Integrand size = 11, antiderivative size = 13 \[ \int \frac {\cosh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {1}{b (a+b \sinh (x))} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2747, 32} \[ \int \frac {\cosh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {1}{b (a+b \sinh (x))} \]
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Rule 32
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{(a+x)^2} \, dx,x,b \sinh (x)\right )}{b} \\ & = -\frac {1}{b (a+b \sinh (x))} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {1}{b (a+b \sinh (x))} \]
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Time = 1.36 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(-\frac {1}{b \left (a +b \sinh \left (x \right )\right )}\) | \(14\) |
default | \(-\frac {1}{b \left (a +b \sinh \left (x \right )\right )}\) | \(14\) |
risch | \(-\frac {2 \,{\mathrm e}^{x}}{b \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}\) | \(25\) |
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (13) = 26\).
Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 3.92 \[ \int \frac {\cosh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )} \]
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Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \frac {\cosh (x)}{(a+b \sinh (x))^2} \, dx=\begin {cases} - \frac {1}{a b + b^{2} \sinh {\left (x \right )}} & \text {for}\: b \neq 0 \\\frac {\sinh {\left (x \right )}}{a^{2}} & \text {otherwise} \end {cases} \]
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none
Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {1}{{\left (b \sinh \left (x\right ) + a\right )} b} \]
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none
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int \frac {\cosh (x)}{(a+b \sinh (x))^2} \, dx=\frac {2}{{\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )} b} \]
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Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {\cosh (x)}{(a+b \sinh (x))^2} \, dx=\frac {\mathrm {sinh}\left (x\right )}{a\,\left (a+b\,\mathrm {sinh}\left (x\right )\right )} \]
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