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

Optimal. Leaf size=110 \[ \frac{2 \cot (e+f x) (4-5 \sec (e+f x)) \sqrt{\frac{1-\sec (e+f x)}{4-5 \sec (e+f x)}} \sqrt{\frac{\sec (e+f x)+1}{4-5 \sec (e+f x)}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{3 \sec (e+f x)+2}}{\sqrt{5} \sqrt{5 \sec (e+f x)-4}}\right ),45\right )}{f} \]

[Out]

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

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Rubi [A]  time = 0.12409, antiderivative size = 110, 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} \[ \frac{2 \cot (e+f x) (4-5 \sec (e+f x)) \sqrt{\frac{1-\sec (e+f x)}{4-5 \sec (e+f x)}} \sqrt{\frac{\sec (e+f x)+1}{4-5 \sec (e+f x)}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3 \sec (e+f x)+2}}{\sqrt{5} \sqrt{5 \sec (e+f x)-4}}\right )\right |45\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[2 + 3*Sec[e + f*x]]/(Sqrt[5]*Sqrt[-4 + 5*Sec[e + f*x]])], 45]*(4 - 5*Sec
[e + f*x])*Sqrt[(1 - Sec[e + f*x])/(4 - 5*Sec[e + f*x])]*Sqrt[(1 + Sec[e + f*x])/(4 - 5*Sec[e + f*x])])/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{2+3 \sec (e+f x)} \sqrt{-4+5 \sec (e+f x)}} \, dx &=\frac{2 \cot (e+f x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{2+3 \sec (e+f x)}}{\sqrt{5} \sqrt{-4+5 \sec (e+f x)}}\right )\right |45\right ) (4-5 \sec (e+f x)) \sqrt{\frac{1-\sec (e+f x)}{4-5 \sec (e+f x)}} \sqrt{\frac{1+\sec (e+f x)}{4-5 \sec (e+f x)}}}{f}\\ \end{align*}

Mathematica [A]  time = 1.75091, size = 176, normalized size = 1.6 \[ -\frac{4 \sin ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{-\cot ^2\left (\frac{1}{2} (e+f x)\right )} \csc (e+f x) \sec (e+f x) \sqrt{-(2 \cos (e+f x)+3) \csc ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{-(4 \cos (e+f x)-5) \csc ^2\left (\frac{1}{2} (e+f x)\right )} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{5}{22}} \sqrt{\frac{4 \cos (e+f x)-5}{\cos (e+f x)-1}}\right ),\frac{44}{45}\right )}{3 \sqrt{5} f \sqrt{3 \sec (e+f x)+2} \sqrt{5 \sec (e+f x)-4}} \]

Antiderivative was successfully verified.

[In]

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

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Maple [C]  time = 0.339, size = 177, normalized size = 1.6 \begin{align*}{\frac{{\frac{i}{5}}\sqrt{5}\sqrt{10}\cos \left ( fx+e \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{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{2\,\cos \left ( fx+e \right ) +3}{1+\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}{\cos \left ( fx+e \right ) }}}\sqrt{-{\frac{4\,\cos \left ( fx+e \right ) -5}{\cos \left ( fx+e \right ) }}}{\it EllipticF} \left ({\frac{{\frac{i}{5}} \left ( -1+\cos \left ( fx+e \right ) \right ) \sqrt{5}}{\sin \left ( fx+e \right ) }},3\,\sqrt{5} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*I/f*5^(1/2)*10^(1/2)*((2*cos(f*x+e)+3)/(1+cos(f*x+e)))^(1/2)*(-2*(4*cos(f*x+e)-5)/(1+cos(f*x+e)))^(1/2)*co
s(f*x+e)*sin(f*x+e)^2*((2*cos(f*x+e)+3)/cos(f*x+e))^(1/2)*(-(4*cos(f*x+e)-5)/cos(f*x+e))^(1/2)*EllipticF(1/5*I
*(-1+cos(f*x+e))*5^(1/2)/sin(f*x+e),3*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{5 \, \sec \left (f x + e\right ) - 4} \sqrt{3 \, \sec \left (f x + e\right ) + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{5 \, \sec \left (f x + e\right ) - 4} \sqrt{3 \, \sec \left (f x + e\right ) + 2} \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)/(2+3*sec(f*x+e))^(1/2)/(-4+5*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(5*sec(f*x + e) - 4)*sqrt(3*sec(f*x + e) + 2)*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{3 \sec{\left (e + f x \right )} + 2} \sqrt{5 \sec{\left (e + f x \right )} - 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sec(e + f*x)/(sqrt(3*sec(e + f*x) + 2)*sqrt(5*sec(e + f*x) - 4)), 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{5 \, \sec \left (f x + e\right ) - 4} \sqrt{3 \, \sec \left (f x + e\right ) + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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