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 = 0.0067568, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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*}
Mathematica [A] time = 0.0017605, size = 23, normalized size = 1.1 \[ -\frac{5 \cos (x)}{8}+\frac{5}{48} \cos (3 x)-\frac{1}{80} \cos (5 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 17, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( x \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( x \right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.928642, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97677, size = 53, normalized size = 2.52 \begin{align*} -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.06298, size = 17, normalized size = 0.81 \begin{align*} - \frac{\cos ^{5}{\left (x \right )}}{5} + \frac{2 \cos ^{3}{\left (x \right )}}{3} - \cos{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04548, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{5} \, \cos \left (x\right )^{5} + \frac{2}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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