Integrand size = 8, antiderivative size = 27 \[ \int \sinh ^3(a+b x) \, dx=-\frac {\cosh (a+b x)}{b}+\frac {\cosh ^3(a+b x)}{3 b} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2713} \[ \int \sinh ^3(a+b x) \, dx=\frac {\cosh ^3(a+b x)}{3 b}-\frac {\cosh (a+b x)}{b} \]
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Rule 2713
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{b} \\ & = -\frac {\cosh (a+b x)}{b}+\frac {\cosh ^3(a+b x)}{3 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \sinh ^3(a+b x) \, dx=-\frac {3 \cosh (a+b x)}{4 b}+\frac {\cosh (3 (a+b x))}{12 b} \]
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Time = 1.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {\left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}\) | \(23\) |
default | \(\frac {\left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}\) | \(23\) |
parallelrisch | \(\frac {\cosh \left (3 b x +3 a \right )-9 \cosh \left (b x +a \right )-8}{12 b}\) | \(25\) |
risch | \(\frac {{\mathrm e}^{3 b x +3 a}}{24 b}-\frac {3 \,{\mathrm e}^{b x +a}}{8 b}-\frac {3 \,{\mathrm e}^{-b x -a}}{8 b}+\frac {{\mathrm e}^{-3 b x -3 a}}{24 b}\) | \(55\) |
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none
Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \sinh ^3(a+b x) \, dx=\frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 9 \, \cosh \left (b x + a\right )}{12 \, b} \]
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Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \sinh ^3(a+b x) \, dx=\begin {cases} \frac {\sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 \cosh ^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sinh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).
Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \sinh ^3(a+b x) \, dx=\frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} - \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac {3 \, e^{\left (-b x - a\right )}}{8 \, b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \sinh ^3(a+b x) \, dx=\frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} - \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac {3 \, e^{\left (-b x - a\right )}}{8 \, b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \]
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Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \sinh ^3(a+b x) \, dx=-\frac {3\,\mathrm {cosh}\left (a+b\,x\right )-{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b} \]
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