Optimal. Leaf size=17 \[ \frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, size = 17, normalized size = 1.00 \[ \frac {\sinh (x)}{a (a \cosh (x)+b \sinh (x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 62, normalized size = 3.65 \[ -\frac {2}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \relax (x)^{2} + a^{2} - b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 26, normalized size = 1.53 \[ -\frac {2}{{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )} {\left (a + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 29, normalized size = 1.71 \[ \frac {2 \tanh \left (\frac {x}{2}\right )}{a \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 29, normalized size = 1.71 \[ \frac {2}{a^{2} - b^{2} + {\left (a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-2 \, x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 22, normalized size = 1.29 \[ -\frac {2}{\left (a+b\right )\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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