Optimal. Leaf size=33 \[ -\frac {1}{7} \cos ^7(x)+\frac {\cos ^9(x)}{3}-\frac {3 \cos ^{11}(x)}{11}+\frac {\cos ^{13}(x)}{13} \]
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Rubi [A]
time = 0.02, 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 = {2645, 276}
\begin {gather*} \frac {\cos ^{13}(x)}{13}-\frac {3 \cos ^{11}(x)}{11}+\frac {\cos ^9(x)}{3}-\frac {\cos ^7(x)}{7} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2645
Rubi steps
\begin {align*} \int \cos ^6(x) \sin ^7(x) \, dx &=-\text {Subst}\left (\int x^6 \left (1-x^2\right )^3 \, dx,x,\cos (x)\right )\\ &=-\text {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 = 0.03, size = 55, normalized size = 1.67 \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 38, normalized size = 1.15
method | result | size |
default | \(-\frac {\left (\cos ^{7}\left (x \right )\right ) \left (\sin ^{6}\left (x \right )\right )}{13}-\frac {6 \left (\sin ^{4}\left (x \right )\right ) \left (\cos ^{7}\left (x \right )\right )}{143}-\frac {8 \left (\sin ^{2}\left (x \right )\right ) \left (\cos ^{7}\left (x \right )\right )}{429}-\frac {16 \left (\cos ^{7}\left (x \right )\right )}{3003}\) | \(38\) |
risch | \(-\frac {5 \cos \left (x \right )}{1024}+\frac {\cos \left (13 x \right )}{53248}-\frac {\cos \left (11 x \right )}{45056}-\frac {\cos \left (9 x \right )}{6144}+\frac {3 \cos \left (7 x \right )}{14336}+\frac {3 \cos \left (5 x \right )}{4096}-\frac {5 \cos \left (3 x \right )}{4096}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.20, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{13} \, \cos \left (x\right )^{13} - \frac {3}{11} \, \cos \left (x\right )^{11} + \frac {1}{3} \, \cos \left (x\right )^{9} - \frac {1}{7} \, \cos \left (x\right )^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.05, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{13} \, \cos \left (x\right )^{13} - \frac {3}{11} \, \cos \left (x\right )^{11} + \frac {1}{3} \, \cos \left (x\right )^{9} - \frac {1}{7} \, \cos \left (x\right )^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 27, normalized size = 0.82 \begin {gather*} \frac {\cos ^{13}{\left (x \right )}}{13} - \frac {3 \cos ^{11}{\left (x \right )}}{11} + \frac {\cos ^{9}{\left (x \right )}}{3} - \frac {\cos ^{7}{\left (x \right )}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.28, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{13} \, \cos \left (x\right )^{13} - \frac {3}{11} \, \cos \left (x\right )^{11} + \frac {1}{3} \, \cos \left (x\right )^{9} - \frac {1}{7} \, \cos \left (x\right )^{7} \end {gather*}
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 \begin {gather*} \frac {{\cos \left (x\right )}^{13}}{13}-\frac {3\,{\cos \left (x\right )}^{11}}{11}+\frac {{\cos \left (x\right )}^9}{3}-\frac {{\cos \left (x\right )}^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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