Optimal. Leaf size=5 \[ -\sinh ^{-1}(\cot (x)) \]
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Rubi [A] time = 0.0066315, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4122, 215} \[ -\sinh ^{-1}(\cot (x)) \]
Antiderivative was successfully verified.
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Rule 4122
Rule 215
Rubi steps
\begin{align*} \int \sqrt{\csc ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\cot (x)\right )\\ &=-\sinh ^{-1}(\cot (x))\\ \end{align*}
Mathematica [B] time = 0.0133144, size = 28, normalized size = 5.6 \[ \sin (x) \sqrt{\csc ^2(x)} \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.181, size = 31, normalized size = 6.2 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{2}\sqrt{- \left ( \left ( \cos \left ( x \right ) \right ) ^{2}-1 \right ) ^{-1}}\ln \left ( -{\frac{-1+\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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Maxima [B] time = 1.64639, size = 47, normalized size = 9.4 \begin{align*} -\frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.470925, size = 77, normalized size = 15.4 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\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 )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24539, size = 16, normalized size = 3.2 \begin{align*} \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \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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