Integrand size = 11, antiderivative size = 20 \[ \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx=\frac {\log (\cosh (x))}{a}-\frac {\log (a+b \cosh (x))}{a} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2800, 36, 29, 31} \[ \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx=\frac {\log (\cosh (x))}{a}-\frac {\log (a+b \cosh (x))}{a} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 2800
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x (a+x)} \, dx,x,b \cosh (x)\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,b \cosh (x)\right )}{a}-\frac {\text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (x)\right )}{a} \\ & = \frac {\log (\cosh (x))}{a}-\frac {\log (a+b \cosh (x))}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx=\frac {\log (\cosh (x))}{a}-\frac {\log (a+b \cosh (x))}{a} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65
method | result | size |
risch | \(\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{a}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right )}{a}\) | \(33\) |
default | \(\frac {\ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b -a -b \right )}{a}\) | \(45\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.00 \[ \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx=-\frac {\log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a} \]
[In]
[Out]
\[ \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx=\int \frac {\tanh {\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx=-\frac {\log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a} + \frac {\log \left (e^{\left (-2 \, x\right )} + 1\right )}{a} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx=\frac {\log \left (e^{\left (-x\right )} + e^{x}\right )}{a} - \frac {\log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 201, normalized size of antiderivative = 10.05 \[ \int \frac {\tanh (x)}{a+b \cosh (x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {-a^2}+b\,{\mathrm {e}}^x\,\sqrt {-a^2}+2\,a\,{\mathrm {e}}^{2\,x}\,\sqrt {-a^2}+b\,{\mathrm {e}}^{3\,x}\,\sqrt {-a^2}}{a^2}\right )}{\sqrt {-a^2}}-\frac {2\,\mathrm {atan}\left (\left (4\,a^4\,b\,\sqrt {-a^2}-4\,a^2\,b^3\,\sqrt {-a^2}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {1}{16\,b^2\,{\left (a^2-b^2\right )}^2}-\frac {{\left (a^2-2\,b^2\right )}^2}{16\,a^4\,b^2\,{\left (a^2-b^2\right )}^2}\right )+\frac {1}{8\,a\,b\,{\left (a^2-b^2\right )}^2}+\frac {a^2-2\,b^2}{8\,a^3\,b\,{\left (a^2-b^2\right )}^2}\right )\right )}{\sqrt {-a^2}} \]
[In]
[Out]