Optimal. Leaf size=34 \[ \frac{x}{2}-\frac{4}{3} \sin ^3(x) \cos ^3(x)-\sin (x) \cos ^3(x)+\frac{1}{2} \sin (x) \cos (x) \]
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Rubi [A] time = 0.0502599, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {12, 2568, 2635, 8} \[ \frac{x}{2}-\frac{4}{3} \sin ^3(x) \cos ^3(x)-\sin (x) \cos ^3(x)+\frac{1}{2} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int 8 \cos ^2(x) \sin ^4(x) \, dx &=8 \int \cos ^2(x) \sin ^4(x) \, dx\\ &=-\frac{4}{3} \cos ^3(x) \sin ^3(x)+4 \int \cos ^2(x) \sin ^2(x) \, dx\\ &=-\cos ^3(x) \sin (x)-\frac{4}{3} \cos ^3(x) \sin ^3(x)+\int \cos ^2(x) \, dx\\ &=\frac{1}{2} \cos (x) \sin (x)-\cos ^3(x) \sin (x)-\frac{4}{3} \cos ^3(x) \sin ^3(x)+\frac{\int 1 \, dx}{2}\\ &=\frac{x}{2}+\frac{1}{2} \cos (x) \sin (x)-\cos ^3(x) \sin (x)-\frac{4}{3} \cos ^3(x) \sin ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0100087, size = 32, normalized size = 0.94 \[ 8 \left (\frac{x}{16}-\frac{1}{64} \sin (2 x)-\frac{1}{64} \sin (4 x)+\frac{1}{192} \sin (6 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 29, normalized size = 0.9 \begin{align*}{\frac{x}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}}- \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) -{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{3} \left ( \sin \left ( x \right ) \right ) ^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960433, size = 24, normalized size = 0.71 \begin{align*} -\frac{1}{6} \, \sin \left (2 \, x\right )^{3} + \frac{1}{2} \, x - \frac{1}{8} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39833, size = 78, normalized size = 2.29 \begin{align*} \frac{1}{6} \,{\left (8 \, \cos \left (x\right )^{5} - 14 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.061079, size = 32, normalized size = 0.94 \begin{align*} \frac{x}{2} + \frac{4 \sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{3} - \frac{\sin ^{3}{\left (x \right )} \cos{\left (x \right )}}{3} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08186, size = 30, normalized size = 0.88 \begin{align*} \frac{1}{2} \, x + \frac{1}{24} \, \sin \left (6 \, x\right ) - \frac{1}{8} \, \sin \left (4 \, x\right ) - \frac{1}{8} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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