Optimal. Leaf size=14 \[ \frac {x}{2}-\frac {1}{2} \cos (x) \sin (x) \]
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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2715, 8}
\begin {gather*} \frac {x}{2}-\frac {1}{2} \sin (x) \cos (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rubi steps
\begin {align*} \int \sin ^2(x) \, dx &=-\frac {1}{2} \cos (x) \sin (x)+\frac {\int 1 \, dx}{2}\\ &=\frac {x}{2}-\frac {1}{2} \cos (x) \sin (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{2}-\frac {1}{4} \sin (2 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 11, normalized size = 0.79
method | result | size |
default | \(\frac {x}{2}-\frac {\cos \left (x \right ) \sin \left (x \right )}{2}\) | \(11\) |
risch | \(\frac {x}{2}-\frac {\sin \left (2 x \right )}{4}\) | \(11\) |
meijerg | \(\frac {\sqrt {\pi }\, \left (\frac {2 x}{\sqrt {\pi }}-\frac {\sin \left (2 x \right )}{\sqrt {\pi }}\right )}{4}\) | \(22\) |
norman | \(\frac {\tan ^{3}\left (\frac {x}{2}\right )+x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {x}{2}+\frac {x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}-\tan \left (\frac {x}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2}}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.54, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, x - \frac {1}{4} \, \sin \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.59, size = 10, normalized size = 0.71 \begin {gather*} -\frac {1}{2} \, \cos \left (x\right ) \sin \left (x\right ) + \frac {1}{2} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{2} - \frac {\sin {\left (x \right )} \cos {\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.91, size = 10, normalized size = 0.71 \begin {gather*} \frac {1}{2} \, x - \frac {1}{4} \, \sin \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{2}-\frac {\sin \left (2\,x\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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