\(\int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx\) [266]

Optimal result

Integrand size = 35, antiderivative size = 192 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{\sqrt {c+d} (b c-a d) f} \]

[Out]

2*cot(f*x+e)*EllipticF((c+d)^(1/2)*(a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sec(f*x+e))^(1/2),((a+b)*(c-d)/(a-b
)/(c+d))^(1/2))*(c+d*sec(f*x+e))*(a+b)^(1/2)*((-a*d+b*c)*(1-sec(f*x+e))/(a+b)/(c+d*sec(f*x+e)))^(1/2)*(-(-a*d+
b*c)*(1+sec(f*x+e))/(a-b)/(c+d*sec(f*x+e)))^(1/2)/(-a*d+b*c)/f/(c+d)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 192, 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} \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \sqrt {a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f \sqrt {c+d} (b c-a d)} \]

[In]

Int[Sec[e + f*x]/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]

[Out]

(2*Sqrt[a + b]*Cot[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[a + b]*Sqrt[c + d*Se
c[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))]*Sqrt[((b*c - a*d)*(1 - Sec[e + f*x]))/((a + b)*(c + d*Sec[
e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sec[e + f*x]))/((a - b)*(c + d*Sec[e + f*x])))]*(c + d*Sec[e + f*x]))/(Sq
rt[c + d]*(b*c - a*d)*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 \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{\sqrt {c+d} (b c-a d) f} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.20 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.21 \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\frac {4 \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} (e+f x)\right )}{c-d}} \sqrt {\frac {(a+b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{-b c+a d}} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b) (d+c \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{-b c+a d}}}{\sqrt {2}}\right ),\frac {2 (b c-a d)}{(a+b) (c-d)}\right ) \sqrt {a+b \sec (e+f x)} \sin ^2\left (\frac {1}{2} (e+f x)\right )}{(a+b) f \sqrt {\frac {(c+d) (b+a \cos (e+f x)) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{b c-a d}} \sqrt {c+d \sec (e+f x)}} \]

[In]

Integrate[Sec[e + f*x]/(Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]]),x]

[Out]

(4*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(-(b*c) +
a*d)]*Csc[e + f*x]*EllipticF[ArcSin[Sqrt[((a + b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(-(b*c) + a*d)]/Sq
rt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sqrt[a + b*Sec[e + f*x]]*Sin[(e + f*x)/2]^2)/((a + b)*f*Sqrt[((c +
d)*(b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[c + d*Sec[e + f*x]])

Maple [A] (verified)

Time = 10.71 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {2 \sqrt {c +d \sec \left (f x +e \right )}\, \sqrt {a +b \sec \left (f x +e \right )}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) \left (\cos \left (f x +e \right )^{2}+\cos \left (f x +e \right )\right )}{f \sqrt {\frac {a -b}{a +b}}\, \left (d +c \cos \left (f x +e \right )\right ) \left (b +a \cos \left (f x +e \right )\right )}\)	\(191\)

[In]

int(sec(f*x+e)/(a+b*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/f/((a-b)/(a+b))^(1/2)*(c+d*sec(f*x+e))^(1/2)*(a+b*sec(f*x+e))^(1/2)*(1/(a+b)*(b+a*cos(f*x+e))/(cos(f*x+e)+1)
)^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*EllipticF(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e))
,((a+b)*(c-d)/(a-b)/(c+d))^(1/2))/(d+c*cos(f*x+e))/(b+a*cos(f*x+e))*(cos(f*x+e)^2+cos(f*x+e))

Fricas [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]

[In]

integrate(sec(f*x+e)/(a+b*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c)*sec(f*x + e)/(b*d*sec(f*x + e)^2 + a*c + (b*c + a*d
)*sec(f*x + e)), x)

Sympy [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}} \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]

[In]

integrate(sec(f*x+e)/(a+b*sec(f*x+e))**(1/2)/(c+d*sec(f*x+e))**(1/2),x)

[Out]

Integral(sec(e + f*x)/(sqrt(a + b*sec(e + f*x))*sqrt(c + d*sec(e + f*x))), x)

Maxima [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]

[In]

integrate(sec(f*x+e)/(a+b*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sec(f*x + e)/(sqrt(b*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c)), x)

Giac [F]

\[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )}{\sqrt {b \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}} \,d x } \]

[In]

integrate(sec(f*x+e)/(a+b*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sec(f*x + e)/(sqrt(b*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx=\int \frac {1}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]

[In]

int(1/(cos(e + f*x)*(a + b/cos(e + f*x))^(1/2)*(c + d/cos(e + f*x))^(1/2)),x)

[Out]

int(1/(cos(e + f*x)*(a + b/cos(e + f*x))^(1/2)*(c + d/cos(e + f*x))^(1/2)), x)