Optimal. Leaf size=68 \[ \frac {5 x}{1024}+\frac {5 \cos (x) \sin (x)}{1024}+\frac {5 \cos ^3(x) \sin (x)}{1536}+\frac {1}{384} \cos ^5(x) \sin (x)-\frac {1}{64} \cos ^7(x) \sin (x)-\frac {1}{24} \cos ^7(x) \sin ^3(x)-\frac {1}{12} \cos ^7(x) \sin ^5(x) \]
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Rubi [A]
time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 = {2648, 2715, 8}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2715
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*}
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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.44 \begin {gather*} \frac {5 x}{1024}-\frac {15 \sin (4 x)}{8192}+\frac {3 \sin (8 x)}{8192}-\frac {\sin (12 x)}{24576} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 52, normalized size = 0.76
method | result | size |
risch | \(\frac {5 x}{1024}-\frac {\sin \left (12 x \right )}{24576}+\frac {3 \sin \left (8 x \right )}{8192}-\frac {15 \sin \left (4 x \right )}{8192}\) | \(23\) |
default | \(-\frac {\left (\cos ^{7}\left (x \right )\right ) \left (\sin ^{5}\left (x \right )\right )}{12}-\frac {\left (\cos ^{7}\left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{24}-\frac {\left (\cos ^{7}\left (x \right )\right ) \sin \left (x \right )}{64}+\frac {\left (\cos ^{5}\left (x \right )+\frac {5 \left (\cos ^{3}\left (x \right )\right )}{4}+\frac {15 \cos \left (x \right )}{8}\right ) \sin \left (x \right )}{384}+\frac {5 x}{1024}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.59, size = 24, normalized size = 0.35 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 43, normalized size = 0.63 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 46, normalized size = 0.68 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.71, size = 22, normalized size = 0.32 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 44, normalized size = 0.65 \begin {gather*} \left (\frac {{\cos \left (x\right )}^5}{12}+\frac {{\cos \left (x\right )}^3}{24}+\frac {\cos \left (x\right )}{64}\right )\,{\sin \left (x\right )}^7+\frac {5\,x}{1024}-\frac {15\,\sin \left (2\,x\right )}{4096}+\frac {3\,\sin \left (4\,x\right )}{4096}-\frac {\sin \left (6\,x\right )}{12288} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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