Optimal. Leaf size=11 \[ \frac{1}{2} (\sinh (x)+\cosh (x))^2 \]
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Rubi [A] time = 0.0389334, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {4385} \[ \frac{1}{2} (\sinh (x)+\cosh (x))^2 \]
Antiderivative was successfully verified.
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Rule 4385
Rubi steps
\begin{align*} \int \frac{\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx &=\frac{1}{2} (\cosh (x)+\sinh (x))^2\\ \end{align*}
Mathematica [A] time = 0.0034965, size = 17, normalized size = 1.55 \[ \frac{1}{2} \sinh (2 x)+\frac{1}{2} \cosh (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 17, normalized size = 1.6 \begin{align*}{\frac{\cosh \left ( x \right ) +\sinh \left ( x \right ) }{2\,\cosh \left ( x \right ) -2\,\sinh \left ( x \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02184, size = 8, normalized size = 0.73 \begin{align*} \frac{1}{2} \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2091, size = 61, normalized size = 5.55 \begin{align*} \frac{\cosh \left (x\right ) + \sinh \left (x\right )}{2 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.376119, size = 8, normalized size = 0.73 \begin{align*} \frac{\cosh{\left (x \right )}}{- \sinh{\left (x \right )} + \cosh{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14802, size = 8, normalized size = 0.73 \begin{align*} \frac{1}{2} \, e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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