Optimal. Leaf size=27 \[ -\frac{2 \sqrt{a \cosh (c+d x)-a \sinh (c+d x)}}{d} \]
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Rubi [A] time = 0.0158502, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {3071} \[ -\frac{2 \sqrt{a \cosh (c+d x)-a \sinh (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3071
Rubi steps
\begin{align*} \int \sqrt{a \cosh (c+d x)-a \sinh (c+d x)} \, dx &=-\frac{2 \sqrt{a \cosh (c+d x)-a \sinh (c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.0220003, size = 26, normalized size = 0.96 \[ -\frac{2 \sqrt{a (\cosh (c+d x)-\sinh (c+d x))}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 26, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{a\cosh \left ( dx+c \right ) -a\sinh \left ( dx+c \right ) }}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03767, size = 23, normalized size = 0.85 \begin{align*} -\frac{2 \, \sqrt{a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26464, size = 62, normalized size = 2.3 \begin{align*} -\frac{2 \, \sqrt{\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- a \sinh{\left (c + d x \right )} + a \cosh{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15519, size = 23, normalized size = 0.85 \begin{align*} -\frac{2 \, \sqrt{a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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