3.683 \(\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-3+2 \sec (c+d x)}} \, dx\)

Optimal. Leaf size=62 \[ \frac{2 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x+\pi ),\frac{6}{5}\right )}{\sqrt{5} d \sqrt{2 \sec (c+d x)-3}} \]

[Out]

(2*Sqrt[2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-3 + 2*Sec[c
+ d*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0571196, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {3858, 2662} \[ \frac{2 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right )}{\sqrt{5} d \sqrt{2 \sec (c+d x)-3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sec[c + d*x]]/Sqrt[-3 + 2*Sec[c + d*x]],x]

[Out]

(2*Sqrt[2 - 3*Cos[c + d*x]]*EllipticF[(c + Pi + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-3 + 2*Sec[c
+ d*x]])

Rule 3858

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(Sqrt[d*
Csc[e + f*x]]*Sqrt[b + a*Sin[e + f*x]])/Sqrt[a + b*Csc[e + f*x]], Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; Fr
eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2662

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c + Pi/2 + d*x))/2, (-2*b
)/(a - b)])/(d*Sqrt[a - b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-3+2 \sec (c+d x)}} \, dx &=\frac{\left (\sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{2-3 \cos (c+d x)}} \, dx}{\sqrt{-3+2 \sec (c+d x)}}\\ &=\frac{2 \sqrt{2-3 \cos (c+d x)} F\left (\frac{1}{2} (c+\pi +d x)|\frac{6}{5}\right ) \sqrt{\sec (c+d x)}}{\sqrt{5} d \sqrt{-3+2 \sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0441874, size = 54, normalized size = 0.87 \[ \frac{2 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),6\right )}{d \sqrt{2 \sec (c+d x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sec[c + d*x]]/Sqrt[-3 + 2*Sec[c + d*x]],x]

[Out]

(2*Sqrt[-2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-3 + 2*Sec[c + d*x]])

________________________________________________________________________________________

Maple [C]  time = 0.241, size = 136, normalized size = 2.2 \begin{align*}{\frac{2\,i\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-5\,\cos \left ( dx+c \right ) +2 \right ) }\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{-{\frac{-2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},i\sqrt{5} \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{{\frac{-2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^(1/2)/(-3+2*sec(d*x+c))^(1/2),x)

[Out]

2*I/d*cos(d*x+c)*(1/cos(d*x+c))^(1/2)*(-(-2+3*cos(d*x+c))/cos(d*x+c))^(1/2)*sin(d*x+c)^2*EllipticF(I*(-1+cos(d
*x+c))/sin(d*x+c),I*5^(1/2))*(1/(cos(d*x+c)+1))^(1/2)*((-2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)/(3*cos(d*x+c)^2
-5*cos(d*x+c)+2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sec \left (d x + c\right )}}{\sqrt{2 \, \sec \left (d x + c\right ) - 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(1/2)/(-3+2*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sec(d*x + c))/sqrt(2*sec(d*x + c) - 3), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\sec \left (d x + c\right )}}{\sqrt{2 \, \sec \left (d x + c\right ) - 3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(1/2)/(-3+2*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(sec(d*x + c))/sqrt(2*sec(d*x + c) - 3), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sec{\left (c + d x \right )}}}{\sqrt{2 \sec{\left (c + d x \right )} - 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**(1/2)/(-3+2*sec(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(sec(c + d*x))/sqrt(2*sec(c + d*x) - 3), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sec \left (d x + c\right )}}{\sqrt{2 \, \sec \left (d x + c\right ) - 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(1/2)/(-3+2*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(sec(d*x + c))/sqrt(2*sec(d*x + c) - 3), x)