Integrand size = 11, antiderivative size = 20 \[ \int \frac {\cosh (x)}{a+b \text {csch}(x)} \, dx=-\frac {b \log (b+a \sinh (x))}{a^2}+\frac {\sinh (x)}{a} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3957, 2912, 12, 45} \[ \int \frac {\cosh (x)}{a+b \text {csch}(x)} \, dx=\frac {\sinh (x)}{a}-\frac {b \log (a \sinh (x)+b)}{a^2} \]
[In]
[Out]
Rule 12
Rule 45
Rule 2912
Rule 3957
Rubi steps \begin{align*} \text {integral}& = i \int \frac {\cosh (x) \sinh (x)}{i b+i a \sinh (x)} \, dx \\ & = -\frac {i \text {Subst}\left (\int \frac {x}{a (i b+x)} \, dx,x,i a \sinh (x)\right )}{a} \\ & = -\frac {i \text {Subst}\left (\int \frac {x}{i b+x} \, dx,x,i a \sinh (x)\right )}{a^2} \\ & = -\frac {i \text {Subst}\left (\int \left (1-\frac {b}{b-i x}\right ) \, dx,x,i a \sinh (x)\right )}{a^2} \\ & = -\frac {b \log (b+a \sinh (x))}{a^2}+\frac {\sinh (x)}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\cosh (x)}{a+b \text {csch}(x)} \, dx=\frac {-b \log (b+a \sinh (x))+a \sinh (x)}{a^2} \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55
method | result | size |
derivativedivides | \(\frac {1}{a \,\operatorname {csch}\left (x \right )}+\frac {b \ln \left (\operatorname {csch}\left (x \right )\right )}{a^{2}}-\frac {b \ln \left (a +b \,\operatorname {csch}\left (x \right )\right )}{a^{2}}\) | \(31\) |
default | \(\frac {1}{a \,\operatorname {csch}\left (x \right )}+\frac {b \ln \left (\operatorname {csch}\left (x \right )\right )}{a^{2}}-\frac {b \ln \left (a +b \,\operatorname {csch}\left (x \right )\right )}{a^{2}}\) | \(31\) |
risch | \(\frac {b x}{a^{2}}+\frac {{\mathrm e}^{x}}{2 a}-\frac {{\mathrm e}^{-x}}{2 a}-\frac {b \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a^{2}}\) | \(45\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.00 \[ \int \frac {\cosh (x)}{a+b \text {csch}(x)} \, dx=\frac {2 \, b x \cosh \left (x\right ) + a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} - 2 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (b x + a \cosh \left (x\right )\right )} \sinh \left (x\right ) - a}{2 \, {\left (a^{2} \cosh \left (x\right ) + a^{2} \sinh \left (x\right )\right )}} \]
[In]
[Out]
\[ \int \frac {\cosh (x)}{a+b \text {csch}(x)} \, dx=\int \frac {\cosh {\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.17 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.40 \[ \int \frac {\cosh (x)}{a+b \text {csch}(x)} \, dx=-\frac {b x}{a^{2}} - \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} - \frac {b \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{2}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \frac {\cosh (x)}{a+b \text {csch}(x)} \, dx=-\frac {e^{\left (-x\right )} - e^{x}}{2 \, a} - \frac {b \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{2}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{a+b \text {csch}(x)} \, dx=\frac {\mathrm {sinh}\left (x\right )}{a}-\frac {b\,\ln \left (b+a\,\mathrm {sinh}\left (x\right )\right )}{a^2} \]
[In]
[Out]