Optimal. Leaf size=12 \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]
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Rubi [A] time = 0.0205382, antiderivative size = 12, 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 = {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 = 0.0092309, size = 12, normalized size = 1. \[ -\frac{\cot (x)}{\sqrt{\csc ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 13, 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 = 1.49327, size = 14, normalized size = 1.17 \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 [A] time = 1.744, size = 63, normalized size = 5.25 \begin{align*} -\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.723615, size = 14, normalized size = 1.17 \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 [B] time = 1.18212, size = 38, normalized size = 3.17 \begin{align*} \frac{2}{{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )} \mathrm{sgn}\left (\sin \left (x\right )\right )} + 2 \, \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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