Optimal. Leaf size=19 \[ \frac{\sin ^5(x)}{5}-\frac{2 \sin ^3(x)}{3}+\sin (x) \]
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Rubi [A] time = 0.0086871, antiderivative size = 19, 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} \[ \frac{\sin ^5(x)}{5}-\frac{2 \sin ^3(x)}{3}+\sin (x) \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin{align*} \int \cos ^5(x) \, dx &=-\operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )\\ &=\sin (x)-\frac{2 \sin ^3(x)}{3}+\frac{\sin ^5(x)}{5}\\ \end{align*}
Mathematica [A] time = 0.0019552, size = 23, normalized size = 1.21 \[ \frac{5 \sin (x)}{8}+\frac{5}{48} \sin (3 x)+\frac{1}{80} \sin (5 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 17, normalized size = 0.9 \begin{align*}{\frac{\sin \left ( x \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \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 = 0.942157, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \sin \left (x\right )^{5} - \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9634, size = 58, normalized size = 3.05 \begin{align*} \frac{1}{15} \,{\left (3 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 8\right )} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.058083, size = 17, normalized size = 0.89 \begin{align*} \frac{\sin ^{5}{\left (x \right )}}{5} - \frac{2 \sin ^{3}{\left (x \right )}}{3} + \sin{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07823, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{5} \, \sin \left (x\right )^{5} - \frac{2}{3} \, \sin \left (x\right )^{3} + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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