Optimal. Leaf size=13 \[ -\frac{1}{b (a+b \sinh (x))} \]
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Rubi [A] time = 0.0253128, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2668, 32} \[ -\frac{1}{b (a+b \sinh (x))} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \frac{\cosh (x)}{(a+b \sinh (x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{(a+x)^2} \, dx,x,b \sinh (x)\right )}{b}\\ &=-\frac{1}{b (a+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.0131484, size = 13, normalized size = 1. \[ -\frac{1}{b (a+b \sinh (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 14, normalized size = 1.1 \begin{align*} -{\frac{1}{b \left ( a+b\sinh \left ( x \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2087, size = 18, normalized size = 1.38 \begin{align*} -\frac{1}{{\left (b \sinh \left (x\right ) + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.95935, size = 149, normalized size = 11.46 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.01477, size = 32, normalized size = 2.46 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sinh{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\tilde{\infty } \sinh{\left (x \right )} & \text{for}\: a = - b \sinh{\left (x \right )} \\\frac{\sinh{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\- \frac{1}{a b + b^{2} \sinh{\left (x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12229, size = 30, normalized size = 2.31 \begin{align*} \frac{2}{{\left (b{\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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