Optimal. Leaf size=68 \[ \frac{5 x}{1024}-\frac{1}{12} \sin ^5(x) \cos ^7(x)-\frac{1}{24} \sin ^3(x) \cos ^7(x)-\frac{1}{64} \sin (x) \cos ^7(x)+\frac{1}{384} \sin (x) \cos ^5(x)+\frac{5 \sin (x) \cos ^3(x)}{1536}+\frac{5 \sin (x) \cos (x)}{1024} \]
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Rubi [A] time = 0.084813, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2568, 2635, 8} \[ \frac{5 x}{1024}-\frac{1}{12} \sin ^5(x) \cos ^7(x)-\frac{1}{24} \sin ^3(x) \cos ^7(x)-\frac{1}{64} \sin (x) \cos ^7(x)+\frac{1}{384} \sin (x) \cos ^5(x)+\frac{5 \sin (x) \cos ^3(x)}{1536}+\frac{5 \sin (x) \cos (x)}{1024} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^6(x) \sin ^6(x) \, dx &=-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{5}{12} \int \cos ^6(x) \sin ^4(x) \, dx\\ &=-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{1}{8} \int \cos ^6(x) \sin ^2(x) \, dx\\ &=-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{1}{64} \int \cos ^6(x) \, dx\\ &=\frac{1}{384} \cos ^5(x) \sin (x)-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{5}{384} \int \cos ^4(x) \, dx\\ &=\frac{5 \cos ^3(x) \sin (x)}{1536}+\frac{1}{384} \cos ^5(x) \sin (x)-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{5}{512} \int \cos ^2(x) \, dx\\ &=\frac{5 \cos (x) \sin (x)}{1024}+\frac{5 \cos ^3(x) \sin (x)}{1536}+\frac{1}{384} \cos ^5(x) \sin (x)-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)+\frac{5 \int 1 \, dx}{1024}\\ &=\frac{5 x}{1024}+\frac{5 \cos (x) \sin (x)}{1024}+\frac{5 \cos ^3(x) \sin (x)}{1536}+\frac{1}{384} \cos ^5(x) \sin (x)-\frac{1}{64} \cos ^7(x) \sin (x)-\frac{1}{24} \cos ^7(x) \sin ^3(x)-\frac{1}{12} \cos ^7(x) \sin ^5(x)\\ \end{align*}
Mathematica [A] time = 0.0152357, size = 30, normalized size = 0.44 \[ \frac{5 x}{1024}-\frac{15 \sin (4 x)}{8192}+\frac{3 \sin (8 x)}{8192}-\frac{\sin (12 x)}{24576} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 52, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7} \left ( \sin \left ( x \right ) \right ) ^{5}}{12}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7} \left ( \sin \left ( x \right ) \right ) ^{3}}{24}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) }{64}}+{\frac{\sin \left ( x \right ) }{384} \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{5\,x}{1024}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.938378, size = 32, normalized size = 0.47 \begin{align*} \frac{1}{6144} \, \sin \left (4 \, x\right )^{3} + \frac{5}{1024} \, x + \frac{3}{8192} \, \sin \left (8 \, x\right ) - \frac{1}{512} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01354, size = 151, normalized size = 2.22 \begin{align*} -\frac{1}{3072} \,{\left (256 \, \cos \left (x\right )^{11} - 640 \, \cos \left (x\right )^{9} + 432 \, \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{5}{1024} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.06577, size = 46, normalized size = 0.68 \begin{align*} \frac{5 x}{1024} - \frac{\sin ^{5}{\left (2 x \right )} \cos{\left (2 x \right )}}{768} - \frac{5 \sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{3072} - \frac{5 \sin{\left (2 x \right )} \cos{\left (2 x \right )}}{2048} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05155, size = 30, normalized size = 0.44 \begin{align*} \frac{5}{1024} \, x - \frac{1}{24576} \, \sin \left (12 \, x\right ) + \frac{3}{8192} \, \sin \left (8 \, x\right ) - \frac{15}{8192} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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