Optimal. Leaf size=27 \[ -\frac{2 a \sinh (c+d x)}{d \sqrt{a-a \cosh (c+d x)}} \]
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Rubi [A] time = 0.0144789, antiderivative size = 27, 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 = {2646} \[ -\frac{2 a \sinh (c+d x)}{d \sqrt{a-a \cosh (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a-a \cosh (c+d x)} \, dx &=-\frac{2 a \sinh (c+d x)}{d \sqrt{a-a \cosh (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0321064, size = 30, normalized size = 1.11 \[ \frac{2 \coth \left (\frac{1}{2} (c+d x)\right ) \sqrt{a-a \cosh (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 41, normalized size = 1.5 \begin{align*} -4\,{\frac{\sinh \left ( 1/2\,dx+c/2 \right ) a\cosh \left ( 1/2\,dx+c/2 \right ) }{\sqrt{-2\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.59228, size = 78, normalized size = 2.89 \begin{align*} -\frac{\sqrt{2} \sqrt{a} e^{\left (-d x - c\right )}}{d \sqrt{-e^{\left (-d x - c\right )}}} - \frac{\sqrt{2} \sqrt{a}}{d \sqrt{-e^{\left (-d x - c\right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74937, size = 124, normalized size = 4.59 \begin{align*} \frac{2 \, \sqrt{\frac{1}{2}} \sqrt{-\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}{\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\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 \cosh{\left (c + d x \right )} + a}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19146, size = 85, normalized size = 3.15 \begin{align*} -\frac{\sqrt{2}{\left (\sqrt{-a e^{\left (d x + c\right )}} a \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - \frac{a^{2} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right )}{\sqrt{-a e^{\left (d x + c\right )}}}\right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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