Optimal. Leaf size=14 \[ \frac{x}{2}-\frac{1}{2} \sin (x) \cos (x) \]
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Rubi [A] time = 0.0045568, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2635, 8} \[ \frac{x}{2}-\frac{1}{2} \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
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*}
Mathematica [A] time = 0.0015989, size = 14, normalized size = 1. \[ \frac{x}{2}-\frac{1}{4} \sin (2 x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 11, normalized size = 0.8 \begin{align*}{\frac{x}{2}}-{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.934652, size = 14, normalized size = 1. \begin{align*} \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98156, size = 38, normalized size = 2.71 \begin{align*} -\frac{1}{2} \, \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{2} \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.058461, size = 10, normalized size = 0.71 \begin{align*} \frac{x}{2} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07458, size = 14, normalized size = 1. \begin{align*} \frac{1}{2} \, x - \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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