Optimal. Leaf size=19 \[ \frac {\cosh ^5(x)}{5}-\frac {2 \cosh ^3(x)}{3}+\cosh (x) \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2633} \[ \frac {\cosh ^5(x)}{5}-\frac {2 \cosh ^3(x)}{3}+\cosh (x) \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin {align*} \int \sinh ^5(x) \, dx &=\operatorname {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*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.21 \[ \frac {5 \cosh (x)}{8}-\frac {5}{48} \cosh (3 x)+\frac {1}{80} \cosh (5 x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh ^5(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.95, size = 42, normalized size = 2.21 \[ \frac {1}{80} \, \cosh \relax (x)^{5} + \frac {1}{16} \, \cosh \relax (x) \sinh \relax (x)^{4} - \frac {5}{48} \, \cosh \relax (x)^{3} + \frac {1}{16} \, {\left (2 \, \cosh \relax (x)^{3} - 5 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + \frac {5}{8} \, \cosh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 37, normalized size = 1.95 \[ \frac {1}{480} \, {\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac {1}{160} \, e^{\left (5 \, x\right )} - \frac {5}{96} \, e^{\left (3 \, x\right )} + \frac {5}{16} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 18, normalized size = 0.95
method | result | size |
default | \(\left (\frac {8}{15}+\frac {\left (\sinh ^{4}\relax (x )\right )}{5}-\frac {4 \left (\sinh ^{2}\relax (x )\right )}{15}\right ) \cosh \relax (x )\) | \(18\) |
risch | \(\frac {{\mathrm e}^{5 x}}{160}-\frac {5 \,{\mathrm e}^{3 x}}{96}+\frac {5 \,{\mathrm e}^{x}}{16}+\frac {5 \,{\mathrm e}^{-x}}{16}-\frac {5 \,{\mathrm e}^{-3 x}}{96}+\frac {{\mathrm e}^{-5 x}}{160}\) | \(36\) |
meijerg | error in int/gbinthm/express: improper op or subscript selector\ | N/A |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 35, normalized size = 1.84 \[ \frac {1}{160} \, e^{\left (5 \, x\right )} - \frac {5}{96} \, e^{\left (3 \, x\right )} + \frac {5}{16} \, e^{\left (-x\right )} - \frac {5}{96} \, e^{\left (-3 \, x\right )} + \frac {1}{160} \, e^{\left (-5 \, x\right )} + \frac {5}{16} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 15, normalized size = 0.79 \[ \frac {{\mathrm {cosh}\relax (x)}^5}{5}-\frac {2\,{\mathrm {cosh}\relax (x)}^3}{3}+\mathrm {cosh}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.12, size = 29, normalized size = 1.53 \[ \sinh ^{4}{\relax (x )} \cosh {\relax (x )} - \frac {4 \sinh ^{2}{\relax (x )} \cosh ^{3}{\relax (x )}}{3} + \frac {8 \cosh ^{5}{\relax (x )}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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