Optimal. Leaf size=14 \[ -\cot (x) \sqrt{a \sin ^2(x)} \]
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Rubi [A] time = 0.0098667, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 2638} \[ -\cot (x) \sqrt{a \sin ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2638
Rubi steps
\begin{align*} \int \sqrt{a \sin ^2(x)} \, dx &=\left (\csc (x) \sqrt{a \sin ^2(x)}\right ) \int \sin (x) \, dx\\ &=-\cot (x) \sqrt{a \sin ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0043304, size = 14, normalized size = 1. \[ -\cot (x) \sqrt{a \sin ^2(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.294, size = 16, normalized size = 1.1 \begin{align*} -{a\cos \left ( x \right ) \sin \left ( x \right ){\frac{1}{\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42444, size = 18, normalized size = 1.29 \begin{align*} -\frac{\sqrt{a}}{\sqrt{\tan \left (x\right )^{2} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5783, size = 51, normalized size = 3.64 \begin{align*} -\frac{\sqrt{-a \cos \left (x\right )^{2} + a} \cos \left (x\right )}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.719972, size = 20, normalized size = 1.43 \begin{align*} - \frac{\sqrt{a} \sqrt{\sin ^{2}{\left (x \right )}} \cos{\left (x \right )}}{\sin{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24183, size = 23, normalized size = 1.64 \begin{align*} -{\left (\cos \left (x\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) - \mathrm{sgn}\left (\sin \left (x\right )\right )\right )} \sqrt{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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