Optimal. Leaf size=16 \[ \tan (x) \sqrt {-\cot ^2(x)} \log (\sin (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 16, 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} \[ \tan (x) \sqrt {-\cot ^2(x)} \log (\sin (x)) \]
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 = 16, 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 = 10, normalized size = 0.62 \[ x + \arctan \left (\frac {\cos \relax (x)}{\sin \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.26, size = 48, normalized size = 3.00 \[ -\frac {1}{2} \, {\left (2 i \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{4} + 1\right ) - i \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{4} + 1\right )\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 46, normalized size = 2.88 \[ \frac {\left (\ln \left (-\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right )-\ln \left (\frac {2}{\cos \relax (x )+1}\right )\right ) \sin \relax (x ) \sqrt {\frac {\cos ^{2}\relax (x )}{-1+\cos ^{2}\relax (x )}}}{\cos \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 15, normalized size = 0.94 \[ -\frac {1}{2} i \, \log \left (\tan \relax (x)^{2} + 1\right ) + i \, \log \left (\tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \sqrt {1-\frac {1}{{\sin \relax (x)}^2}} \,d x \]
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 \[ \int \sqrt {1 - \csc ^{2}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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