Optimal. Leaf size=90 \[ \frac{35 x}{32768}-\frac{1}{16} \sin ^7(x) \cos ^9(x)-\frac{1}{32} \sin ^5(x) \cos ^9(x)-\frac{5}{384} \sin ^3(x) \cos ^9(x)-\frac{1}{256} \sin (x) \cos ^9(x)+\frac{\sin (x) \cos ^7(x)}{2048}+\frac{7 \sin (x) \cos ^5(x)}{12288}+\frac{35 \sin (x) \cos ^3(x)}{49152}+\frac{35 \sin (x) \cos (x)}{32768} \]
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Rubi [A] time = 0.136664, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2568, 2635, 8} \[ \frac{35 x}{32768}-\frac{1}{16} \sin ^7(x) \cos ^9(x)-\frac{1}{32} \sin ^5(x) \cos ^9(x)-\frac{5}{384} \sin ^3(x) \cos ^9(x)-\frac{1}{256} \sin (x) \cos ^9(x)+\frac{\sin (x) \cos ^7(x)}{2048}+\frac{7 \sin (x) \cos ^5(x)}{12288}+\frac{35 \sin (x) \cos ^3(x)}{49152}+\frac{35 \sin (x) \cos (x)}{32768} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^8(x) \sin ^8(x) \, dx &=-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{7}{16} \int \cos ^8(x) \sin ^6(x) \, dx\\ &=-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{5}{32} \int \cos ^8(x) \sin ^4(x) \, dx\\ &=-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{5}{128} \int \cos ^8(x) \sin ^2(x) \, dx\\ &=-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{1}{256} \int \cos ^8(x) \, dx\\ &=\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{7 \int \cos ^6(x) \, dx}{2048}\\ &=\frac{7 \cos ^5(x) \sin (x)}{12288}+\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{35 \int \cos ^4(x) \, dx}{12288}\\ &=\frac{35 \cos ^3(x) \sin (x)}{49152}+\frac{7 \cos ^5(x) \sin (x)}{12288}+\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{35 \int \cos ^2(x) \, dx}{16384}\\ &=\frac{35 \cos (x) \sin (x)}{32768}+\frac{35 \cos ^3(x) \sin (x)}{49152}+\frac{7 \cos ^5(x) \sin (x)}{12288}+\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)+\frac{35 \int 1 \, dx}{32768}\\ &=\frac{35 x}{32768}+\frac{35 \cos (x) \sin (x)}{32768}+\frac{35 \cos ^3(x) \sin (x)}{49152}+\frac{7 \cos ^5(x) \sin (x)}{12288}+\frac{\cos ^7(x) \sin (x)}{2048}-\frac{1}{256} \cos ^9(x) \sin (x)-\frac{5}{384} \cos ^9(x) \sin ^3(x)-\frac{1}{32} \cos ^9(x) \sin ^5(x)-\frac{1}{16} \cos ^9(x) \sin ^7(x)\\ \end{align*}
Mathematica [A] time = 0.0148192, size = 38, normalized size = 0.42 \[ \frac{35 x}{32768}-\frac{7 \sin (4 x)}{16384}+\frac{7 \sin (8 x)}{65536}-\frac{\sin (12 x)}{49152}+\frac{\sin (16 x)}{524288} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 68, normalized size = 0.8 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{9} \left ( \sin \left ( x \right ) \right ) ^{7}}{16}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{9} \left ( \sin \left ( x \right ) \right ) ^{5}}{32}}-{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{9} \left ( \sin \left ( x \right ) \right ) ^{3}}{384}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{9}\sin \left ( x \right ) }{256}}+{\frac{\sin \left ( x \right ) }{2048} \left ( \left ( \cos \left ( x \right ) \right ) ^{7}+{\frac{7\, \left ( \cos \left ( x \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \cos \left ( x \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( x \right ) }{16}} \right ) }+{\frac{35\,x}{32768}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944744, size = 41, normalized size = 0.46 \begin{align*} \frac{1}{12288} \, \sin \left (4 \, x\right )^{3} + \frac{35}{32768} \, x + \frac{1}{524288} \, \sin \left (16 \, x\right ) + \frac{7}{65536} \, \sin \left (8 \, x\right ) - \frac{1}{2048} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00414, size = 208, normalized size = 2.31 \begin{align*} \frac{1}{98304} \,{\left (6144 \, \cos \left (x\right )^{15} - 21504 \, \cos \left (x\right )^{13} + 25856 \, \cos \left (x\right )^{11} - 10880 \, \cos \left (x\right )^{9} + 48 \, \cos \left (x\right )^{7} + 56 \, \cos \left (x\right )^{5} + 70 \, \cos \left (x\right )^{3} + 105 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{35}{32768} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.069368, size = 61, normalized size = 0.68 \begin{align*} \frac{35 x}{32768} - \frac{\sin ^{7}{\left (2 x \right )} \cos{\left (2 x \right )}}{4096} - \frac{7 \sin ^{5}{\left (2 x \right )} \cos{\left (2 x \right )}}{24576} - \frac{35 \sin ^{3}{\left (2 x \right )} \cos{\left (2 x \right )}}{98304} - \frac{35 \sin{\left (2 x \right )} \cos{\left (2 x \right )}}{65536} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05709, size = 38, normalized size = 0.42 \begin{align*} \frac{35}{32768} \, x + \frac{1}{524288} \, \sin \left (16 \, x\right ) - \frac{1}{49152} \, \sin \left (12 \, x\right ) + \frac{7}{65536} \, \sin \left (8 \, x\right ) - \frac{7}{16384} \, \sin \left (4 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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