Optimal. Leaf size=16 \[ \tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x)) \]
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Rubi [A] time = 0.0276965, antiderivative size = 16, 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 = {4121, 3658, 3475} \[ \tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \sqrt{1-\csc ^2(x)} \, dx &=\int \sqrt{-\cot ^2(x)} \, dx\\ &=\left (\sqrt{-\cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=\sqrt{-\cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end{align*}
Mathematica [A] time = 0.005678, size = 16, normalized size = 1. \[ \tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.125, size = 46, normalized size = 2.9 \begin{align*}{\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) } \left ( -\ln \left ( 2\, \left ( \cos \left ( x \right ) +1 \right ) ^{-1} \right ) +\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ) \sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{ \left ( \cos \left ( x \right ) \right ) ^{2}-1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.51252, size = 20, normalized size = 1.25 \begin{align*} -\frac{1}{2} i \, \log \left (\tan \left (x\right )^{2} + 1\right ) + i \, \log \left (\tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.489414, size = 36, normalized size = 2.25 \begin{align*} x + \arctan \left (\frac{\cos \left (x\right )}{\sin \left (x\right )}\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 - \csc ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.63763, size = 65, normalized size = 4.06 \begin{align*} -\frac{1}{2} \,{\left (2 i \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{4} + 1\right ) - i \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2}\right ) \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{4} + 1\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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