Optimal. Leaf size=17 \[ \frac{\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]
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Rubi [A] time = 0.0145181, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3075} \[ \frac{\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]
Antiderivative was successfully verified.
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Rule 3075
Rubi steps
\begin{align*} \int \frac{1}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac{\sinh (x)}{a (a \cosh (x)+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.0240173, size = 17, normalized size = 1. \[ \frac{\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 29, normalized size = 1.7 \begin{align*} 2\,{\frac{\tanh \left ( x/2 \right ) }{a \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23529, size = 39, normalized size = 2.29 \begin{align*} \frac{2}{a^{2} - b^{2} +{\left (a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-2 \, x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.36344, size = 162, normalized size = 9.53 \begin{align*} -\frac{2}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 151.74, size = 206, normalized size = 12.12 \begin{align*} \begin{cases} \tilde{\infty } \left (- \frac{\tanh{\left (\frac{x}{2} \right )}}{2} - \frac{1}{2 \tanh{\left (\frac{x}{2} \right )}}\right ) & \text{for}\: a = 0 \wedge b = 0 \\\frac{- \frac{\tanh{\left (\frac{x}{2} \right )}}{2} - \frac{1}{2 \tanh{\left (\frac{x}{2} \right )}}}{b^{2}} & \text{for}\: a = 0 \\\frac{x \tanh ^{2}{\left (x \right )}}{2 a^{2} \sinh ^{2}{\left (x \right )} - 4 a^{2} \sinh{\left (x \right )} \cosh{\left (x \right )} \tanh{\left (x \right )} + 2 a^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (x \right )}} - \frac{x}{2 a^{2} \sinh ^{2}{\left (x \right )} - 4 a^{2} \sinh{\left (x \right )} \cosh{\left (x \right )} \tanh{\left (x \right )} + 2 a^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (x \right )}} + \frac{\tanh{\left (x \right )}}{2 a^{2} \sinh ^{2}{\left (x \right )} - 4 a^{2} \sinh{\left (x \right )} \cosh{\left (x \right )} \tanh{\left (x \right )} + 2 a^{2} \cosh ^{2}{\left (x \right )} \tanh ^{2}{\left (x \right )}} & \text{for}\: b = - \frac{a}{\tanh{\left (x \right )}} \\\frac{2 \tanh{\left (\frac{x}{2} \right )}}{a^{2} \tanh ^{2}{\left (\frac{x}{2} \right )} + a^{2} + 2 a b \tanh{\left (\frac{x}{2} \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17418, size = 35, normalized size = 2.06 \begin{align*} -\frac{2}{{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}{\left (a + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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