Optimal. Leaf size=14 \[ -\frac{\cot (x)}{\sqrt{-\csc ^2(x)}} \]
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Rubi [A] time = 0.0226727, antiderivative size = 14, 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 = {3657, 4122, 191} \[ -\frac{\cot (x)}{\sqrt{-\csc ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-\cot ^2(x)}} \, dx &=\int \frac{1}{\sqrt{-\csc ^2(x)}} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{\left (-1-x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.006912, size = 14, normalized size = 1. \[ -\frac{\cot (x)}{\sqrt{-\csc ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 15, normalized size = 1.1 \begin{align*} -{\cot \left ( x \right ){\frac{1}{\sqrt{-1- \left ( \cot \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46695, size = 16, normalized size = 1.14 \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 [C] time = 1.50184, size = 45, normalized size = 3.21 \begin{align*} \frac{1}{2} \,{\left (-i \, e^{\left (2 i \, x\right )} - i\right )} e^{\left (-i \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.547312, size = 15, normalized size = 1.07 \begin{align*} - \frac{\cot{\left (x \right )}}{\sqrt{- \cot ^{2}{\left (x \right )} - 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.17405, size = 38, normalized size = 2.71 \begin{align*} -\frac{2 i}{{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )} - 2 i \, \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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