Optimal. Leaf size=26 \[ \frac{2 \sqrt{a \sinh (c+d x)+a \cosh (c+d x)}}{d} \]
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Rubi [A] time = 0.0161512, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3071} \[ \frac{2 \sqrt{a \sinh (c+d x)+a \cosh (c+d x)}}{d} \]
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 = 0.0223419, size = 24, normalized size = 0.92 \[ \frac{2 \sqrt{a (\sinh (c+d x)+\cosh (c+d x))}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 25, 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 = 1.01627, size = 23, normalized size = 0.88 \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 = 2.00343, size = 61, normalized size = 2.35 \begin{align*} \frac{2 \, \sqrt{a \cosh \left (d x + c\right ) + a \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 = 1.16006, size = 23, normalized size = 0.88 \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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