Optimal. Leaf size=34 \[ -\frac{\sqrt{b^2-c^2} \sinh (x)+c}{c (b \sinh (x)+c \cosh (x))} \]
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Rubi [A] time = 0.0367554, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {3114} \[ -\frac{\sqrt{b^2-c^2} \sinh (x)+c}{c (b \sinh (x)+c \cosh (x))} \]
Antiderivative was successfully verified.
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Rule 3114
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b^2-c^2}+b \cosh (x)+c \sinh (x)} \, dx &=-\frac{c+\sqrt{b^2-c^2} \sinh (x)}{c (c \cosh (x)+b \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.0734254, size = 36, normalized size = 1.06 \[ \frac{-\sqrt{b^2-c^2} \sinh (x)-c}{c (b \sinh (x)+c \cosh (x))} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.254, size = 596, normalized size = 17.5 \begin{align*} \text{result too large to display} \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.98676, size = 235, normalized size = 6.91 \begin{align*} -\frac{2 \,{\left ({\left (b + c\right )} \cosh \left (x\right ) +{\left (b + c\right )} \sinh \left (x\right ) - \sqrt{b^{2} - c^{2}}\right )}}{{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{2} - b^{2} + c^{2}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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