Optimal. Leaf size=8 \[ -\frac {1}{4} \cos ^4(x) \]
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Rubi [A] time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2565, 30} \[ -\frac {1}{4} \cos ^4(x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2565
Rubi steps
\begin {align*} \int \cos ^3(x) \sin (x) \, dx &=-\operatorname {Subst}\left (\int x^3 \, dx,x,\cos (x)\right )\\ &=-\frac {1}{4} \cos ^4(x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 8, normalized size = 1.00 \[ -\frac {1}{4} \cos ^4(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^3(x) \sin (x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.33, size = 6, normalized size = 0.75 \[ -\frac {1}{4} \, \cos \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 6, normalized size = 0.75 \[ -\frac {1}{4} \, \cos \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 7, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {\left (\cos ^{4}\relax (x )\right )}{4}\) | \(7\) |
default | \(-\frac {\left (\cos ^{4}\relax (x )\right )}{4}\) | \(7\) |
risch | \(-\frac {\cos \left (4 x \right )}{32}-\frac {\cos \left (2 x \right )}{8}\) | \(14\) |
norman | \(\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4}}\) | \(29\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (2 x \right )}{\sqrt {\pi }}\right )}{8}+\frac {\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (4 x \right )}{\sqrt {\pi }}\right )}{32}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 6, normalized size = 0.75 \[ -\frac {1}{4} \, \cos \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 12, normalized size = 1.50 \[ -\frac {{\sin \relax (x)}^2\,\left ({\sin \relax (x)}^2-2\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.06, size = 7, normalized size = 0.88 \[ - \frac {\cos ^{4}{\relax (x )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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