Optimal. Leaf size=24 \[ \frac{1}{d (a \cosh (c+d x)-a \sinh (c+d x))} \]
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Rubi [A] time = 0.0155538, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {3071} \[ \frac{1}{d (a \cosh (c+d x)-a \sinh (c+d x))} \]
Antiderivative was successfully verified.
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Rule 3071
Rubi steps
\begin{align*} \int \frac{1}{a \cosh (c+d x)-a \sinh (c+d x)} \, dx &=\frac{1}{d (a \cosh (c+d x)-a \sinh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0074059, size = 22, normalized size = 0.92 \[ \frac{1}{a d \cosh (c+d x)-a d \sinh (c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 25, normalized size = 1. \begin{align*}{\frac{1}{da \left ( \cosh \left ( dx+c \right ) -\sinh \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03732, size = 18, normalized size = 0.75 \begin{align*} \frac{e^{\left (d x + c\right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2483, size = 53, normalized size = 2.21 \begin{align*} \frac{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.475574, size = 32, normalized size = 1.33 \begin{align*} \begin{cases} \frac{1}{- a d \sinh{\left (c + d x \right )} + a d \cosh{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x}{- a \sinh{\left (c \right )} + a \cosh{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12332, size = 18, normalized size = 0.75 \begin{align*} \frac{e^{\left (d x + c\right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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