Integrand size = 10, antiderivative size = 15 \[ \int (a+b \sinh (c+d x)) \, dx=a x+\frac {b \cosh (c+d x)}{d} \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2718} \[ \int (a+b \sinh (c+d x)) \, dx=a x+\frac {b \cosh (c+d x)}{d} \]
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Rule 2718
Rubi steps \begin{align*} \text {integral}& = a x+b \int \sinh (c+d x) \, dx \\ & = a x+\frac {b \cosh (c+d x)}{d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73 \[ \int (a+b \sinh (c+d x)) \, dx=a x+\frac {b \cosh (c) \cosh (d x)}{d}+\frac {b \sinh (c) \sinh (d x)}{d} \]
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Time = 0.59 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
default | \(a x +\frac {b \cosh \left (d x +c \right )}{d}\) | \(16\) |
parts | \(a x +\frac {b \cosh \left (d x +c \right )}{d}\) | \(16\) |
parallelrisch | \(\frac {b \left (1+\cosh \left (d x +c \right )\right )}{d}+a x\) | \(18\) |
derivativedivides | \(\frac {\left (d x +c \right ) a +b \cosh \left (d x +c \right )}{d}\) | \(21\) |
risch | \(a x +\frac {b \,{\mathrm e}^{d x +c}}{2 d}+\frac {b \,{\mathrm e}^{-d x -c}}{2 d}\) | \(32\) |
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none
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int (a+b \sinh (c+d x)) \, dx=\frac {a d x + b \cosh \left (d x + c\right )}{d} \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int (a+b \sinh (c+d x)) \, dx=a x + b \left (\begin {cases} \frac {\cosh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \sinh {\left (c \right )} & \text {otherwise} \end {cases}\right ) \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int (a+b \sinh (c+d x)) \, dx=a x + \frac {b \cosh \left (d x + c\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int (a+b \sinh (c+d x)) \, dx=a x + \frac {1}{2} \, b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} \]
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Time = 1.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int (a+b \sinh (c+d x)) \, dx=a\,x+\frac {b\,\mathrm {cosh}\left (c+d\,x\right )}{d} \]
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