Integrand size = 11, antiderivative size = 31 \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx=\frac {2}{3} \sec ^{\frac {3}{2}}(x)-\frac {4}{7} \sec ^{\frac {7}{2}}(x)+\frac {2}{11} \sec ^{\frac {11}{2}}(x) \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2702, 276} \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx=\frac {2}{11} \sec ^{\frac {11}{2}}(x)-\frac {4}{7} \sec ^{\frac {7}{2}}(x)+\frac {2}{3} \sec ^{\frac {3}{2}}(x) \]
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Rule 276
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \sqrt {x} \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right ) \\ & = \text {Subst}\left (\int \left (\sqrt {x}-2 x^{5/2}+x^{9/2}\right ) \, dx,x,\sec (x)\right ) \\ & = \frac {2}{3} \sec ^{\frac {3}{2}}(x)-\frac {4}{7} \sec ^{\frac {7}{2}}(x)+\frac {2}{11} \sec ^{\frac {11}{2}}(x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx=\frac {1}{924} (135+44 \cos (2 x)+77 \cos (4 x)) \sec ^{\frac {11}{2}}(x) \]
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Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(\frac {2 \left (\sec ^{\frac {3}{2}}\left (x \right )\right )}{3}-\frac {4 \left (\sec ^{\frac {7}{2}}\left (x \right )\right )}{7}+\frac {2 \left (\sec ^{\frac {11}{2}}\left (x \right )\right )}{11}\) | \(20\) |
default | \(\frac {2 \left (\sec ^{\frac {3}{2}}\left (x \right )\right )}{3}-\frac {4 \left (\sec ^{\frac {7}{2}}\left (x \right )\right )}{7}+\frac {2 \left (\sec ^{\frac {11}{2}}\left (x \right )\right )}{11}\) | \(20\) |
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Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx=\frac {2 \, {\left (77 \, \cos \left (x\right )^{4} - 66 \, \cos \left (x\right )^{2} + 21\right )}}{231 \, \cos \left (x\right )^{\frac {11}{2}}} \]
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Time = 13.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx=\frac {2 \tan ^{4}{\left (x \right )} \sec ^{\frac {3}{2}}{\left (x \right )}}{11} - \frac {16 \tan ^{2}{\left (x \right )} \sec ^{\frac {3}{2}}{\left (x \right )}}{77} + \frac {64 \sec ^{\frac {3}{2}}{\left (x \right )}}{231} \]
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Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61 \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx=\frac {2}{3 \, \cos \left (x\right )^{\frac {3}{2}}} - \frac {4}{7 \, \cos \left (x\right )^{\frac {7}{2}}} + \frac {2}{11 \, \cos \left (x\right )^{\frac {11}{2}}} \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx=\frac {2 \, {\left (77 \, \cos \left (x\right )^{4} - 66 \, \cos \left (x\right )^{2} + 21\right )}}{231 \, \cos \left (x\right )^{\frac {11}{2}}} \]
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Time = 1.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx=\frac {2\,{\left (\frac {1}{\cos \left (x\right )}\right )}^{11/2}\,\left (77\,{\cos \left (x\right )}^4-66\,{\cos \left (x\right )}^2+21\right )}{231} \]
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