Optimal. Leaf size=16 \[ \frac{a x}{2}+\frac{1}{2} a \sin (x) \cos (x) \]
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Rubi [A] time = 0.0085589, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2635, 8} \[ \frac{a x}{2}+\frac{1}{2} a \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (a-a \sin ^2(x)\right ) \, dx &=a x-a \int \sin ^2(x) \, dx\\ &=a x+\frac{1}{2} a \cos (x) \sin (x)-\frac{1}{2} a \int 1 \, dx\\ &=\frac{a x}{2}+\frac{1}{2} a \cos (x) \sin (x)\\ \end{align*}
Mathematica [A] time = 0.0028456, size = 16, normalized size = 1. \[ a \left (\frac{x}{2}+\frac{1}{4} \sin (2 x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 18, normalized size = 1.1 \begin{align*} ax-a \left ( -{\frac{\sin \left ( x \right ) \cos \left ( x \right ) }{2}}+{\frac{x}{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97762, size = 23, normalized size = 1.44 \begin{align*} -\frac{1}{4} \, a{\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64897, size = 42, normalized size = 2.62 \begin{align*} \frac{1}{2} \, a \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{2} \, a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.155767, size = 15, normalized size = 0.94 \begin{align*} a x - a \left (\frac{x}{2} - \frac{\sin{\left (x \right )} \cos{\left (x \right )}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1417, size = 23, normalized size = 1.44 \begin{align*} -\frac{1}{4} \, a{\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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