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

Optimal. Leaf size=61 \[ \frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{6}{5}\right )}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}} \]

[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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Rubi [A]  time = 0.0570815, antiderivative size = 61, 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, 2661} \[ \frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left (\frac{1}{2} (c+d x)|\frac{6}{5}\right )}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}} \]

Antiderivative was successfully verified.

[In]

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 + 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 2661

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+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 = 0.0572409, size = 61, normalized size = 1. \[ \frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),\frac{6}{5}\right )}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}} \]

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 = 0.215, size = 145, normalized size = 2.4 \begin{align*}{\frac{\sqrt{5} \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) \sqrt{10}\sqrt{2}}{5\,d \left ( 3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\cos \left ( dx+c \right ) -2 \right ) }{\it EllipticF} \left ({\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{5\,\sin \left ( dx+c \right ) }},i\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{{\frac{2+3\,\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{ \left ( \cos \left ( dx+c \right ) +1 \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/d*5^(1/2)*EllipticF(1/5*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),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)*10^(1/2)*((2+3*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*2^(1/2)*(1/(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{\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)

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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)