Integrand size = 4, antiderivative size = 19 \[ \int \sinh ^5(x) \, dx=\cosh (x)-\frac {2 \cosh ^3(x)}{3}+\frac {\cosh ^5(x)}{5} \]
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Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2713} \[ \int \sinh ^5(x) \, dx=\frac {\cosh ^5(x)}{5}-\frac {2 \cosh ^3(x)}{3}+\cosh (x) \]
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Rule 2713
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (x)\right ) \\ & = \cosh (x)-\frac {2 \cosh ^3(x)}{3}+\frac {\cosh ^5(x)}{5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \sinh ^5(x) \, dx=\frac {5 \cosh (x)}{8}-\frac {5}{48} \cosh (3 x)+\frac {1}{80} \cosh (5 x) \]
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Time = 0.49 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
method | result | size |
default | \(\left (\frac {8}{15}+\frac {\left (\sinh ^{4}\left (x \right )\right )}{5}-\frac {4 \left (\sinh ^{2}\left (x \right )\right )}{15}\right ) \cosh \left (x \right )\) | \(18\) |
risch | \(\frac {{\mathrm e}^{5 x}}{160}-\frac {5 \,{\mathrm e}^{3 x}}{96}+\frac {5 \,{\mathrm e}^{x}}{16}+\frac {5 \,{\mathrm e}^{-x}}{16}-\frac {5 \,{\mathrm e}^{-3 x}}{96}+\frac {{\mathrm e}^{-5 x}}{160}\) | \(36\) |
parallelrisch | \(\frac {-18 \cosh \left (7 x \right )+90 \cosh \left (5 x \right )-162 \cosh \left (3 x \right )+90 \cosh \left (x \right )+3 \cosh \left (8 x \right )-180 \cosh \left (4 x \right )+20 \cosh \left (6 x \right )+492 \cosh \left (2 x \right )-335}{480 \cosh \left (3 x \right )+7200 \cosh \left (x \right )-2880 \cosh \left (2 x \right )-4800}\) | \(70\) |
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (15) = 30\).
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \sinh ^5(x) \, dx=\frac {1}{80} \, \cosh \left (x\right )^{5} + \frac {1}{16} \, \cosh \left (x\right ) \sinh \left (x\right )^{4} - \frac {5}{48} \, \cosh \left (x\right )^{3} + \frac {1}{16} \, {\left (2 \, \cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + \frac {5}{8} \, \cosh \left (x\right ) \]
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Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.53 \[ \int \sinh ^5(x) \, dx=\sinh ^{4}{\left (x \right )} \cosh {\left (x \right )} - \frac {4 \sinh ^{2}{\left (x \right )} \cosh ^{3}{\left (x \right )}}{3} + \frac {8 \cosh ^{5}{\left (x \right )}}{15} \]
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (15) = 30\).
Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \sinh ^5(x) \, dx=\frac {1}{160} \, e^{\left (5 \, x\right )} - \frac {5}{96} \, e^{\left (3 \, x\right )} + \frac {5}{16} \, e^{\left (-x\right )} - \frac {5}{96} \, e^{\left (-3 \, x\right )} + \frac {1}{160} \, e^{\left (-5 \, x\right )} + \frac {5}{16} \, e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \sinh ^5(x) \, dx=\frac {1}{480} \, {\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{160} \, e^{\left (5 \, x\right )} - \frac {5}{96} \, e^{\left (3 \, x\right )} + \frac {5}{16} \, e^{x} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \sinh ^5(x) \, dx=\frac {{\mathrm {cosh}\left (x\right )}^5}{5}-\frac {2\,{\mathrm {cosh}\left (x\right )}^3}{3}+\mathrm {cosh}\left (x\right ) \]
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