Integrand size = 17, antiderivative size = 23 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \cosh (c+d x)}{d}+\frac {a \sinh (c+d x)}{d} \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2717, 2718} \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \sinh (c+d x)}{d}+\frac {a \cosh (c+d x)}{d} \]
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Rule 2717
Rule 2718
Rubi steps \begin{align*} \text {integral}& = 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*}
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \cosh (c) \cosh (d x)}{d}+\frac {a \cosh (d x) \sinh (c)}{d}+\frac {a \cosh (c) \sinh (d x)}{d}+\frac {a \sinh (c) \sinh (d x)}{d} \]
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Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52
method | result | size |
risch | \(\frac {{\mathrm e}^{d x +c} a}{d}\) | \(12\) |
gosper | \(\frac {a \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )}{d}\) | \(19\) |
derivativedivides | \(\frac {a \cosh \left (d x +c \right )+a \sinh \left (d x +c \right )}{d}\) | \(22\) |
default | \(\frac {a \cosh \left (d x +c \right )}{d}+\frac {a \sinh \left (d x +c \right )}{d}\) | \(24\) |
parts | \(\frac {a \cosh \left (d x +c \right )}{d}+\frac {a \sinh \left (d x +c \right )}{d}\) | \(24\) |
meijerg | \(\frac {\left (\cosh \left (c \right ) \sqrt {\pi }\, a +\sqrt {\pi }\, \sinh \left (c \right ) a \right ) \sinh \left (d x \right )}{d \sqrt {\pi }}-\frac {i \left (-i \cosh \left (c \right ) \sqrt {\pi }\, a -i \sqrt {\pi }\, \sinh \left (c \right ) a \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (d x \right )}{\sqrt {\pi }}\right )}{d}\) | \(66\) |
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none
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )}{d} \]
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Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=a \left (\begin {cases} \frac {\sinh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cosh {\left (c \right )} & \text {otherwise} \end {cases}\right ) + a \left (\begin {cases} \frac {\cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sinh {\left (c \right )} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a \cosh \left (d x + c\right )}{d} + \frac {a \sinh \left (d x + c\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {1}{2} \, a {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{2} \, a {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} \]
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Time = 0.06 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int (a \cosh (c+d x)+a \sinh (c+d x)) \, dx=\frac {a\,{\mathrm {e}}^{c+d\,x}}{d} \]
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