Optimal. Leaf size=26 \[ -\frac{2}{d \sqrt{a \sinh (c+d x)+a \cosh (c+d x)}} \]
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Rubi [A] time = 0.0170826, 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} \[ -\frac{2}{d \sqrt{a \sinh (c+d x)+a \cosh (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3071
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \cosh (c+d x)+a \sinh (c+d x)}} \, dx &=-\frac{2}{d \sqrt{a \cosh (c+d x)+a \sinh (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0344189, size = 24, normalized size = 0.92 \[ -\frac{2}{d \sqrt{a (\sinh (c+d x)+\cosh (c+d x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 1. \begin{align*} -2\,{\frac{1}{d\sqrt{a\cosh \left ( dx+c \right ) +a\sinh \left ( dx+c \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07291, size = 23, normalized size = 0.88 \begin{align*} -\frac{2 \, e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{\sqrt{a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95519, size = 113, normalized size = 4.35 \begin{align*} -\frac{2 \, \sqrt{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )}}{a d \cosh \left (d x + c\right ) + a d \sinh \left (d x + c\right )} \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 \frac{1}{\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 = 1.14864, size = 23, normalized size = 0.88 \begin{align*} -\frac{2 \, e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{\sqrt{a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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