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 = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.38, size = 22, normalized size = 0.92 \[ -\frac {a}{d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 56, normalized size = 2.33 \[ -\frac {1}{2} \, a {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{2} \, a {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 21, normalized size = 0.88 \[ \frac {a \left (\sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 24, normalized size = 1.00 \[ -\frac {a \cosh \left (d x + c\right )}{d} + \frac {a \sinh \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 15, normalized size = 0.62 \[ -\frac {a\,{\mathrm {e}}^{-c-d\,x}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 29, normalized size = 1.21 \[ a \left (\begin {cases} \frac {\sinh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cosh {\relax (c )} & \text {otherwise} \end {cases}\right ) - a \left (\begin {cases} \frac {\cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sinh {\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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