Optimal. Leaf size=15 \[ -\frac {\cot (x) \log (\cos (x))}{\sqrt {\cot ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 = 15, 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 = 5, normalized size = 0.33 \[ \log \left (-\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.76, size = 22, normalized size = 1.47 \[ -\log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + \log \left ({\left | \tan \left (\frac {1}{2} \, x\right )^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.61, size = 68, normalized size = 4.53 \[ -\frac {\left (\ln \left (-\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}\right )+\ln \left (-\frac {-\sin \relax (x )-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (\frac {2}{\cos \relax (x )+1}\right )\right ) \cos \relax (x ) \sqrt {4}}{2 \sqrt {-\frac {\cos ^{2}\relax (x )}{-1+\cos ^{2}\relax (x )}}\, \sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 9, normalized size = 0.60 \[ \frac {1}{2} \, \log \left (\tan \relax (x)^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {\frac {1}{{\sin \relax (x)}^2}-1}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\csc ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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