Optimal. Leaf size=19 \[ \frac {\sin ^5(x)}{5}-\frac {2 \sin ^3(x)}{3}+\sin (x) \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2633} \[ \frac {\sin ^5(x)}{5}-\frac {2 \sin ^3(x)}{3}+\sin (x) \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin {align*} \int \cos ^5(x) \, dx &=-\operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (x)\right )\\ &=\sin (x)-\frac {2 \sin ^3(x)}{3}+\frac {\sin ^5(x)}{5}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.21 \[ \frac {5 \sin (x)}{8}+\frac {5}{48} \sin (3 x)+\frac {1}{80} \sin (5 x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^5(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.89, size = 18, normalized size = 0.95 \[ \frac {1}{15} \, {\left (3 \, \cos \relax (x)^{4} + 4 \, \cos \relax (x)^{2} + 8\right )} \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 15, normalized size = 0.79 \[ \frac {1}{5} \, \sin \relax (x)^{5} - \frac {2}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 17, normalized size = 0.89
method | result | size |
default | \(\frac {\left (\frac {8}{3}+\cos ^{4}\relax (x )+\frac {4 \left (\cos ^{2}\relax (x )\right )}{3}\right ) \sin \relax (x )}{5}\) | \(17\) |
risch | \(\frac {5 \sin \relax (x )}{8}+\frac {\sin \left (5 x \right )}{80}+\frac {5 \sin \left (3 x \right )}{48}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 15, normalized size = 0.79 \[ \frac {1}{5} \, \sin \relax (x)^{5} - \frac {2}{3} \, \sin \relax (x)^{3} + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 21, normalized size = 1.11 \[ \frac {\sin \relax (x)\,{\cos \relax (x)}^4}{5}+\frac {4\,\sin \relax (x)\,{\cos \relax (x)}^2}{15}+\frac {8\,\sin \relax (x)}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 17, normalized size = 0.89 \[ \frac {\sin ^{5}{\relax (x )}}{5} - \frac {2 \sin ^{3}{\relax (x )}}{3} + \sin {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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