Optimal. Leaf size=14 \[ \frac{2}{9} (3 \sec (x)+4)^{3/2} \]
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Rubi [A] time = 0.0438361, antiderivative size = 14, normalized size of antiderivative = 1., 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 4339
Rule 261
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*}
Mathematica [A] time = 0.0540786, size = 14, normalized size = 1. \[ \frac{2}{9} (3 \sec (x)+4)^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 11, normalized size = 0.8 \begin{align*}{\frac{2}{9} \left ( 4+3\,\sec \left ( x \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95489, size = 14, normalized size = 1. \begin{align*} \frac{2}{9} \,{\left (3 \, \sec \left (x\right ) + 4\right )}^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.04178, size = 74, normalized size = 5.29 \begin{align*} \frac{2 \, \sqrt{\frac{4 \, \cos \left (x\right ) + 3}{\cos \left (x\right )}}{\left (4 \, \cos \left (x\right ) + 3\right )}}{9 \, \cos \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.910715, size = 29, normalized size = 2.07 \begin{align*} \frac{2 \sqrt{3 \sec{\left (x \right )} + 4} \sec{\left (x \right )}}{3} + \frac{8 \sqrt{3 \sec{\left (x \right )} + 4}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09666, size = 92, normalized size = 6.57 \begin{align*} \frac{2 \,{\left (4 \,{\left (\sqrt{4 \, \cos \left (x\right )^{2} + 3 \, \cos \left (x\right )} - 2 \, \cos \left (x\right )\right )}^{2} - 6 \, \sqrt{4 \, \cos \left (x\right )^{2} + 3 \, \cos \left (x\right )} + 12 \, \cos \left (x\right ) + 3\right )} \mathrm{sgn}\left (\cos \left (x\right )\right )}{{\left (\sqrt{4 \, \cos \left (x\right )^{2} + 3 \, \cos \left (x\right )} - 2 \, \cos \left (x\right )\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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