Optimal. Leaf size=22 \[ \frac{b \sinh (x)+c \cosh (x)}{a+b \cosh (x)+c \sinh (x)} \]
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Rubi [A] time = 0.0829528, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031, Rules used = {3150} \[ \frac{b \sinh (x)+c \cosh (x)}{a+b \cosh (x)+c \sinh (x)} \]
Antiderivative was successfully verified.
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Rule 3150
Rubi steps
\begin{align*} \int \frac{b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx &=\frac{c \cosh (x)+b \sinh (x)}{a+b \cosh (x)+c \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0831417, size = 34, normalized size = 1.55 \[ \frac{-a c+b^2 \sinh (x)-c^2 \sinh (x)}{b (a+b \cosh (x)+c \sinh (x))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.082, size = 73, normalized size = 3.3 \begin{align*} 2\,{\frac{1}{a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b-2\,c\tanh \left ( x/2 \right ) -a-b} \left ( -{\frac{ \left ( ab-{b}^{2}+{c}^{2} \right ) \tanh \left ( x/2 \right ) }{a-b}}-{\frac{ac}{a-b}} \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 = 2.3252, size = 178, normalized size = 8.09 \begin{align*} -\frac{2 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b - c\right )}}{{\left (b + c\right )} \cosh \left (x\right )^{2} +{\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left ({\left (b + c\right )} \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b - c} \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 [A] time = 1.1841, size = 47, normalized size = 2.14 \begin{align*} -\frac{2 \,{\left (a e^{x} + b - c\right )}}{b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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