Optimal. Leaf size=22 \[ -\frac{1}{2 b (\sinh (a+b x)+\cosh (a+b x))^2} \]
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Rubi [A] time = 0.0503993, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {4385} \[ -\frac{1}{2 b (\sinh (a+b x)+\cosh (a+b x))^2} \]
Antiderivative was successfully verified.
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Rule 4385
Rubi steps
\begin{align*} \int \frac{\cosh (a+b x)-\sinh (a+b x)}{\cosh (a+b x)+\sinh (a+b x)} \, dx &=-\frac{1}{2 b (\cosh (a+b x)+\sinh (a+b x))^2}\\ \end{align*}
Mathematica [B] time = 0.0240751, size = 65, normalized size = 2.95 \[ -\frac{\sinh (2 a) \sinh (2 b x)}{2 b}-\frac{\cosh (2 a) \cosh (2 b x)}{2 b}+\frac{\sinh (2 a) \cosh (2 b x)}{2 b}+\frac{\cosh (2 a) \sinh (2 b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 36, normalized size = 1.6 \begin{align*}{\frac{-\cosh \left ( bx+a \right ) +\sinh \left ( bx+a \right ) }{2\,b \left ( \cosh \left ( bx+a \right ) +\sinh \left ( bx+a \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0248, size = 19, normalized size = 0.86 \begin{align*} -\frac{e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00972, size = 108, normalized size = 4.91 \begin{align*} -\frac{1}{2 \,{\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.649011, size = 39, normalized size = 1.77 \begin{align*} \begin{cases} - \frac{\cosh{\left (a + b x \right )}}{b \sinh{\left (a + b x \right )} + b \cosh{\left (a + b x \right )}} & \text{for}\: b \neq 0 \\\frac{x \left (- \sinh{\left (a \right )} + \cosh{\left (a \right )}\right )}{\sinh{\left (a \right )} + \cosh{\left (a \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13724, size = 19, normalized size = 0.86 \begin{align*} -\frac{e^{\left (-2 \, b x - 2 \, a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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