Optimal. Leaf size=16 \[ \frac {a x}{2}+\frac {1}{2} a \sin (x) \cos (x) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 8
Rule 2635
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*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ a \left (\frac {x}{2}+\frac {1}{4} \sin (2 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 12, normalized size = 0.75 \[ \frac {1}{2} \, a \cos \relax (x) \sin \relax (x) + \frac {1}{2} \, a x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 17, normalized size = 1.06 \[ -\frac {1}{4} \, a {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 18, normalized size = 1.12 \[ a x -a \left (-\frac {\sin \relax (x ) \cos \relax (x )}{2}+\frac {x}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 17, normalized size = 1.06 \[ -\frac {1}{4} \, a {\left (2 \, x - \sin \left (2 \, x\right )\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.58, size = 11, normalized size = 0.69 \[ \frac {a\,\left (2\,x+\sin \left (2\,x\right )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 15, normalized size = 0.94 \[ a x - a \left (\frac {x}{2} - \frac {\sin {\relax (x )} \cos {\relax (x )}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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