Optimal. Leaf size=14 \[ \tan (x) \sqrt{\cot ^2(x)} \log (\sin (x)) \]
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Rubi [A] time = 0.0236293, 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 = {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 = 0.0059525, size = 14, normalized size = 1. \[ \tan (x) \sqrt{\cot ^2(x)} \log (\sin (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.137, size = 51, normalized size = 3.6 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{2\,\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 [A] time = 1.69768, size = 18, normalized size = 1.29 \begin{align*} -\frac{1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) + \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 = 0.48569, size = 24, normalized size = 1.71 \begin{align*} -\log \left (\frac{1}{2} \, \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{\csc ^{2}{\left (x \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39093, size = 59, normalized size = 4.21 \begin{align*} \frac{1}{2} \,{\left (2 \, \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 ) - \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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