Optimal. Leaf size=8 \[ \frac {\sin ^2(x)}{2} \]
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Rubi [A]
time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2644, 30}
\begin {gather*} \frac {\sin ^2(x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2644
Rubi steps
\begin {align*} \int \cos (x) \sin (x) \, dx &=\text {Subst}(\int x \, dx,x,\sin (x))\\ &=\frac {\sin ^2(x)}{2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \cos ^2(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 7, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\left (\sin ^{2}\left (x \right )\right )}{2}\) | \(7\) |
default | \(\frac {\left (\sin ^{2}\left (x \right )\right )}{2}\) | \(7\) |
risch | \(-\frac {\cos \left (2 x \right )}{4}\) | \(7\) |
norman | \(\frac {2 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}\) | \(19\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (2 x \right )}{\sqrt {\pi }}\right )}{4}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.40, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{2} \, \cos \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{2} \, \cos \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 5, normalized size = 0.62 \begin {gather*} \frac {\sin ^{2}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 6, normalized size = 0.75 \begin {gather*} -\frac {1}{2} \, \cos \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 6, normalized size = 0.75 \begin {gather*} \frac {{\sin \left (x\right )}^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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