Optimal. Leaf size=13 \[ \tan (x) \sqrt{a \cos ^2(x)} \]
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Rubi [A] time = 0.0259045, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3176, 3207, 2637} \[ \tan (x) \sqrt{a \cos ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 2637
Rubi steps
\begin{align*} \int \sqrt{a-a \sin ^2(x)} \, dx &=\int \sqrt{a \cos ^2(x)} \, dx\\ &=\left (\sqrt{a \cos ^2(x)} \sec (x)\right ) \int \cos (x) \, dx\\ &=\sqrt{a \cos ^2(x)} \tan (x)\\ \end{align*}
Mathematica [A] time = 0.0042642, size = 13, normalized size = 1. \[ \tan (x) \sqrt{a \cos ^2(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.392, size = 15, normalized size = 1.2 \begin{align*}{a\cos \left ( x \right ) \sin \left ( x \right ){\frac{1}{\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59161, size = 8, normalized size = 0.62 \begin{align*} \sqrt{a} \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62571, size = 43, normalized size = 3.31 \begin{align*} \frac{\sqrt{a \cos \left (x\right )^{2}} \sin \left (x\right )}{\cos \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- a \sin ^{2}{\left (x \right )} + a}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14487, size = 36, normalized size = 2.77 \begin{align*} -\frac{2 \, \sqrt{a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, x\right )^{4} - 1\right )}{\frac{1}{\tan \left (\frac{1}{2} \, x\right )} + \tan \left (\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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