Integrand size = 6, antiderivative size = 9 \[ \int \log (\cosh (x)) \tanh (x) \, dx=\frac {1}{2} \log ^2(\cosh (x)) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3556, 4426, 2338} \[ \int \log (\cosh (x)) \tanh (x) \, dx=\frac {1}{2} \log ^2(\cosh (x)) \]
[In]
[Out]
Rule 2338
Rule 3556
Rule 4426
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\cosh (x)\right ) \\ & = \frac {1}{2} \log ^2(\cosh (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \log (\cosh (x)) \tanh (x) \, dx=\frac {1}{2} \log ^2(\cosh (x)) \]
[In]
[Out]
Time = 1.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\ln \left (\cosh \left (x \right )\right )^{2}}{2}\) | \(8\) |
default | \(\frac {\ln \left (\cosh \left (x \right )\right )^{2}}{2}\) | \(8\) |
risch | \(\left (x -\ln \left (1+{\mathrm e}^{2 x}\right )\right ) \ln \left ({\mathrm e}^{x}\right )+\frac {\ln \left (1+{\mathrm e}^{2 x}\right )^{2}}{2}+\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right ) x}{2}+\ln \left (2\right ) x -\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2} x}{2}-\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{3}}{2}-\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i \left (1+{\mathrm e}^{2 x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )}{2}-\frac {x^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{3} x}{2}+\frac {i \ln \left (1+{\mathrm e}^{2 x}\right ) \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right )\right )^{2}}{2}-\ln \left (1+{\mathrm e}^{2 x}\right ) \ln \left (2\right )\) | \(308\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \log (\cosh (x)) \tanh (x) \, dx=\frac {1}{2} \, \log \left (\cosh \left (x\right )\right )^{2} \]
[In]
[Out]
Time = 1.63 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \log (\cosh (x)) \tanh (x) \, dx=\frac {\log {\left (\cosh {\left (x \right )} \right )}^{2}}{2} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \log (\cosh (x)) \tanh (x) \, dx=\frac {1}{2} \, \log \left (\cosh \left (x\right )\right )^{2} \]
[In]
[Out]
\[ \int \log (\cosh (x)) \tanh (x) \, dx=\int { \log \left (\cosh \left (x\right )\right ) \tanh \left (x\right ) \,d x } \]
[In]
[Out]
Time = 0.51 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.78 \[ \int \log (\cosh (x)) \tanh (x) \, dx=\frac {{\ln \left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^2}{2} \]
[In]
[Out]