Optimal. Leaf size=14 \[ \frac {2}{9} (3 \sec (x)+4)^{3/2} \]
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Rubi [A] time = 0.04, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4339, 261} \[ \frac {2}{9} (3 \sec (x)+4)^{3/2} \]
Antiderivative was successfully verified.
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Rule 261
Rule 4339
Rubi steps
\begin {align*} \int \sec (x) \sqrt {4+3 \sec (x)} \tan (x) \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {4+\frac {3}{x}}}{x^2} \, dx,x,\cos (x)\right )\\ &=\frac {2}{9} (4+3 \sec (x))^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 14, normalized size = 1.00 \[ \frac {2}{9} (3 \sec (x)+4)^{3/2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.26, size = 25, normalized size = 1.79 \[ \frac {2 \, \sqrt {\frac {4 \, \cos \relax (x) + 3}{\cos \relax (x)}} {\left (4 \, \cos \relax (x) + 3\right )}}{9 \, \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 68, normalized size = 4.86 \[ \frac {2 \, {\left (4 \, {\left (\sqrt {4 \, \cos \relax (x)^{2} + 3 \, \cos \relax (x)} - 2 \, \cos \relax (x)\right )}^{2} - 6 \, \sqrt {4 \, \cos \relax (x)^{2} + 3 \, \cos \relax (x)} + 12 \, \cos \relax (x) + 3\right )} \mathrm {sgn}\left (\cos \relax (x)\right )}{{\left (\sqrt {4 \, \cos \relax (x)^{2} + 3 \, \cos \relax (x)} - 2 \, \cos \relax (x)\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 11, normalized size = 0.79 \[ \frac {2 \left (4+3 \sec \relax (x )\right )^{\frac {3}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 10, normalized size = 0.71 \[ \frac {2}{9} \, {\left (3 \, \sec \relax (x) + 4\right )}^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.23, size = 29, normalized size = 2.07 \[ \frac {8\,\sqrt {\frac {3}{\cos \relax (x)}+4}}{9}+\frac {2\,\sqrt {\frac {3}{\cos \relax (x)}+4}}{3\,\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.68, size = 29, normalized size = 2.07 \[ \frac {2 \sqrt {3 \sec {\relax (x )} + 4} \sec {\relax (x )}}{3} + \frac {8 \sqrt {3 \sec {\relax (x )} + 4}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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