Optimal. Leaf size=33 \[ \frac {\cos ^{13}(x)}{13}-\frac {3 \cos ^{11}(x)}{11}+\frac {\cos ^9(x)}{3}-\frac {\cos ^7(x)}{7} \]
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Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2565, 270} \[ \frac {\cos ^{13}(x)}{13}-\frac {3 \cos ^{11}(x)}{11}+\frac {\cos ^9(x)}{3}-\frac {\cos ^7(x)}{7} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2565
Rubi steps
\begin {align*} \int \cos ^6(x) \sin ^7(x) \, dx &=-\operatorname {Subst}\left (\int x^6 \left (1-x^2\right )^3 \, dx,x,\cos (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (x^6-3 x^8+3 x^{10}-x^{12}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac {1}{7} \cos ^7(x)+\frac {\cos ^9(x)}{3}-\frac {3 \cos ^{11}(x)}{11}+\frac {\cos ^{13}(x)}{13}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 1.67 \[ -\frac {5 \cos (x)}{1024}-\frac {5 \cos (3 x)}{4096}+\frac {3 \cos (5 x)}{4096}+\frac {3 \cos (7 x)}{14336}-\frac {\cos (9 x)}{6144}-\frac {\cos (11 x)}{45056}+\frac {\cos (13 x)}{53248} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^6(x) \sin ^7(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.92, size = 25, normalized size = 0.76 \[ \frac {1}{13} \, \cos \relax (x)^{13} - \frac {3}{11} \, \cos \relax (x)^{11} + \frac {1}{3} \, \cos \relax (x)^{9} - \frac {1}{7} \, \cos \relax (x)^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.83, size = 25, normalized size = 0.76 \[ \frac {1}{13} \, \cos \relax (x)^{13} - \frac {3}{11} \, \cos \relax (x)^{11} + \frac {1}{3} \, \cos \relax (x)^{9} - \frac {1}{7} \, \cos \relax (x)^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 1.15
method | result | size |
default | \(-\frac {\left (\cos ^{7}\relax (x )\right ) \left (\sin ^{6}\relax (x )\right )}{13}-\frac {6 \left (\sin ^{4}\relax (x )\right ) \left (\cos ^{7}\relax (x )\right )}{143}-\frac {8 \left (\sin ^{2}\relax (x )\right ) \left (\cos ^{7}\relax (x )\right )}{429}-\frac {16 \left (\cos ^{7}\relax (x )\right )}{3003}\) | \(38\) |
risch | \(-\frac {5 \cos \relax (x )}{1024}+\frac {\cos \left (13 x \right )}{53248}-\frac {\cos \left (11 x \right )}{45056}-\frac {\cos \left (9 x \right )}{6144}+\frac {3 \cos \left (7 x \right )}{14336}+\frac {3 \cos \left (5 x \right )}{4096}-\frac {5 \cos \left (3 x \right )}{4096}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 25, normalized size = 0.76 \[ \frac {1}{13} \, \cos \relax (x)^{13} - \frac {3}{11} \, \cos \relax (x)^{11} + \frac {1}{3} \, \cos \relax (x)^{9} - \frac {1}{7} \, \cos \relax (x)^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 25, normalized size = 0.76 \[ \frac {{\cos \relax (x)}^{13}}{13}-\frac {3\,{\cos \relax (x)}^{11}}{11}+\frac {{\cos \relax (x)}^9}{3}-\frac {{\cos \relax (x)}^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 27, normalized size = 0.82 \[ \frac {\cos ^{13}{\relax (x )}}{13} - \frac {3 \cos ^{11}{\relax (x )}}{11} + \frac {\cos ^{9}{\relax (x )}}{3} - \frac {\cos ^{7}{\relax (x )}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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