3.268 \(\int \frac{\sec (e+f x)}{\sqrt{4-5 \sec (e+f x)} \sqrt{2+3 \sec (e+f x)}} \, dx\)

Optimal. Leaf size=125 \[ \frac{2 i \cot (e+f x) \sqrt{\frac{1-\sec (e+f x)}{3 \sec (e+f x)+2}} \sqrt{\frac{\sec (e+f x)+1}{3 \sec (e+f x)+2}} (3 \sec (e+f x)+2) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{5} \sqrt{4-5 \sec (e+f x)}}{\sqrt{3 \sec (e+f x)+2}}\right ),\frac{1}{45}\right )}{3 \sqrt{5} f} \]

[Out]

(((2*I)/3)*Cot[e + f*x]*EllipticF[I*ArcSinh[(Sqrt[5]*Sqrt[4 - 5*Sec[e + f*x]])/Sqrt[2 + 3*Sec[e + f*x]]], 1/45
]*Sqrt[(1 - Sec[e + f*x])/(2 + 3*Sec[e + f*x])]*Sqrt[(1 + Sec[e + f*x])/(2 + 3*Sec[e + f*x])]*(2 + 3*Sec[e + f
*x]))/(Sqrt[5]*f)

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Rubi [A]  time = 0.119524, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {3984} \[ \frac{2 i \cot (e+f x) \sqrt{\frac{1-\sec (e+f x)}{3 \sec (e+f x)+2}} \sqrt{\frac{\sec (e+f x)+1}{3 \sec (e+f x)+2}} (3 \sec (e+f x)+2) F\left (i \sinh ^{-1}\left (\frac{\sqrt{5} \sqrt{4-5 \sec (e+f x)}}{\sqrt{3 \sec (e+f x)+2}}\right )|\frac{1}{45}\right )}{3 \sqrt{5} f} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]/(Sqrt[4 - 5*Sec[e + f*x]]*Sqrt[2 + 3*Sec[e + f*x]]),x]

[Out]

(((2*I)/3)*Cot[e + f*x]*EllipticF[I*ArcSinh[(Sqrt[5]*Sqrt[4 - 5*Sec[e + f*x]])/Sqrt[2 + 3*Sec[e + f*x]]], 1/45
]*Sqrt[(1 - Sec[e + f*x])/(2 + 3*Sec[e + f*x])]*Sqrt[(1 + Sec[e + f*x])/(2 + 3*Sec[e + f*x])]*(2 + 3*Sec[e + f
*x]))/(Sqrt[5]*f)

Rule 3984

Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (
c_)]), x_Symbol] :> Simp[(-2*(c + d*Csc[e + f*x])*Sqrt[((b*c - a*d)*(1 - Csc[e + f*x]))/((a + b)*(c + d*Csc[e
+ f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Csc[e + f*x]))/((a - b)*(c + d*Csc[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)
/(a + b), 2]*(Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b
*c - a*d)*Rt[(c + d)/(a + b), 2]*Cot[e + f*x]), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ
[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps

\begin{align*} \int \frac{\sec (e+f x)}{\sqrt{4-5 \sec (e+f x)} \sqrt{2+3 \sec (e+f x)}} \, dx &=\frac{2 i \cot (e+f x) F\left (i \sinh ^{-1}\left (\frac{\sqrt{5} \sqrt{4-5 \sec (e+f x)}}{\sqrt{2+3 \sec (e+f x)}}\right )|\frac{1}{45}\right ) \sqrt{\frac{1-\sec (e+f x)}{2+3 \sec (e+f x)}} \sqrt{\frac{1+\sec (e+f x)}{2+3 \sec (e+f x)}} (2+3 \sec (e+f x))}{3 \sqrt{5} f}\\ \end{align*}

Mathematica [A]  time = 0.444602, size = 176, normalized size = 1.41 \[ -\frac{4 \sin ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{-\cot ^2\left (\frac{1}{2} (e+f x)\right )} \csc (e+f x) \sec (e+f x) \sqrt{-(2 \cos (e+f x)+3) \csc ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{-(4 \cos (e+f x)-5) \csc ^2\left (\frac{1}{2} (e+f x)\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{5}{22}} \sqrt{\frac{4 \cos (e+f x)-5}{\cos (e+f x)-1}}\right ),\frac{44}{45}\right )}{3 \sqrt{5} f \sqrt{4-5 \sec (e+f x)} \sqrt{3 \sec (e+f x)+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]/(Sqrt[4 - 5*Sec[e + f*x]]*Sqrt[2 + 3*Sec[e + f*x]]),x]

