Optimal. Leaf size=26 \[ \frac {2 a \sinh (c+d x)}{d \sqrt {a \cosh (c+d x)+a}} \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2646} \[ \frac {2 a \sinh (c+d x)}{d \sqrt {a \cosh (c+d x)+a}} \]
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 = 29, normalized size = 1.12 \[ \frac {2 \tanh \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\cosh (c+d x)+1)}}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 41, normalized size = 1.58 \[ \frac {2 \, \sqrt {\frac {1}{2}} \sqrt {\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 35, normalized size = 1.35 \[ \frac {\sqrt {2} {\left (\sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 43, normalized size = 1.65 \[ \frac {2 a \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2}}{\sqrt {a \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 40, normalized size = 1.54 \[ \frac {\sqrt {2} \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{d} - \frac {\sqrt {2} \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 26, normalized size = 1.00 \[ \frac {2\,\mathrm {tanh}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a+a\,\mathrm {cosh}\left (c+d\,x\right )}}{d} \]
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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