Optimal. Leaf size=14 \[ \sqrt{-\sin ^2(x)} (-\cot (x)) \]
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Rubi [A] time = 0.0158867, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3176, 3207, 2638} \[ \sqrt{-\sin ^2(x)} (-\cot (x)) \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 2638
Rubi steps
\begin{align*} \int \sqrt{-1+\cos ^2(x)} \, dx &=\int \sqrt{-\sin ^2(x)} \, dx\\ &=\left (\csc (x) \sqrt{-\sin ^2(x)}\right ) \int \sin (x) \, dx\\ &=-\cot (x) \sqrt{-\sin ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0046514, size = 14, normalized size = 1. \[ \sqrt{-\sin ^2(x)} (-\cot (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.231, size = 14, normalized size = 1. \begin{align*}{\cos \left ( x \right ) \sin \left ( x \right ){\frac{1}{\sqrt{- \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 = 1.45688, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{\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 = 1.77924, size = 4, normalized size = 0.29 \begin{align*} 0 \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{\cos ^{2}{\left (x \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.1781, size = 38, normalized size = 2.71 \begin{align*} \frac{2 i \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{3} - \tan \left (\frac{1}{2} \, x\right )\right )}{\tan \left (\frac{1}{2} \, x\right )^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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