Optimal. Leaf size=90 \[ \frac {35 x}{32768}+\frac {35 \cos (x) \sin (x)}{32768}+\frac {35 \cos ^3(x) \sin (x)}{49152}+\frac {7 \cos ^5(x) \sin (x)}{12288}+\frac {\cos ^7(x) \sin (x)}{2048}-\frac {1}{256} \cos ^9(x) \sin (x)-\frac {5}{384} \cos ^9(x) \sin ^3(x)-\frac {1}{32} \cos ^9(x) \sin ^5(x)-\frac {1}{16} \cos ^9(x) \sin ^7(x) \]
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Rubi [A]
time = 0.08, 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 = {2648, 2715, 8}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2715
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 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 68, normalized size = 0.76
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 | \(-\frac {\left (\cos ^{9}\left (x \right )\right ) \left (\sin ^{7}\left (x \right )\right )}{16}-\frac {\left (\cos ^{9}\left (x \right )\right ) \left (\sin ^{5}\left (x \right )\right )}{32}-\frac {5 \left (\cos ^{9}\left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{384}-\frac {\left (\cos ^{9}\left (x \right )\right ) \sin \left (x \right )}{256}+\frac {\left (\cos ^{7}\left (x \right )+\frac {7 \left (\cos ^{5}\left (x \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (x \right )\right )}{24}+\frac {35 \cos \left (x \right )}{16}\right ) \sin \left (x \right )}{2048}+\frac {35 x}{32768}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.18, size = 30, normalized size = 0.33 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.85, size = 55, normalized size = 0.61 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 61, normalized size = 0.68 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 28, normalized size = 0.31 \begin {gather*} \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 {gather*}
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 \begin {gather*} \left (\frac {{\cos \left (x\right )}^7}{16}+\frac {{\cos \left (x\right )}^5}{32}+\frac {5\,{\cos \left (x\right )}^3}{384}+\frac {\cos \left (x\right )}{256}\right )\,{\sin \left (x\right )}^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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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