Optimal. Leaf size=24 \[ -\frac {a \cosh (c+d x)}{d}+\frac {a \sinh (c+d x)}{d} \]
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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 = {2717, 2718}
\begin {gather*} \frac {a \sinh (c+d x)}{d}-\frac {a \cosh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 2718
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 \begin {gather*} -\frac {a \cosh (c) \cosh (d x)}{d}+\frac {a \cosh (d x) \sinh (c)}{d}+\frac {a \cosh (c) \sinh (d x)}{d}-\frac {a \sinh (c) \sinh (d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.01, size = 25, normalized size = 1.04
method | result | size |
risch | \(-\frac {a \,{\mathrm e}^{-d x -c}}{d}\) | \(16\) |
gosper | \(\frac {a \left (\sinh \left (d x +c \right )-\cosh \left (d x +c \right )\right )}{d}\) | \(21\) |
derivativedivides | \(\frac {a \sinh \left (d x +c \right )-a \cosh \left (d x +c \right )}{d}\) | \(23\) |
default | \(-\frac {a \cosh \left (d x +c \right )}{d}+\frac {a \sinh \left (d x +c \right )}{d}\) | \(25\) |
meijerg | \(\frac {\left (\cosh \left (c \right ) \sqrt {\pi }\, a -\sqrt {\pi }\, \sinh \left (c \right ) a \right ) \sinh \left (d x \right )}{d \sqrt {\pi }}-\frac {i \left (i \cosh \left (c \right ) \sqrt {\pi }\, a -i \sqrt {\pi }\, \sinh \left (c \right ) a \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 24, normalized size = 1.00 \begin {gather*} -\frac {a \cosh \left (d x + c\right )}{d} + \frac {a \sinh \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 22, normalized size = 0.92 \begin {gather*} -\frac {a}{d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 29, normalized size = 1.21 \begin {gather*} 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (24) = 48\).
time = 0.41, size = 56, normalized size = 2.33 \begin {gather*} -\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 {gather*}
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 \begin {gather*} -\frac {a\,{\mathrm {e}}^{-c-d\,x}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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