Optimal. Leaf size=26 \[ -\frac{1}{2} \sin (\cos (x))+\frac{1}{44} \sin (11 \cos (x))+\frac{1}{52} \sin (13 \cos (x)) \]
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Rubi [A] time = 0.0464201, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4335, 4354, 2637} \[ -\frac{1}{2} \sin (\cos (x))+\frac{1}{44} \sin (11 \cos (x))+\frac{1}{52} \sin (13 \cos (x)) \]
Antiderivative was successfully verified.
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Rule 4335
Rule 4354
Rule 2637
Rubi steps
\begin{align*} \int \cos (\cos (x)) \sin (x) \sin ^2(6 \cos (x)) \, dx &=-\operatorname{Subst}\left (\int \cos (x) \sin ^2(6 x) \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{\cos (x)}{2}-\frac{1}{4} \cos (11 x)-\frac{1}{4} \cos (13 x)\right ) \, dx,x,\cos (x)\right )\\ &=\frac{1}{4} \operatorname{Subst}(\int \cos (11 x) \, dx,x,\cos (x))+\frac{1}{4} \operatorname{Subst}(\int \cos (13 x) \, dx,x,\cos (x))-\frac{1}{2} \operatorname{Subst}(\int \cos (x) \, dx,x,\cos (x))\\ &=-\frac{1}{2} \sin (\cos (x))+\frac{1}{44} \sin (11 \cos (x))+\frac{1}{52} \sin (13 \cos (x))\\ \end{align*}
Mathematica [A] time = 4.68363, size = 26, normalized size = 1. \[ -\frac{1}{2} \sin (\cos (x))+\frac{1}{44} \sin (11 \cos (x))+\frac{1}{52} \sin (13 \cos (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 21, normalized size = 0.8 \begin{align*} -{\frac{\sin \left ( \cos \left ( x \right ) \right ) }{2}}+{\frac{\sin \left ( 11\,\cos \left ( x \right ) \right ) }{44}}+{\frac{\sin \left ( 13\,\cos \left ( x \right ) \right ) }{52}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961604, size = 27, normalized size = 1.04 \begin{align*} \frac{1}{52} \, \sin \left (13 \, \cos \left (x\right )\right ) + \frac{1}{44} \, \sin \left (11 \, \cos \left (x\right )\right ) - \frac{1}{2} \, \sin \left (\cos \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.37467, size = 494, normalized size = 19. \begin{align*} -\frac{4}{143} \,{\left (2816 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{12} - 6912 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{10} + 6048 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{8} - 2240 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{6} + 315 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{4} - 9 \, \cos \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right )^{2} - 18\right )} \sin \left (\frac{\tan \left (\frac{1}{2} \, x\right )^{2} - 1}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 70.251, size = 54, normalized size = 2.08 \begin{align*} - \frac{71 \sin{\left (\cos{\left (x \right )} \right )} \sin ^{2}{\left (6 \cos{\left (x \right )} \right )}}{143} - \frac{72 \sin{\left (\cos{\left (x \right )} \right )} \cos ^{2}{\left (6 \cos{\left (x \right )} \right )}}{143} + \frac{12 \sin{\left (6 \cos{\left (x \right )} \right )} \cos{\left (\cos{\left (x \right )} \right )} \cos{\left (6 \cos{\left (x \right )} \right )}}{143} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09441, size = 27, normalized size = 1.04 \begin{align*} \frac{1}{52} \, \sin \left (13 \, \cos \left (x\right )\right ) + \frac{1}{44} \, \sin \left (11 \, \cos \left (x\right )\right ) - \frac{1}{2} \, \sin \left (\cos \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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