\(\int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx\) [184]

Optimal result

Integrand size = 29, antiderivative size = 123 \[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} \sqrt {c} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f}+\frac {2 \sqrt {a} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f} \]

[Out]

2*arctan(a^(1/2)*c^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2))*a^(1/2)*c^(1/2)/f+2*arctanh
(a^(1/2)*d^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2))*a^(1/2)*d^(1/2)/f

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4017, 4019, 209, 4065, 212} \[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} \sqrt {c} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{f}+\frac {2 \sqrt {a} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{f} \]

[In]

Int[Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]

[Out]

(2*Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])
/f + (2*Sqrt[a]*Sqrt[d]*ArcTanh[(Sqrt[a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*
x]])])/f

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 4017

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

Rule 4019

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

Rule 4065

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

Rubi steps \begin{align*} \text {integral}& = c \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx+d \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx \\ & = -\frac {(2 a c) \text {Subst}\left (\int \frac {1}{1+a c x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f}-\frac {(2 a d) \text {Subst}\left (\int \frac {1}{1-a d x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f} \\ & = \frac {2 \sqrt {a} \sqrt {c} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f}+\frac {2 \sqrt {a} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 14.89 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.95 \[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=-\frac {2 \cot (e+f x) \sqrt {a (1+\sec (e+f x))} \sqrt {c+d \sec (e+f x)} \left (-2 \sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+c \cos (e+f x)}}{\sqrt {d} \sqrt {c-c \cos (e+f x)}}\right ) \sqrt {c (1+\cos (e+f x))} \sin ^2\left (\frac {1}{2} (e+f x)\right )+\arctan \left (\frac {\sqrt {c (1+\cos (e+f x))} \sqrt {d+c \cos (e+f x)}}{\sqrt {c^2 \sin ^2(e+f x)}}\right ) \sqrt {c-c \cos (e+f x)} \sqrt {c^2 \sin ^2(e+f x)}\right )}{f \sqrt {c (1+\cos (e+f x))} \sqrt {c-c \cos (e+f x)} \sqrt {d+c \cos (e+f x)}} \]

[In]

Integrate[Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]],x]

[Out]

(-2*Cot[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]]*(-2*Sqrt[c]*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt
[d + c*Cos[e + f*x]])/(Sqrt[d]*Sqrt[c - c*Cos[e + f*x]])]*Sqrt[c*(1 + Cos[e + f*x])]*Sin[(e + f*x)/2]^2 + ArcT
an[(Sqrt[c*(1 + Cos[e + f*x])]*Sqrt[d + c*Cos[e + f*x]])/Sqrt[c^2*Sin[e + f*x]^2]]*Sqrt[c - c*Cos[e + f*x]]*Sq
rt[c^2*Sin[e + f*x]^2]))/(f*Sqrt[c*(1 + Cos[e + f*x])]*Sqrt[c - c*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1495\) vs. \(2(99)=198\).

Time = 5.00 (sec) , antiderivative size = 1496, normalized size of antiderivative = 12.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1496\)

[In]

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

[Out]

