Optimal. Leaf size=12 \[ \sqrt{\sin ^2(x)} (-\cot (x)) \]
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Rubi [A] time = 0.0158364, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 = 0.0061805, size = 12, normalized size = 1. \[ \sqrt{\sin ^2(x)} (-\cot (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.463, size = 13, normalized size = 1.1 \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 = 1.44539, size = 14, normalized size = 1.17 \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 = 1.89958, size = 12, normalized size = 1. \begin{align*} -\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{1 - \cos ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.271, size = 32, normalized size = 2.67 \begin{align*} -\frac{2 \, \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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