Optimal. Leaf size=61 \[ a \sin (x) \cos (x) \sqrt {a \sec ^4(x)}+\frac {1}{5} a \sin ^2(x) \tan ^3(x) \sqrt {a \sec ^4(x)}+\frac {2}{3} a \sin ^2(x) \tan (x) \sqrt {a \sec ^4(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4123, 3767} \[ a \sin (x) \cos (x) \sqrt {a \sec ^4(x)}+\frac {1}{5} a \sin ^2(x) \tan ^3(x) \sqrt {a \sec ^4(x)}+\frac {2}{3} a \sin ^2(x) \tan (x) \sqrt {a \sec ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 4123
Rubi steps
\begin {align*} \int \left (a \sec ^4(x)\right )^{3/2} \, dx &=\left (a \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \sec ^6(x) \, dx\\ &=-\left (\left (a \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )\right )\\ &=a \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+\frac {2}{3} a \sqrt {a \sec ^4(x)} \sin ^2(x) \tan (x)+\frac {1}{5} a \sqrt {a \sec ^4(x)} \sin ^2(x) \tan ^3(x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 30, normalized size = 0.49 \[ \frac {1}{15} \sin (x) \cos (x) (6 \cos (2 x)+\cos (4 x)+8) \left (a \sec ^4(x)\right )^{3/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 34, normalized size = 0.56 \[ \frac {{\left (8 \, a \cos \relax (x)^{4} + 4 \, a \cos \relax (x)^{2} + 3 \, a\right )} \sqrt {\frac {a}{\cos \relax (x)^{4}}} \sin \relax (x)}{15 \, \cos \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 22, normalized size = 0.36 \[ \frac {1}{15} \, {\left (3 \, \tan \relax (x)^{5} + 10 \, \tan \relax (x)^{3} + 15 \, \tan \relax (x)\right )} a^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 29, normalized size = 0.48 \[ \frac {\left (8 \left (\cos ^{4}\relax (x )\right )+4 \left (\cos ^{2}\relax (x )\right )+3\right ) \cos \relax (x ) \sin \relax (x ) \left (\frac {a}{\cos \relax (x )^{4}}\right )^{\frac {3}{2}}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 25, normalized size = 0.41 \[ \frac {1}{5} \, a^{\frac {3}{2}} \tan \relax (x)^{5} + \frac {2}{3} \, a^{\frac {3}{2}} \tan \relax (x)^{3} + a^{\frac {3}{2}} \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.56, size = 36, normalized size = 0.59 \[ \frac {4\,a^{3/2}\,\sin \relax (x)}{5\,{\cos \relax (x)}^3}+\frac {a^{3/2}\,\sin \relax (x)}{5\,{\cos \relax (x)}^5}-\frac {8\,a^{3/2}\,{\sin \relax (x)}^3}{15\,{\cos \relax (x)}^3} \]
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 \[ \int \left (a \sec ^{4}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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