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 = 0.14, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 8
Rule 2568
Rule 2635
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*}
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Mathematica [A] time = 0.02, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^8(x) \sin ^8(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.78, size = 55, normalized size = 0.61 \[ \frac {1}{98304} \, {\left (6144 \, \cos \relax (x)^{15} - 21504 \, \cos \relax (x)^{13} + 25856 \, \cos \relax (x)^{11} - 10880 \, \cos \relax (x)^{9} + 48 \, \cos \relax (x)^{7} + 56 \, \cos \relax (x)^{5} + 70 \, \cos \relax (x)^{3} + 105 \, \cos \relax (x)\right )} \sin \relax (x) + \frac {35}{32768} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 28, normalized size = 0.31 \[ \frac {35}{32768} \, x + \frac {1}{524288} \, \sin \left (16 \, x\right ) - \frac {1}{49152} \, \sin \left (12 \, x\right ) + \frac {7}{65536} \, \sin \left (8 \, x\right ) - \frac {7}{16384} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 29, normalized size = 0.32
method | result | size |
risch | \(\frac {35 x}{32768}+\frac {\sin \left (16 x \right )}{524288}-\frac {\sin \left (12 x \right )}{49152}+\frac {7 \sin \left (8 x \right )}{65536}-\frac {7 \sin \left (4 x \right )}{16384}\) | \(29\) |
default | \(-\frac {\left (\cos ^{9}\relax (x )\right ) \left (\sin ^{7}\relax (x )\right )}{16}-\frac {\left (\cos ^{9}\relax (x )\right ) \left (\sin ^{5}\relax (x )\right )}{32}-\frac {5 \left (\cos ^{9}\relax (x )\right ) \left (\sin ^{3}\relax (x )\right )}{384}-\frac {\left (\cos ^{9}\relax (x )\right ) \sin \relax (x )}{256}+\frac {\left (\cos ^{7}\relax (x )+\frac {7 \left (\cos ^{5}\relax (x )\right )}{6}+\frac {35 \left (\cos ^{3}\relax (x )\right )}{24}+\frac {35 \cos \relax (x )}{16}\right ) \sin \relax (x )}{2048}+\frac {35 x}{32768}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 30, normalized size = 0.33 \[ \frac {1}{12288} \, \sin \left (4 \, x\right )^{3} + \frac {35}{32768} \, x + \frac {1}{524288} \, \sin \left (16 \, x\right ) + \frac {7}{65536} \, \sin \left (8 \, x\right ) - \frac {1}{2048} \, \sin \left (4 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 56, normalized size = 0.62 \[ \left (\frac {{\cos \relax (x)}^7}{16}+\frac {{\cos \relax (x)}^5}{32}+\frac {5\,{\cos \relax (x)}^3}{384}+\frac {\cos \relax (x)}{256}\right )\,{\sin \relax (x)}^9+\frac {35\,x}{32768}-\frac {7\,\sin \left (2\,x\right )}{8192}+\frac {7\,\sin \left (4\,x\right )}{32768}-\frac {\sin \left (6\,x\right )}{24576}+\frac {\sin \left (8\,x\right )}{262144} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 61, normalized size = 0.68 \[ \frac {35 x}{32768} - \frac {\sin ^{7}{\left (2 x \right )} \cos {\left (2 x \right )}}{4096} - \frac {7 \sin ^{5}{\left (2 x \right )} \cos {\left (2 x \right )}}{24576} - \frac {35 \sin ^{3}{\left (2 x \right )} \cos {\left (2 x \right )}}{98304} - \frac {35 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{65536} \]
Verification of antiderivative is not currently implemented for this CAS.
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