+2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (1-i {\mathrm e}^{x}\right )\) | \(480\) |
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Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.03 \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=-x^{2} + x \log \left (\frac {4 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + 3 \, \cosh \left (x\right )}\right ) + 2 \, x \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) + 2 \, x \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + 2 \, {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) + 2 \, {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) \]
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\[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=\int \log {\left (a \operatorname {sech}^{2}{\left (x \right )} \right )}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=-x^{2} + x \log \left (a \operatorname {sech}\left (x\right )^{2}\right ) + 2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) \]
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\[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=\int { \log \left (a \operatorname {sech}\left (x\right )^{2}\right ) \,d x } \]
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Timed out. \[ \int \log \left (a \text {sech}^2(x)\right ) \, dx=-\int 2\,\ln \left (\mathrm {cosh}\left (x\right )\right )-\ln \left (a\right ) \,d x \]
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