Optimal. Leaf size=31 \[ \frac {2}{3} \sec ^{\frac {3}{2}}(x)-\frac {4}{7} \sec ^{\frac {7}{2}}(x)+\frac {2}{11} \sec ^{\frac {11}{2}}(x) \]
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Rubi [A]
time = 0.02, 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 = {2702, 276}
\begin {gather*} \frac {2}{11} \sec ^{\frac {11}{2}}(x)-\frac {4}{7} \sec ^{\frac {7}{2}}(x)+\frac {2}{3} \sec ^{\frac {3}{2}}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2702
Rubi steps
\begin {align*} \int \sec ^{\frac {13}{2}}(x) \sin ^5(x) \, dx &=\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*}
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Mathematica [A]
time = 0.04, size = 24, normalized size = 0.77 \begin {gather*} \frac {1}{924} (135+44 \cos (2 x)+77 \cos (4 x)) \sec ^{\frac {11}{2}}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs.
\(2(19)=38\).
time = 0.11, 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 = 1.61, size = 19, normalized size = 0.61 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.29, size = 20, normalized size = 0.65 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.08, size = 39, normalized size = 1.26 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 20, normalized size = 0.65 \begin {gather*} \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}}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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