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

Optimal. Leaf size=55 \[ \frac{2 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x+\pi ),6\right )}{d \sqrt{-2 \sec (c+d x)-3}} \]

[Out]

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

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Rubi [A]  time = 0.0580804, antiderivative size = 55, 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} \[ \frac{2 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x+\pi )\right |6\right )}{d \sqrt{-2 \sec (c+d x)-3}} \]

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]*Sqrt[Sec[c + d*x]])/(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 (\left .\frac{1}{2} (c+\pi +d x)\right |6\right ) \sqrt{\sec (c+d x)}}{d \sqrt{-3-2 \sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0466407, size = 61, normalized size = 1.11 \[ \frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{6}{5}\right )}{\sqrt{5} d \sqrt{-2 \sec (c+d x)-3}} \]

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

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Maple [C]  time = 0.235, size = 142, normalized size = 2.6 \begin{align*}{\frac{{\frac{i}{5}} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) \sqrt{2}\sqrt{10}}{d \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\cos \left ( dx+c \right ) -2 \right ) }{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{-{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \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]

1/5*I/d*EllipticF(I*(-1+cos(d*x+c))/sin(d*x+c),1/5*I*5^(1/2))*sin(d*x+c)^2*cos(d*x+c)*(1/cos(d*x+c))^(1/2)*(-(
2+3*cos(d*x+c))/cos(d*x+c))^(1/2)*2^(1/2)*(1/(cos(d*x+c)+1))^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(cos(d*x+c)+1))^
(1/2)/(3*cos(d*x+c)^2-cos(d*x+c)-2)

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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)

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

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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)

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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)