Optimal. Leaf size=19 \[ \cosh (x)-\frac {2 \cosh ^3(x)}{3}+\frac {\cosh ^5(x)}{5} \]
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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 = {2713}
\begin {gather*} \frac {\cosh ^5(x)}{5}-\frac {2 \cosh ^3(x)}{3}+\cosh (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rubi steps
\begin {align*} \int \sinh ^5(x) \, dx &=\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*}
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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.21 \begin {gather*} \frac {5 \cosh (x)}{8}-\frac {5}{48} \cosh (3 x)+\frac {1}{80} \cosh (5 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 18, normalized size = 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 | \(\frac {{\mathrm e}^{5 x}}{160}-\frac {5 \,{\mathrm e}^{3 x}}{96}+\frac {5 \,{\mathrm e}^{x}}{16}+\frac {5 \,{\mathrm e}^{-x}}{16}-\frac {5 \,{\mathrm e}^{-3 x}}{96}+\frac {{\mathrm e}^{-5 x}}{160}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs.
\(2 (15) = 30\).
time = 1.54, size = 35, normalized size = 1.84 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (15) = 30\).
time = 0.39, size = 42, normalized size = 2.21 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 29, normalized size = 1.53 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (15) = 30\).
time = 0.89, size = 37, normalized size = 1.95 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \frac {{\mathrm {cosh}\left (x\right )}^5}{5}-\frac {2\,{\mathrm {cosh}\left (x\right )}^3}{3}+\mathrm {cosh}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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