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 = 0.013316, antiderivative size = 26, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0302019, 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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Maple [A] time = 0.026, size = 43, normalized size = 1.7 \begin{align*} 2\,{\frac{a\cosh \left ( 1/2\,dx+c/2 \right ) \sinh \left ( 1/2\,dx+c/2 \right ) \sqrt{2}}{\sqrt{a \left ( \cosh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78412, size = 54, normalized size = 2.08 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85192, size = 123, normalized size = 4.73 \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 [A] time = 1.15514, size = 47, normalized size = 1.81 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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