Optimal. Leaf size=15 \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{\cot ^2(x)}} \]
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Rubi [A] time = 0.0185607, antiderivative size = 15, 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} \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{\cot ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 4121
Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+\csc ^2(x)}} \, dx &=\int \frac{1}{\sqrt{\cot ^2(x)}} \, dx\\ &=\frac{\cot (x) \int \tan (x) \, dx}{\sqrt{\cot ^2(x)}}\\ &=-\frac{\cot (x) \log (\cos (x))}{\sqrt{\cot ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0087518, size = 15, normalized size = 1. \[ -\frac{\cot (x) \log (\cos (x))}{\sqrt{\cot ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.134, size = 68, normalized size = 4.5 \begin{align*} -{\frac{\sqrt{4}\cos \left ( x \right ) }{2\,\sin \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 ) }{\sin \left ( x \right ) }} \right ) +\ln \left ( -{\frac{-1+\cos \left ( x \right ) +\sin \left ( x \right ) }{\sin \left ( x \right ) }} \right ) \right ){\frac{1}{\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 = 1.68719, size = 12, normalized size = 0.8 \begin{align*} \frac{1}{2} \, \log \left (\tan \left (x\right )^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.485637, size = 19, normalized size = 1.27 \begin{align*} \log \left (-\cos \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 \frac{1}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\csc \left (x\right )^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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