-1/f/(c-d)^(1/2)*2^(1/2)/(-d)^(1/2)/(c^2-2*c*d+d^2)*(a*(sec(f*x+e)+1))^(1/2)*(c+d*sec(f*x+e))^(1/2)*(2^(1/2)*(
-d)^(1/2)*ln(-(c*cot(f*x+e)-d*cot(f*x+e)-c*csc(f*x+e)+d*csc(f*x+e)-(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*
(c-d)^(1/2))/(c-d)^(1/2))*c^3-3*2^(1/2)*(-d)^(1/2)*ln(-(c*cot(f*x+e)-d*cot(f*x+e)-c*csc(f*x+e)+d*csc(f*x+e)-(-
2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(c-d)^(1/2))/(c-d)^(1/2))*c^2*d+3*2^(1/2)*(-d)^(1/2)*ln(-(c*cot(f*x+e
)-d*cot(f*x+e)-c*csc(f*x+e)+d*csc(f*x+e)-(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(c-d)^(1/2))/(c-d)^(1/2))*
c*d^2-2^(1/2)*(-d)^(1/2)*ln(-(c*cot(f*x+e)-d*cot(f*x+e)-c*csc(f*x+e)+d*csc(f*x+e)-(-2*(d+c*cos(f*x+e))/(cos(f*
x+e)+1))^(1/2)*(c-d)^(1/2))/(c-d)^(1/2))*d^3-2^(1/2)*(-d)^(1/2)*ln((-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)-
(c-d)^(1/2)*cot(f*x+e)+(c-d)^(1/2)*csc(f*x+e))*c^3+3*2^(1/2)*(-d)^(1/2)*ln((-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1)
)^(1/2)-(c-d)^(1/2)*cot(f*x+e)+(c-d)^(1/2)*csc(f*x+e))*c^2*d-3*2^(1/2)*(-d)^(1/2)*ln((-2*(d+c*cos(f*x+e))/(cos
(f*x+e)+1))^(1/2)-(c-d)^(1/2)*cot(f*x+e)+(c-d)^(1/2)*csc(f*x+e))*c*d^2+2^(1/2)*(-d)^(1/2)*ln((-2*(d+c*cos(f*x+
e))/(cos(f*x+e)+1))^(1/2)-(c-d)^(1/2)*cot(f*x+e)+(c-d)^(1/2)*csc(f*x+e))*d^3+(c-d)^(1/2)*ln(-2*(2^(1/2)*(-d)^(
1/2)*(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)-sin(f*x+e)*c-sin(f*x+e)*d-c*cos(f*x+e)+d*cos(f*x+e)
+c-d)/(cos(f*x+e)-1+sin(f*x+e)))*c^2*d-2*(c-d)^(1/2)*ln(-2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(cos(f*x+e
)+1))^(1/2)*sin(f*x+e)-sin(f*x+e)*c-sin(f*x+e)*d-c*cos(f*x+e)+d*cos(f*x+e)+c-d)/(cos(f*x+e)-1+sin(f*x+e)))*c*d
^2+(c-d)^(1/2)*ln(-2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)-sin(f*x+e)*c-si
n(f*x+e)*d-c*cos(f*x+e)+d*cos(f*x+e)+c-d)/(cos(f*x+e)-1+sin(f*x+e)))*d^3-(c-d)^(1/2)*ln(2*(2^(1/2)*(-d)^(1/2)*
(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)-sin(f*x+e)*c-sin(f*x+e)*d+c*cos(f*x+e)-d*cos(f*x+e)-c+d)
/(-cos(f*x+e)+1+sin(f*x+e)))*c^2*d+2*(c-d)^(1/2)*ln(2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))
^(1/2)*sin(f*x+e)-sin(f*x+e)*c-sin(f*x+e)*d+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*c*d^2-(
c-d)^(1/2)*ln(2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)-sin(f*x+e)*c-sin(f*x
+e)*d+c*cos(f*x+e)-d*cos(f*x+e)-c+d)/(-cos(f*x+e)+1+sin(f*x+e)))*d^3+2*(-(c-d)^4*c)^(1/2)*arctan((c-d)^2*c*2^(
1/2)/(-(c-d)^4*c)^(1/2)*sin(f*x+e)/(cos(f*x+e)+1)/(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2))*(c-d)^(1/2)*(-d)
^(1/2))*cos(f*x+e)/(cos(f*x+e)+1)/(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 806, normalized size of antiderivative = 6.55 \[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\left [\frac {\sqrt {a d} \log \left (\frac {2 \, \sqrt {a d} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (a c - a d\right )} \cos \left (f x + e\right )^{2} + 2 \, a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )}\right ) + \sqrt {-a c} \log \left (\frac {2 \, a c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c + a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{f}, -\frac {2 \, \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a c \sin \left (f x + e\right )}\right ) - \sqrt {a d} \log \left (\frac {2 \, \sqrt {a d} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (a c - a d\right )} \cos \left (f x + e\right )^{2} + 2 \, a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )}\right )}{f}, -\frac {2 \, \sqrt {-a d} \arctan \left (\frac {\sqrt {-a d} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a d \sin \left (f x + e\right )}\right ) - \sqrt {-a c} \log \left (\frac {2 \, a c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c + a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{f}, -\frac {2 \, {\left (\sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a c \sin \left (f x + e\right )}\right ) + \sqrt {-a d} \arctan \left (\frac {\sqrt {-a d} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a d \sin \left (f x + e\right )}\right )\right )}}{f}\right ] \]

[In]

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

[Out]

[(sqrt(a*d)*log((2*sqrt(a*d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*c
os(f*x + e)*sin(f*x + e) + (a*c - a*d)*cos(f*x + e)^2 + 2*a*d + (a*c + a*d)*cos(f*x + e))/(cos(f*x + e)^2 + co
s(f*x + e))) + sqrt(-a*c)*log((2*a*c*cos(f*x + e)^2 - 2*sqrt(-a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqr
t((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) - a*c + a*d + (a*c + a*d)*cos(f*x + e))/(cos(f*
x + e) + 1)))/f, -(2*sqrt(a*c)*arctan(sqrt(a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) +
d)/cos(f*x + e))*cos(f*x + e)/(a*c*sin(f*x + e))) - sqrt(a*d)*log((2*sqrt(a*d)*sqrt((a*cos(f*x + e) + a)/cos(
f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + (a*c - a*d)*cos(f*x + e)^2 + 2*a
*d + (a*c + a*d)*cos(f*x + e))/(cos(f*x + e)^2 + cos(f*x + e))))/f, -(2*sqrt(-a*d)*arctan(sqrt(-a*d)*sqrt((a*c
os(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)/(a*d*sin(f*x + e))) - sqrt
(-a*c)*log((2*a*c*cos(f*x + e)^2 - 2*sqrt(-a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) +
d)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) - a*c + a*d + (a*c + a*d)*cos(f*x + e))/(cos(f*x + e) + 1)))/f, -2
*(sqrt(a*c)*arctan(sqrt(a*c)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*c
os(f*x + e)/(a*c*sin(f*x + e))) + sqrt(-a*d)*arctan(sqrt(-a*d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c
*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)/(a*d*sin(f*x + e))))/f]

Sympy [F]

\[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\int \sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \sqrt {c + d \sec {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(a*(sec(e + f*x) + 1))*sqrt(c + d*sec(e + f*x)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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