Integrand size = 11, antiderivative size = 12 \[ \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx=\frac {2 \cos (x)}{\sqrt {\cos ^3(x)}} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3286, 2645, 30} \[ \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx=\frac {2 \cos (x)}{\sqrt {\cos ^3(x)}} \]
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Rule 30
Rule 2645
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \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) \text {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*}
Time = 0.12 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx=\frac {2 \cos (x)}{\sqrt {\cos ^3(x)}} \]
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Time = 0.12 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {4 \cos \left (x \right )}{\sqrt {\cos \left (3 x \right )+3 \cos \left (x \right )}}\) | \(11\) |
default | \(\frac {4 \cos \left (x \right )}{\sqrt {\cos \left (3 x \right )+3 \cos \left (x \right )}}\) | \(11\) |
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none
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx=\frac {2 \, \sqrt {\cos \left (x\right )^{3}}}{\cos \left (x\right )^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx=\frac {2 \cos {\left (x \right )}}{\sqrt {\cos ^{3}{\left (x \right )}}} \]
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none
Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx=\frac {2 \, \cos \left (x\right )}{\sqrt {\cos \left (x\right )^{3}}} \]
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none
Time = 0.27 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.50 \[ \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx=\frac {2}{\sqrt {\cos \left (x\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {\sin (x)}{\sqrt {\cos ^3(x)}} \, dx=\frac {2\,\left |\cos \left (x\right )\right |}{{\cos \left (x\right )}^{3/2}} \]
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