Optimal. Leaf size=13 \[ \tan (x) \sqrt{a \cos ^2(x)} \]
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Rubi [A] time = 0.0112618, antiderivative size = 13, 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, 2637} \[ \tan (x) \sqrt{a \cos ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2637
Rubi steps
\begin{align*} \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 = 0.0041028, size = 13, normalized size = 1. \[ \tan (x) \sqrt{a \cos ^2(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.337, 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 = 2.32193, 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 = 1.0715, 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 [A] time = 0.576638, size = 19, normalized size = 1.46 \begin{align*} \frac{\sqrt{a} \sqrt{\cos ^{2}{\left (x \right )}} \sin{\left (x \right )}}{\cos{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23228, size = 12, normalized size = 0.92 \begin{align*} \sqrt{a} \mathrm{sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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