Optimal. Leaf size=16 \[ -\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4331} \[ -\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)} \]
Antiderivative was successfully verified.
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Rule 4331
Rubi steps
\begin {align*} \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx &=-\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 16, normalized size = 1.00 \[ -\frac {\cos (3 x)}{3 \cos ^{\frac {3}{2}}(2 x)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (x)}{\cos ^{\frac {5}{2}}(2 x)} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.33, size = 39, normalized size = 2.44 \[ -\frac {{\left (4 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x)\right )} \sqrt {2 \, \cos \relax (x)^{2} - 1}}{3 \, {\left (4 \, \cos \relax (x)^{4} - 4 \, \cos \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.97, size = 46, normalized size = 2.88 \[ \frac {{\left ({\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 15\right )} \tan \left (\frac {1}{2} \, x\right )^{2} + 15\right )} \tan \left (\frac {1}{2} \, x\right )^{2} - 1}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{4} - 6 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 39, normalized size = 2.44
method | result | size |
default | \(\frac {\sqrt {-2 \left (\sin ^{2}\relax (x )\right )+1}\, \cos \relax (x ) \left (4 \left (\sin ^{2}\relax (x )\right )-1\right )}{12 \left (\sin ^{4}\relax (x )\right )-12 \left (\sin ^{2}\relax (x )\right )+3}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.02, size = 90, normalized size = 5.62 \[ -\frac {\sqrt {2} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right ) \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) + {\left (\sqrt {2} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right )\right )\right ) + \sqrt {2}\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (4 \, x\right ), \cos \left (4 \, x\right ) + 1\right )\right )}{3 \, {\left (\cos \left (4 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 2 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 12, normalized size = 0.75 \[ -\frac {\cos \left (3\,x\right )}{3\,{\cos \left (2\,x\right )}^{3/2}} \]
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 \[ \int \frac {\sin {\relax (x )}}{\cos ^{\frac {5}{2}}{\left (2 x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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