3.862 \(\int \sec (x) \sqrt{4+3 \sec (x)} \tan (x) \, dx\)

Optimal. Leaf size=14 \[ \frac{2}{9} (3 \sec (x)+4)^{3/2} \]

[Out]

(2*(4 + 3*Sec[x])^(3/2))/9

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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.

[In]

Int[Sec[x]*Sqrt[4 + 3*Sec[x]]*Tan[x],x]

[Out]

(2*(4 + 3*Sec[x])^(3/2))/9

Rule 4339

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[(b*
c)^(-1), Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[
c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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.

[In]

Integrate[Sec[x]*Sqrt[4 + 3*Sec[x]]*Tan[x],x]

[Out]

(2*(4 + 3*Sec[x])^(3/2))/9

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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.

[In]

int(sec(x)*(4+3*sec(x))^(1/2)*tan(x),x)

[Out]

2/9*(4+3*sec(x))^(3/2)

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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.

[In]

integrate(sec(x)*(4+3*sec(x))^(1/2)*tan(x),x, algorithm="maxima")

[Out]

2/9*(3*sec(x) + 4)^(3/2)

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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.

[In]

integrate(sec(x)*(4+3*sec(x))^(1/2)*tan(x),x, algorithm="fricas")

[Out]

2/9*sqrt((4*cos(x) + 3)/cos(x))*(4*cos(x) + 3)/cos(x)

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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.

[In]

integrate(sec(x)*(4+3*sec(x))**(1/2)*tan(x),x)

[Out]

2*sqrt(3*sec(x) + 4)*sec(x)/3 + 8*sqrt(3*sec(x) + 4)/9

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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.

[In]

integrate(sec(x)*(4+3*sec(x))^(1/2)*tan(x),x, algorithm="giac")

[Out]

2*(4*(sqrt(4*cos(x)^2 + 3*cos(x)) - 2*cos(x))^2 - 6*sqrt(4*cos(x)^2 + 3*cos(x)) + 12*cos(x) + 3)*sgn(cos(x))/(
sqrt(4*cos(x)^2 + 3*cos(x)) - 2*cos(x))^3