\(\int \frac {\sec (e+f x)}{\sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \, dx\) [267]

Optimal result

Integrand size = 35, antiderivative size = 110 \[ \int \frac {\sec (e+f x)}{\sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \, dx=\frac {2 \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2+3 \sec (e+f x)}}{\sqrt {5} \sqrt {-4+5 \sec (e+f x)}}\right ),45\right ) (4-5 \sec (e+f x)) \sqrt {\frac {1-\sec (e+f x)}{4-5 \sec (e+f x)}} \sqrt {\frac {1+\sec (e+f x)}{4-5 \sec (e+f x)}}}{f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {4069} \[ \int \frac {\sec (e+f x)}{\sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \, dx=\frac {2 \cot (e+f x) (4-5 \sec (e+f x)) \sqrt {\frac {1-\sec (e+f x)}{4-5 \sec (e+f x)}} \sqrt {\frac {\sec (e+f x)+1}{4-5 \sec (e+f x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3 \sec (e+f x)+2}}{\sqrt {5} \sqrt {5 \sec (e+f x)-4}}\right ),45\right )}{f} \]

[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 4069

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])/(f*(b*c - a*d)*Rt[(c + d)/(a + b), 2]*Cot[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)))], 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*} \text {integral}& = \frac {2 \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2+3 \sec (e+f x)}}{\sqrt {5} \sqrt {-4+5 \sec (e+f x)}}\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] (verified)

Time = 2.65 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.60 \[ \int \frac {\sec (e+f x)}{\sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \, dx=-\frac {4 \sqrt {-\cot ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {-\left ((3+2 \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )} \sqrt {-\left ((-5+4 \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )\right )} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{22}} \sqrt {\frac {-5+4 \cos (e+f x)}{-1+\cos (e+f x)}}\right ),\frac {44}{45}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{3 \sqrt {5} f \sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \]

[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]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 7.54 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.35

method	result	size
default	\(-\frac {i \sqrt {5}\, \operatorname {EllipticF}\left (\frac {i \sqrt {5}\, \left (\cot \left (f x +e \right )-\csc \left (f x +e \right )\right )}{5}, 3 \sqrt {5}\right ) \sqrt {2+3 \sec \left (f x +e \right )}\, \sqrt {-4+5 \sec \left (f x +e \right )}\, \sqrt {-\frac {2 \left (4 \cos \left (f x +e \right )-5\right )}{\cos \left (f x +e \right )+1}}\, \sqrt {10}\, \sqrt {\frac {2 \cos \left (f x +e \right )+3}{\cos \left (f x +e \right )+1}}\, \left (\cos \left (f x +e \right )^{2}+\cos \left (f x +e \right )\right )}{5 f \left (8 \cos \left (f x +e \right )^{2}+2 \cos \left (f x +e \right )-15\right )}\)	\(148\)

[In]

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

[Out]

-1/5*I/f*5^(1/2)*EllipticF(1/5*I*5^(1/2)*(cot(f*x+e)-csc(f*x+e)),3*5^(1/2))*(2+3*sec(f*x+e))^(1/2)*(-4+5*sec(f
*x+e))^(1/2)*(-2*(4*cos(f*x+e)-5)/(cos(f*x+e)+1))^(1/2)*10^(1/2)*((2*cos(f*x+e)+3)/(cos(f*x+e)+1))^(1/2)/(8*co
s(f*x+e)^2+2*cos(f*x+e)-15)*(cos(f*x+e)^2+cos(f*x+e))

Fricas [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {5 \, \sec \left (f x + e\right ) - 4} \sqrt {3 \, \sec \left (f x + e\right ) + 2}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \, dx=\int \frac {\sec {\left (e + f x \right )}}{\sqrt {3 \sec {\left (e + f x \right )} + 2} \sqrt {5 \sec {\left (e + f x \right )} - 4}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {5 \, \sec \left (f x + e\right ) - 4} \sqrt {3 \, \sec \left (f x + e\right ) + 2}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {5 \, \sec \left (f x + e\right ) - 4} \sqrt {3 \, \sec \left (f x + e\right ) + 2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec (e+f x)}{\sqrt {2+3 \sec (e+f x)} \sqrt {-4+5 \sec (e+f x)}} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {\frac {3}{\cos \left (e+f\,x\right )}+2}\,\sqrt {\frac {5}{\cos \left (e+f\,x\right )}-4}} \,d x \]

[In]

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

[Out]

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