Optimal. Leaf size=27 \[ \frac{1}{2 d (a \cosh (c+d x)-a \sinh (c+d x))^2} \]
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Rubi [A] time = 0.0153704, antiderivative size = 27, 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} \[ \frac{1}{2 d (a \cosh (c+d x)-a \sinh (c+d x))^2} \]
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))^2} \, dx &=\frac{1}{2 d (a \cosh (c+d x)-a \sinh (c+d x))^2}\\ \end{align*}
Mathematica [A] time = 0.044446, size = 27, normalized size = 1. \[ \frac{1}{2 d (a \cosh (c+d x)-a \sinh (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 26, normalized size = 1. \begin{align*}{\frac{1}{2\,d{a}^{2} \left ( \cosh \left ( dx+c \right ) -\sinh \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06942, size = 23, normalized size = 0.85 \begin{align*} \frac{e^{\left (2 \, d x + 2 \, c\right )}}{2 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17896, size = 109, normalized size = 4.04 \begin{align*} \frac{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}{2 \,{\left (a^{2} d \cosh \left (d x + c\right ) - a^{2} d \sinh \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.923225, size = 65, normalized size = 2.41 \begin{align*} \begin{cases} \frac{1}{2 a^{2} d \sinh ^{2}{\left (c + d x \right )} - 4 a^{2} d \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )} + 2 a^{2} d \cosh ^{2}{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x}{\left (- a \sinh{\left (c \right )} + a \cosh{\left (c \right )}\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11292, size = 23, normalized size = 0.85 \begin{align*} \frac{e^{\left (2 \, d x + 2 \, c\right )}}{2 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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