Optimal. Leaf size=14 \[ \tan (x) \sqrt {\cot ^2(x)} \log (\sin (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 14, 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} \[ \tan (x) \sqrt {\cot ^2(x)} \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rule 4121
Rubi steps
\begin {align*} \int \sqrt {-1+\csc ^2(x)} \, dx &=\int \sqrt {\cot ^2(x)} \, dx\\ &=\left (\sqrt {\cot ^2(x)} \tan (x)\right ) \int \cot (x) \, dx\\ &=\sqrt {\cot ^2(x)} \log (\sin (x)) \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ \tan (x) \sqrt {\cot ^2(x)} \log (\sin (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 7, normalized size = 0.50 \[ -\log \left (\frac {1}{2} \, \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 44, normalized size = 3.14 \[ \frac {1}{2} \, {\left (2 \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right ) - \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 1\right )\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.68, size = 51, normalized size = 3.64 \[ \frac {\left (\ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (\frac {2}{\cos \relax (x )+1}\right )\right ) \sin \relax (x ) \sqrt {-\frac {\cos ^{2}\relax (x )}{-1+\cos ^{2}\relax (x )}}\, \sqrt {4}}{2 \cos \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 13, normalized size = 0.93 \[ -\frac {1}{2} \, \log \left (\tan \relax (x)^{2} + 1\right ) + \log \left (\tan \relax (x)\right ) \]
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 \[ \int \sqrt {\frac {1}{{\sin \relax (x)}^2}-1} \,d x \]
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 \[ \int \sqrt {\csc ^{2}{\relax (x )} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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