Optimal. Leaf size=31 \[ \frac {2}{11} \sec ^{\frac {11}{2}}(x)-\frac {4}{7} \sec ^{\frac {7}{2}}(x)+\frac {2}{3} \sec ^{\frac {3}{2}}(x) \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2622, 270} \[ \frac {2}{11} \sec ^{\frac {11}{2}}(x)-\frac {4}{7} \sec ^{\frac {7}{2}}(x)+\frac {2}{3} \sec ^{\frac {3}{2}}(x) \]
Antiderivative was successfully verified.
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Rule 270
Rule 2622
Rubi steps
\begin {align*} \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx &=\operatorname {Subst}\left (\int \sqrt {x} \left (-1+x^2\right )^2 \, dx,x,\sec (x)\right )\\ &=\operatorname {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*}
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Mathematica [A] time = 0.05, size = 24, normalized size = 0.77 \[ \frac {1}{924} (44 \cos (2 x)+77 \cos (4 x)+135) \sec ^{\frac {11}{2}}(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.01, size = 20, normalized size = 0.65 \[ \frac {2 \, {\left (77 \, \cos \relax (x)^{4} - 66 \, \cos \relax (x)^{2} + 21\right )}}{231 \, \cos \relax (x)^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 20, normalized size = 0.65 \[ \frac {2 \, {\left (77 \, \cos \relax (x)^{4} - 66 \, \cos \relax (x)^{2} + 21\right )}}{231 \, \cos \relax (x)^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 49, normalized size = 1.58
method | result | size |
default | \(\frac {\frac {32 \left (\sin ^{8}\left (\frac {x}{2}\right )\right )}{3}-\frac {64 \left (\sin ^{6}\left (\frac {x}{2}\right )\right )}{3}+\frac {96 \left (\sin ^{4}\left (\frac {x}{2}\right )\right )}{7}-\frac {64 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}{21}+\frac {64}{231}}{\left (-2 \left (\sin ^{2}\left (\frac {x}{2}\right )\right )+1\right )^{\frac {11}{2}}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 19, normalized size = 0.61 \[ \frac {2}{3 \, \cos \relax (x)^{\frac {3}{2}}} - \frac {4}{7 \, \cos \relax (x)^{\frac {7}{2}}} + \frac {2}{11 \, \cos \relax (x)^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 22, normalized size = 0.71 \[ \frac {2\,{\left (\frac {1}{\cos \relax (x)}\right )}^{11/2}\,\left (77\,{\cos \relax (x)}^4-66\,{\cos \relax (x)}^2+21\right )}{231} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 178.37, size = 39, normalized size = 1.26 \[ \frac {2 \tan ^{4}{\relax (x )} \sec ^{\frac {3}{2}}{\relax (x )}}{11} - \frac {16 \tan ^{2}{\relax (x )} \sec ^{\frac {3}{2}}{\relax (x )}}{77} + \frac {64 \sec ^{\frac {3}{2}}{\relax (x )}}{231} \]
Verification of antiderivative is not currently implemented for this CAS.
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