Optimal. Leaf size=11 \[ \frac{\sin ^4(x)}{4}+\sin (x) \]
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Rubi [A] time = 0.0120964, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3223} \[ \frac{\sin ^4(x)}{4}+\sin (x) \]
Antiderivative was successfully verified.
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Rule 3223
Rubi steps
\begin{align*} \int \cos (x) \left (1+\sin ^3(x)\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^3\right ) \, dx,x,\sin (x)\right )\\ &=\sin (x)+\frac{\sin ^4(x)}{4}\\ \end{align*}
Mathematica [A] time = 0.0029038, size = 11, normalized size = 1. \[ \frac{\sin ^4(x)}{4}+\sin (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 10, normalized size = 0.9 \begin{align*} \sin \left ( x \right ) +{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.916528, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{4} \, \sin \left (x\right )^{4} + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01435, size = 51, normalized size = 4.64 \begin{align*} \frac{1}{4} \, \cos \left (x\right )^{4} - \frac{1}{2} \, \cos \left (x\right )^{2} + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.536143, size = 8, normalized size = 0.73 \begin{align*} \frac{\sin ^{4}{\left (x \right )}}{4} + \sin{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04765, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{4} \, \sin \left (x\right )^{4} + \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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