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 = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.40, size = 42, normalized size = 1.56 \[ \frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 61, normalized size = 2.26 \[ -\frac {\sqrt {2} {\left (\sqrt {-a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + \sqrt {-a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 41, normalized size = 1.52 \[ -\frac {4 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) a \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {-2 a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 58, normalized size = 2.15 \[ -\frac {\sqrt {2} \sqrt {a} e^{\left (-d x - c\right )}}{d \sqrt {-e^{\left (-d x - c\right )}}} - \frac {\sqrt {2} \sqrt {a}}{d \sqrt {-e^{\left (-d x - c\right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 27, normalized size = 1.00 \[ \frac {2\,\mathrm {coth}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a-a\,\mathrm {cosh}\left (c+d\,x\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- a \cosh {\left (c + d x \right )} + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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