Optimal. Leaf size=21 \[ \frac {2}{9} \tan ^{\frac {9}{2}}(x)+\frac {2}{5} \tan ^{\frac {5}{2}}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 21, 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 = {2607, 14} \[ \frac {2}{9} \tan ^{\frac {9}{2}}(x)+\frac {2}{5} \tan ^{\frac {5}{2}}(x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rubi steps
\begin {align*} \int \sec ^4(x) \tan ^{\frac {3}{2}}(x) \, dx &=\operatorname {Subst}\left (\int x^{3/2} \left (1+x^2\right ) \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (x^{3/2}+x^{7/2}\right ) \, dx,x,\tan (x)\right )\\ &=\frac {2}{5} \tan ^{\frac {5}{2}}(x)+\frac {2}{9} \tan ^{\frac {9}{2}}(x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 22, normalized size = 1.05 \[ \frac {2}{45} (2 \cos (2 x)+7) \tan ^{\frac {5}{2}}(x) \sec ^2(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec ^4(x) \tan ^{\frac {3}{2}}(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.86, size = 27, normalized size = 1.29 \[ -\frac {2 \, {\left (4 \, \cos \relax (x)^{4} + \cos \relax (x)^{2} - 5\right )} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x)}}}{45 \, \cos \relax (x)^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 13, normalized size = 0.62 \[ \frac {2}{9} \, \tan \relax (x)^{\frac {9}{2}} + \frac {2}{5} \, \tan \relax (x)^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 26, normalized size = 1.24
method | result | size |
default | \(\frac {2 \left (4 \left (\cos ^{2}\relax (x )\right )+5\right ) \sin \relax (x ) \left (\frac {\sin \relax (x )}{\cos \relax (x )}\right )^{\frac {3}{2}}}{45 \cos \relax (x )^{3}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 13, normalized size = 0.62 \[ \frac {2}{9} \, \tan \relax (x)^{\frac {9}{2}} + \frac {2}{5} \, \tan \relax (x)^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 32, normalized size = 1.52 \[ -\frac {4\,\sqrt {\sin \left (2\,x\right )}\,\left (2\,{\cos \left (2\,x\right )}^2+5\,\cos \left (2\,x\right )-7\right )}{45\,{\left (\cos \left (2\,x\right )+1\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \tan ^{\frac {3}{2}}{\relax (x )} \sec ^{4}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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