Optimal. Leaf size=29 \[ \frac {4 \sin (x)}{3 \sqrt {\sin (2 x)}}-\frac {2 \cos (x)}{3 \sin ^{\frac {3}{2}}(2 x)} \]
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Rubi [A] time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4308, 4303, 4292} \[ \frac {4 \sin (x)}{3 \sqrt {\sin (2 x)}}-\frac {2 \cos (x)}{3 \sin ^{\frac {3}{2}}(2 x)} \]
Antiderivative was successfully verified.
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Rule 4292
Rule 4303
Rule 4308
Rubi steps
\begin {align*} \int \frac {\csc (x)}{\sin ^{\frac {3}{2}}(2 x)} \, dx &=2 \int \frac {\cos (x)}{\sin ^{\frac {5}{2}}(2 x)} \, dx\\ &=-\frac {2 \cos (x)}{3 \sin ^{\frac {3}{2}}(2 x)}+\frac {4}{3} \int \frac {\sin (x)}{\sin ^{\frac {3}{2}}(2 x)} \, dx\\ &=-\frac {2 \cos (x)}{3 \sin ^{\frac {3}{2}}(2 x)}+\frac {4 \sin (x)}{3 \sqrt {\sin (2 x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 24, normalized size = 0.83 \[ \sqrt {\sin (2 x)} \left (\frac {\sec (x)}{2}-\frac {1}{6} \cot (x) \csc (x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc (x)}{\sin ^{\frac {3}{2}}(2 x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.61, size = 43, normalized size = 1.48 \[ \frac {4 \, \cos \relax (x)^{3} + \sqrt {2} {\left (4 \, \cos \relax (x)^{2} - 3\right )} \sqrt {\cos \relax (x) \sin \relax (x)} - 4 \, \cos \relax (x)}{6 \, {\left (\cos \relax (x)^{3} - \cos \relax (x)\right )}} \]
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 \[ \int \frac {1}{\sin \left (2 \, x\right )^{\frac {3}{2}} \sin \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 121, normalized size = 4.17
method | result | size |
default | \(-\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (2 \sqrt {\tan \left (\frac {x}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {x}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {x}{2}\right )-\left (\tan ^{4}\left (\frac {x}{2}\right )\right )+1\right )}{12 \tan \left (\frac {x}{2}\right ) \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}}\) | \(121\) |
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 \[ \int \frac {1}{\sin \left (2 \, x\right )^{\frac {3}{2}} \sin \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 29, normalized size = 1.00 \[ -\frac {\sqrt {\sin \left (2\,x\right )}\,\left (2\,\cos \left (2\,x\right )-1\right )}{6\,\left (\cos \relax (x)-{\cos \relax (x)}^3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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