Optimal. Leaf size=12 \[ -\frac{\cosh (x)}{a \sinh (x)+b} \]
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Rubi [A] time = 0.0321646, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2754, 8} \[ -\frac{\cosh (x)}{a \sinh (x)+b} \]
Antiderivative was successfully verified.
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Rule 2754
Rule 8
Rubi steps
\begin{align*} \int \frac{a-b \sinh (x)}{(b+a \sinh (x))^2} \, dx &=-\frac{\cosh (x)}{b+a \sinh (x)}-\frac{\int 0 \, dx}{a^2+b^2}\\ &=-\frac{\cosh (x)}{b+a \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.03527, size = 12, normalized size = 1. \[ -\frac{\cosh (x)}{a \sinh (x)+b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 36, normalized size = 3. \begin{align*} -2\,{\frac{1}{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b-2\,a\tanh \left ( x/2 \right ) -b} \left ( -{\frac{a\tanh \left ( x/2 \right ) }{b}}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63292, size = 158, normalized size = 13.17 \begin{align*} \frac{2 \,{\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) - a\right )}}{a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - a^{2} + 2 \,{\left (a^{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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23304, size = 41, normalized size = 3.42 \begin{align*} \frac{2 \,{\left (b e^{x} - a\right )}}{{\left (a e^{\left (2 \, x\right )} + 2 \, b e^{x} - a\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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