Optimal. Leaf size=12 \[ \frac {2 \cos (x)}{\sqrt {\cos ^3(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 12, 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 = {3207, 2565, 30} \[ \frac {2 \cos (x)}{\sqrt {\cos ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2565
Rule 3207
Rubi steps
\begin {align*} \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx &=\frac {\cos ^{\frac {3}{2}}(x) \int \frac {\sin (x)}{\cos ^{\frac {3}{2}}(x)} \, dx}{\sqrt {\cos ^3(x)}}\\ &=-\frac {\cos ^{\frac {3}{2}}(x) \operatorname {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,\cos (x)\right )}{\sqrt {\cos ^3(x)}}\\ &=\frac {2 \cos (x)}{\sqrt {\cos ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \frac {2 \cos (x)}{\sqrt {\cos ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 12, normalized size = 1.00 \[ \frac {2 \, \sqrt {\cos \relax (x)^{3}}}{\cos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 6, normalized size = 0.50 \[ \frac {2}{\sqrt {\cos \relax (x)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 11, normalized size = 0.92 \[ \frac {4 \cos \relax (x )}{\sqrt {3 \cos \relax (x )+\cos \left (3 x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 10, normalized size = 0.83 \[ \frac {2 \, \cos \relax (x)}{\sqrt {\cos \relax (x)^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 9, normalized size = 0.75 \[ \frac {2\,\left |\cos \relax (x)\right |}{{\cos \relax (x)}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 12, normalized size = 1.00 \[ \frac {2 \cos {\relax (x )}}{\sqrt {\cos ^{3}{\relax (x )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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