Optimal. Leaf size=24 \[ \frac{a \sinh (c+d x)}{d}-\frac{a \cosh (c+d x)}{d} \]
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Rubi [A] time = 0.0148298, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2637, 2638} \[ \frac{a \sinh (c+d x)}{d}-\frac{a \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (a \cosh (c+d x)-a \sinh (c+d x)) \, dx &=a \int \cosh (c+d x) \, dx-a \int \sinh (c+d x) \, dx\\ &=-\frac{a \cosh (c+d x)}{d}+\frac{a \sinh (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0185646, size = 47, normalized size = 1.96 \[ -\frac{a \sinh (c) \sinh (d x)}{d}-\frac{a \cosh (c) \cosh (d x)}{d}+\frac{a \sinh (c) \cosh (d x)}{d}+\frac{a \cosh (c) \sinh (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 0.9 \begin{align*}{\frac{a \left ( \sinh \left ( dx+c \right ) -\cosh \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987686, size = 32, normalized size = 1.33 \begin{align*} -\frac{a \cosh \left (d x + c\right )}{d} + \frac{a \sinh \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84096, size = 54, normalized size = 2.25 \begin{align*} -\frac{a}{d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.198325, size = 29, normalized size = 1.21 \begin{align*} a \left (\begin{cases} \frac{\sinh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \cosh{\left (c \right )} & \text{otherwise} \end{cases}\right ) - a \left (\begin{cases} \frac{\cosh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \sinh{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13443, size = 76, normalized size = 3.17 \begin{align*} -\frac{1}{2} \, a{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{1}{2} \, a{\left (\frac{e^{\left (d x + c\right )}}{d} - \frac{e^{\left (-d x - c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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