Optimal. Leaf size=46 \[ \frac{3 x}{128}-\frac{1}{8} \sin ^3(x) \cos ^5(x)-\frac{1}{16} \sin (x) \cos ^5(x)+\frac{1}{64} \sin (x) \cos ^3(x)+\frac{3}{128} \sin (x) \cos (x) \]
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Rubi [A] time = 0.0516621, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2568, 2635, 8} \[ \frac{3 x}{128}-\frac{1}{8} \sin ^3(x) \cos ^5(x)-\frac{1}{16} \sin (x) \cos ^5(x)+\frac{1}{64} \sin (x) \cos ^3(x)+\frac{3}{128} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^4(x) \sin ^4(x) \, dx &=-\frac{1}{8} \cos ^5(x) \sin ^3(x)+\frac{3}{8} \int \cos ^4(x) \sin ^2(x) \, dx\\ &=-\frac{1}{16} \cos ^5(x) \sin (x)-\frac{1}{8} \cos ^5(x) \sin ^3(x)+\frac{1}{16} \int \cos ^4(x) \, dx\\ &=\frac{1}{64} \cos ^3(x) \sin (x)-\frac{1}{16} \cos ^5(x) \sin (x)-\frac{1}{8} \cos ^5(x) \sin ^3(x)+\frac{3}{64} \int \cos ^2(x) \, dx\\ &=\frac{3}{128} \cos (x) \sin (x)+\frac{1}{64} \cos ^3(x) \sin (x)-\frac{1}{16} \cos ^5(x) \sin (x)-\frac{1}{8} \cos ^5(x) \sin ^3(x)+\frac{3 \int 1 \, dx}{128}\\ &=\frac{3 x}{128}+\frac{3}{128} \cos (x) \sin (x)+\frac{1}{64} \cos ^3(x) \sin (x)-\frac{1}{16} \cos ^5(x) \sin (x)-\frac{1}{8} \cos ^5(x) \sin ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0066726, size = 22, normalized size = 0.48 \[ \frac{3 x}{128}-\frac{1}{128} \sin (4 x)+\frac{\sin (8 x)}{1024} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 36, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5} \left ( \sin \left ( x \right ) \right ) ^{3}}{8}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) }{16}}+{\frac{\sin \left ( x \right ) }{64} \left ( \left ( \cos \left ( x \right ) \right ) ^{3}+{\frac{3\,\cos \left ( x \right ) }{2}} \right ) }+{\frac{3\,x}{128}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.929238, size = 22, normalized size = 0.48 \begin{align*} \frac{3}{128} \, x + \frac{1}{1024} \, \sin \left (8 \, x\right ) - \frac{1}{128} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10534, size = 103, normalized size = 2.24 \begin{align*} \frac{1}{128} \,{\left (16 \, \cos \left (x\right )^{7} - 24 \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{3}{128} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.067331, size = 31, normalized size = 0.67 \begin{align*} \frac{3 x}{128} - \frac{\sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{128} - \frac{3 \sin{\left (2 x \right )} \cos{\left (2 x \right )}}{256} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04786, size = 22, normalized size = 0.48 \begin{align*} \frac{3}{128} \, x + \frac{1}{1024} \, \sin \left (8 \, x\right ) - \frac{1}{128} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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