Optimal. Leaf size=16 \[ -\frac {2 \cos (x) \cot (x)}{3 \sqrt {\sin (2 x)}} \]
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Rubi [A] time = 0.09, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4390, 30} \[ -\frac {2 \cos (x) \cot (x)}{3 \sqrt {\sin (2 x)}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 4390
Rubi steps
\begin {align*} \int \frac {\cot (x) \csc (x)}{\sqrt {\sin (2 x)}} \, dx &=\frac {\sin (x) \int \frac {\csc ^2(x)}{\sqrt {\tan (x)}} \, dx}{\sqrt {\sin (2 x)} \sqrt {\tan (x)}}\\ &=\frac {\sin (x) \operatorname {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,\tan (x)\right )}{\sqrt {\sin (2 x)} \sqrt {\tan (x)}}\\ &=-\frac {2 \cos (x) \cot (x)}{3 \sqrt {\sin (2 x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 16, normalized size = 1.00 \[ -\frac {1}{3} \sqrt {\sin (2 x)} \cot (x) \csc (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.87, size = 29, normalized size = 1.81 \[ \frac {\sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} \cos \relax (x) + \cos \relax (x)^{2} - 1}{3 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \relax (x) \csc \relax (x)}{\sqrt {\sin \left (2 \, x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 119, normalized size = 7.44 \[ \frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (4 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {x}{2}\right )+\tan ^{4}\left (\frac {x}{2}\right )-1\right )}{6 \tan \left (\frac {x}{2}\right ) \sqrt {\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \relax (x) \csc \relax (x)}{\sqrt {\sin \left (2 \, x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.10, size = 14, normalized size = 0.88 \[ -\frac {\sqrt {\sin \left (2\,x\right )}\,\cos \relax (x)}{3\,{\sin \relax (x)}^2} \]
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 \[ \int \frac {\cot {\relax (x )} \csc {\relax (x )}}{\sqrt {\sin {\left (2 x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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