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 = {2645, 30}
\begin {gather*} -\frac {1}{4} \cos ^4(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2645
Rubi steps
\begin {align*} \int \cos ^3(x) \sin (x) \, dx &=-\text {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 \begin {gather*} -\frac {1}{4} \cos ^4(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 7, normalized size = 0.88
method | result | size |
derivativedivides | \(-\frac {\left (\cos ^{4}\left (x \right )\right )}{4}\) | \(7\) |
default | \(-\frac {\left (\cos ^{4}\left (x \right )\right )}{4}\) | \(7\) |
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 = 1.97, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{4} \, \cos \left (x\right )^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{4} \, \cos \left (x\right )^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 7, normalized size = 0.88 \begin {gather*} - \frac {\cos ^{4}{\left (x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{4} \, \cos \left (x\right )^{4} \end {gather*}
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 \begin {gather*} -\frac {{\sin \left (x\right )}^2\,\left ({\sin \left (x\right )}^2-2\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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