Optimal. Leaf size=26 \[ -\frac{1}{2 d (a \sinh (c+d x)+a \cosh (c+d x))^2} \]
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Rubi [A] time = 0.0158443, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3071} \[ -\frac{1}{2 d (a \sinh (c+d x)+a \cosh (c+d x))^2} \]
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 = 0.0425167, size = 26, normalized size = 1. \[ -\frac{1}{2 d (a \sinh (c+d x)+a \cosh (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 0.9 \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 = 1.15344, size = 23, normalized size = 0.88 \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 [B] time = 1.98545, size = 124, normalized size = 4.77 \begin{align*} -\frac{1}{2 \,{\left (a^{2} d \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d \sinh \left (d x + c\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.2183, size = 66, normalized size = 2.54 \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 = 1.13492, size = 23, normalized size = 0.88 \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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