[Out]

(-4*Sqrt[-Cot[(e + f*x)/2]^2]*Sqrt[-((3 + 2*Cos[e + f*x])*Csc[(e + f*x)/2]^2)]*Sqrt[-((-5 + 4*Cos[e + f*x])*Cs
c[(e + f*x)/2]^2)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[5/22]*Sqrt[(-5 + 4*Cos[e + f*x])/(-1 + Cos[e + f*x])]],
44/45]*Sec[e + f*x]*Sin[(e + f*x)/2]^4)/(3*Sqrt[5]*f*Sqrt[4 - 5*Sec[e + f*x]]*Sqrt[2 + 3*Sec[e + f*x]])

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Maple [A]  time = 0.309, size = 170, normalized size = 1.4 \begin{align*}{\frac{-{\frac{i}{15}} \left ( \sin \left ( fx+e \right ) \right ) ^{2}\cos \left ( fx+e \right ) \sqrt{10}}{f \left ( 8\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}-6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-17\,\cos \left ( fx+e \right ) +15 \right ) }\sqrt{{\frac{4\,\cos \left ( fx+e \right ) -5}{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{2\,\cos \left ( fx+e \right ) +3}{\cos \left ( fx+e \right ) }}}\sqrt{-2\,{\frac{4\,\cos \left ( fx+e \right ) -5}{1+\cos \left ( fx+e \right ) }}}\sqrt{{\frac{2\,\cos \left ( fx+e \right ) +3}{1+\cos \left ( fx+e \right ) }}}{\it EllipticF} \left ({\frac{3\,i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},{\frac{\sqrt{5}}{15}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)/(4-5*sec(f*x+e))^(1/2)/(2+3*sec(f*x+e))^(1/2),x)

[Out]

-1/15*I/f*sin(f*x+e)^2*cos(f*x+e)*((4*cos(f*x+e)-5)/cos(f*x+e))^(1/2)*((2*cos(f*x+e)+3)/cos(f*x+e))^(1/2)*(-2*
(4*cos(f*x+e)-5)/(1+cos(f*x+e)))^(1/2)*10^(1/2)*((2*cos(f*x+e)+3)/(1+cos(f*x+e)))^(1/2)*EllipticF(3*I*(-1+cos(
f*x+e))/sin(f*x+e),1/15*5^(1/2))/(8*cos(f*x+e)^3-6*cos(f*x+e)^2-17*cos(f*x+e)+15)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )}{\sqrt{3 \, \sec \left (f x + e\right ) + 2} \sqrt{-5 \, \sec \left (f x + e\right ) + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(4-5*sec(f*x+e))^(1/2)/(2+3*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sec(f*x + e)/(sqrt(3*sec(f*x + e) + 2)*sqrt(-5*sec(f*x + e) + 4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, \sec \left (f x + e\right ) + 2} \sqrt{-5 \, \sec \left (f x + e\right ) + 4} \sec \left (f x + e\right )}{15 \, \sec \left (f x + e\right )^{2} - 2 \, \sec \left (f x + e\right ) - 8}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(4-5*sec(f*x+e))^(1/2)/(2+3*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(3*sec(f*x + e) + 2)*sqrt(-5*sec(f*x + e) + 4)*sec(f*x + e)/(15*sec(f*x + e)^2 - 2*sec(f*x + e)
- 8), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec{\left (e + f x \right )}}{\sqrt{4 - 5 \sec{\left (e + f x \right )}} \sqrt{3 \sec{\left (e + f x \right )} + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(4-5*sec(f*x+e))**(1/2)/(2+3*sec(f*x+e))**(1/2),x)

[Out]

Integral(sec(e + f*x)/(sqrt(4 - 5*sec(e + f*x))*sqrt(3*sec(e + f*x) + 2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (f x + e\right )}{\sqrt{3 \, \sec \left (f x + e\right ) + 2} \sqrt{-5 \, \sec \left (f x + e\right ) + 4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)/(4-5*sec(f*x+e))^(1/2)/(2+3*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sec(f*x + e)/(sqrt(3*sec(f*x + e) + 2)*sqrt(-5*sec(f*x + e) + 4)), x)