Optimal. Leaf size=28 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0191643, antiderivative size = 28, 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 = {3661, 377, 203} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 3661
Rule 377
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-\cot ^2(x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{\cot (x)}{\sqrt{1-\cot ^2(x)}}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0632248, size = 42, normalized size = 1.5 \[ -\frac{\sqrt{\cos (2 x)} \csc (x) \log \left (\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right )}{\sqrt{2-2 \cot ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 31, normalized size = 1.1 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{\sqrt{2}\cot \left ( x \right ) }{-1+ \left ( \cot \left ( x \right ) \right ) ^{2}}\sqrt{1- \left ( \cot \left ( x \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.67901, size = 122, normalized size = 4.36 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left ({\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac{1}{4}} \sin \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (2 \, x\right ),{\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac{1}{4}} \cos \left (\frac{1}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) + \cos \left (2 \, x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.16573, size = 171, normalized size = 6.11 \begin{align*} \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (2 \, \sqrt{2} \cos \left (2 \, x\right ) + \sqrt{2}\right )} \sqrt{\frac{\cos \left (2 \, x\right )}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right )}{4 \,{\left (\cos \left (2 \, x\right )^{2} + \cos \left (2 \, x\right )\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{1 - \cot ^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.43033, size = 46, normalized size = 1.64 \begin{align*} -\frac{1}{2} i \, \sqrt{2} \log \left (i \, \sqrt{2} + i\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) - \frac{\sqrt{2} \arcsin \left (\sqrt{2} \cos \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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