Optimal. Leaf size=21 \[ \frac {2}{5} \tan ^{\frac {5}{2}}(x)+\frac {2}{9} \tan ^{\frac {9}{2}}(x) \]
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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 = {2687, 14}
\begin {gather*} \frac {2}{9} \tan ^{\frac {9}{2}}(x)+\frac {2}{5} \tan ^{\frac {5}{2}}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2687
Rubi steps
\begin {align*} \int \sec ^4(x) \tan ^{\frac {3}{2}}(x) \, dx &=\text {Subst}\left (\int x^{3/2} \left (1+x^2\right ) \, dx,x,\tan (x)\right )\\ &=\text {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 = 0.03, size = 22, normalized size = 1.05 \begin {gather*} \frac {2}{45} (7+2 \cos (2 x)) \sec ^2(x) \tan ^{\frac {5}{2}}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 26, normalized size = 1.24
method | result | size |
default | \(\frac {2 \left (4 \left (\cos ^{2}\left (x \right )\right )+5\right ) \sin \left (x \right ) \left (\frac {\sin \left (x \right )}{\cos \left (x \right )}\right )^{\frac {3}{2}}}{45 \cos \left (x \right )^{3}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.13, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{9} \, \tan \left (x\right )^{\frac {9}{2}} + \frac {2}{5} \, \tan \left (x\right )^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (13) = 26\).
time = 1.07, size = 27, normalized size = 1.29 \begin {gather*} -\frac {2 \, {\left (4 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 5\right )} \sqrt {\frac {\sin \left (x\right )}{\cos \left (x\right )}}}{45 \, \cos \left (x\right )^{4}} \end {gather*}
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 \begin {gather*} \int \tan ^{\frac {3}{2}}{\left (x \right )} \sec ^{4}{\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.27, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{9} \, \tan \left (x\right )^{\frac {9}{2}} + \frac {2}{5} \, \tan \left (x\right )^{\frac {5}{2}} \end {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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