Optimal. Leaf size=56 \[ \frac{3 x}{256}-\frac{1}{10} \sin ^3(x) \cos ^7(x)-\frac{3}{80} \sin (x) \cos ^7(x)+\frac{1}{160} \sin (x) \cos ^5(x)+\frac{1}{128} \sin (x) \cos ^3(x)+\frac{3}{256} \sin (x) \cos (x) \]
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Rubi [A] time = 0.0623641, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2568, 2635, 8} \[ \frac{3 x}{256}-\frac{1}{10} \sin ^3(x) \cos ^7(x)-\frac{3}{80} \sin (x) \cos ^7(x)+\frac{1}{160} \sin (x) \cos ^5(x)+\frac{1}{128} \sin (x) \cos ^3(x)+\frac{3}{256} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^6(x) \sin ^4(x) \, dx &=-\frac{1}{10} \cos ^7(x) \sin ^3(x)+\frac{3}{10} \int \cos ^6(x) \sin ^2(x) \, dx\\ &=-\frac{3}{80} \cos ^7(x) \sin (x)-\frac{1}{10} \cos ^7(x) \sin ^3(x)+\frac{3}{80} \int \cos ^6(x) \, dx\\ &=\frac{1}{160} \cos ^5(x) \sin (x)-\frac{3}{80} \cos ^7(x) \sin (x)-\frac{1}{10} \cos ^7(x) \sin ^3(x)+\frac{1}{32} \int \cos ^4(x) \, dx\\ &=\frac{1}{128} \cos ^3(x) \sin (x)+\frac{1}{160} \cos ^5(x) \sin (x)-\frac{3}{80} \cos ^7(x) \sin (x)-\frac{1}{10} \cos ^7(x) \sin ^3(x)+\frac{3}{128} \int \cos ^2(x) \, dx\\ &=\frac{3}{256} \cos (x) \sin (x)+\frac{1}{128} \cos ^3(x) \sin (x)+\frac{1}{160} \cos ^5(x) \sin (x)-\frac{3}{80} \cos ^7(x) \sin (x)-\frac{1}{10} \cos ^7(x) \sin ^3(x)+\frac{3 \int 1 \, dx}{256}\\ &=\frac{3 x}{256}+\frac{3}{256} \cos (x) \sin (x)+\frac{1}{128} \cos ^3(x) \sin (x)+\frac{1}{160} \cos ^5(x) \sin (x)-\frac{3}{80} \cos ^7(x) \sin (x)-\frac{1}{10} \cos ^7(x) \sin ^3(x)\\ \end{align*}
Mathematica [A] time = 0.0154152, size = 46, normalized size = 0.82 \[ \frac{3 x}{256}+\frac{1}{512} \sin (2 x)-\frac{1}{256} \sin (4 x)-\frac{\sin (6 x)}{1024}+\frac{\sin (8 x)}{2048}+\frac{\sin (10 x)}{5120} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 42, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7} \left ( \sin \left ( x \right ) \right ) ^{3}}{10}}-{\frac{3\, \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) }{80}}+{\frac{\sin \left ( x \right ) }{160} \left ( \left ( \cos \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( x \right ) }{8}} \right ) }+{\frac{3\,x}{256}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.930434, size = 32, normalized size = 0.57 \begin{align*} \frac{1}{320} \, \sin \left (2 \, x\right )^{5} + \frac{3}{256} \, x + \frac{1}{2048} \, \sin \left (8 \, x\right ) - \frac{1}{256} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98124, size = 127, normalized size = 2.27 \begin{align*} \frac{1}{1280} \,{\left (128 \, \cos \left (x\right )^{9} - 176 \, \cos \left (x\right )^{7} + 8 \, \cos \left (x\right )^{5} + 10 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{3}{256} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.06141, size = 56, normalized size = 1. \begin{align*} \frac{3 x}{256} + \frac{\sin{\left (x \right )} \cos ^{9}{\left (x \right )}}{10} - \frac{11 \sin{\left (x \right )} \cos ^{7}{\left (x \right )}}{80} + \frac{\sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{160} + \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )}}{128} + \frac{3 \sin{\left (x \right )} \cos{\left (x \right )}}{256} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05848, size = 46, normalized size = 0.82 \begin{align*} \frac{3}{256} \, x + \frac{1}{5120} \, \sin \left (10 \, x\right ) + \frac{1}{2048} \, \sin \left (8 \, x\right ) - \frac{1}{1024} \, \sin \left (6 \, x\right ) - \frac{1}{256} \, \sin \left (4 \, x\right ) + \frac{1}{512} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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