Optimal. Leaf size=17 \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {-\cot ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4121, 3658, 3475} \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {-\cot ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rule 4121
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*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {-\cot ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 12, normalized size = 0.71 \[ x - \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.64, size = 24, normalized size = 1.41 \[ -i \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + i \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right )^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.69, size = 67, normalized size = 3.94 \[ -\frac {\left (\ln \left (-\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}\right )+\ln \left (-\frac {-\sin \relax (x )-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (\frac {2}{\cos \relax (x )+1}\right )\right ) \cos \relax (x ) \sqrt {4}}{2 \sqrt {\frac {\cos ^{2}\relax (x )}{-1+\cos ^{2}\relax (x )}}\, \sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 9, normalized size = 0.53 \[ -\frac {1}{2} i \, \log \left (\tan \relax (x)^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {1-\frac {1}{{\sin \relax (x)}^2}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - \csc ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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