Optimal. Leaf size=21 \[ -\frac {1}{5} \cos ^5(x)+\frac {2 \cos ^3(x)}{3}-\cos (x) \]
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Rubi [A] time = 0.01, antiderivative size = 21, 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} \[ -\frac {1}{5} \cos ^5(x)+\frac {2 \cos ^3(x)}{3}-\cos (x) \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin {align*} \int \sin ^5(x) \, dx &=-\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (x)\right )\\ &=-\cos (x)+\frac {2 \cos ^3(x)}{3}-\frac {\cos ^5(x)}{5}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.10 \[ -\frac {5 \cos (x)}{8}+\frac {5}{48} \cos (3 x)-\frac {1}{80} \cos (5 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 17, normalized size = 0.81 \[ -\frac {1}{5} \, \cos \relax (x)^{5} + \frac {2}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 17, normalized size = 0.81 \[ -\frac {1}{5} \, \cos \relax (x)^{5} + \frac {2}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.81 \[ -\frac {\left (\sin ^{4}\relax (x )+\frac {4 \left (\sin ^{2}\relax (x )\right )}{3}+\frac {8}{3}\right ) \cos \relax (x )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 17, normalized size = 0.81 \[ -\frac {1}{5} \, \cos \relax (x)^{5} + \frac {2}{3} \, \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 17, normalized size = 0.81 \[ -\frac {{\cos \relax (x)}^5}{5}+\frac {2\,{\cos \relax (x)}^3}{3}-\cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 17, normalized size = 0.81 \[ - \frac {\cos ^{5}{\relax (x )}}{5} + \frac {2 \cos ^{3}{\relax (x )}}{3} - \cos {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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