Integrand size = 27, antiderivative size = 166 \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} c f}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} (c-d) f}+\frac {2 d^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a+a \sec (e+f x)}}\right )}{\sqrt {a} c (c-d) \sqrt {c+d} f} \]
2*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/c/f/a^(1/2)-arctan(1/2 *a^(1/2)*tan(f*x+e)*2^(1/2)/(a+a*sec(f*x+e))^(1/2))*2^(1/2)/(c-d)/f/a^(1/2 )+2*d^(3/2)*arctan(a^(1/2)*d^(1/2)*tan(f*x+e)/(c+d)^(1/2)/(a+a*sec(f*x+e)) ^(1/2))/c/(c-d)/f/a^(1/2)/(c+d)^(1/2)
Time = 3.44 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\frac {2 \left (-c \sqrt {c+d} \arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {2} \left ((c-d) \sqrt {c+d} \arctan \left (\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )+d^{3/2} \arctan \left (\frac {\sqrt {d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d} \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}}}\right )\right )\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} (d+c \cos (e+f x)) \sec ^{\frac {3}{2}}(e+f x) \sqrt {1+\sec (e+f x)}}{c (c-d) \sqrt {c+d} f \sqrt {\sec ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {a (1+\sec (e+f x))} (c+d \sec (e+f x))} \]
(2*(-(c*Sqrt[c + d]*ArcSin[Tan[(e + f*x)/2]]) + Sqrt[2]*((c - d)*Sqrt[c + d]*ArcTan[Tan[(e + f*x)/2]/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]] + d^(3/2 )*ArcTan[(Sqrt[d]*Tan[(e + f*x)/2])/(Sqrt[c + d]*Sqrt[Cos[e + f*x]/(1 + Co s[e + f*x])])]))*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*(d + c*Cos[e + f*x] )*Sec[e + f*x]^(3/2)*Sqrt[1 + Sec[e + f*x]])/(c*(c - d)*Sqrt[c + d]*f*Sqrt [Sec[(e + f*x)/2]^2]*Sqrt[a*(1 + Sec[e + f*x])]*(c + d*Sec[e + f*x]))
Time = 1.15 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {3042, 4417, 3042, 4408, 3042, 4261, 216, 4282, 216, 4455, 218}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{\sqrt {a \sec (e+f x)+a} (c+d \sec (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {a \csc \left (e+f x+\frac {\pi }{2}\right )+a} \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4417 |
| \(\displaystyle \frac {d^2 \int \frac {\sec (e+f x) \sqrt {\sec (e+f x) a+a}}{c+d \sec (e+f x)}dx}{a c (c-d)}+\frac {\int \frac {a (c-d)-a d \sec (e+f x)}{\sqrt {\sec (e+f x) a+a}}dx}{a c (c-d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a c (c-d)}+\frac {\int \frac {a (c-d)-a d \csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{a c (c-d)}\) |
| \(\Big \downarrow \) 4408 |
| \(\displaystyle \frac {d^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a c (c-d)}+\frac {(c-d) \int \sqrt {\sec (e+f x) a+a}dx-a c \int \frac {\sec (e+f x)}{\sqrt {\sec (e+f x) a+a}}dx}{a c (c-d)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a c (c-d)}+\frac {(c-d) \int \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}dx-a c \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{a c (c-d)}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {-\frac {2 a (c-d) \int \frac {1}{\frac {a^2 \tan ^2(e+f x)}{\sec (e+f x) a+a}+a}d\left (-\frac {a \tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{f}-a c \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{a c (c-d)}+\frac {d^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a c (c-d)}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2 \sqrt {a} (c-d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{f}-a c \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}dx}{a c (c-d)}+\frac {d^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a c (c-d)}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle \frac {\frac {2 a c \int \frac {1}{\frac {a^2 \tan ^2(e+f x)}{\sec (e+f x) a+a}+2 a}d\left (-\frac {a \tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{f}+\frac {2 \sqrt {a} (c-d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{f}}{a c (c-d)}+\frac {d^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a c (c-d)}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx}{a c (c-d)}+\frac {\frac {2 \sqrt {a} (c-d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{f}-\frac {\sqrt {2} \sqrt {a} c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{f}}{a c (c-d)}\) |
| \(\Big \downarrow \) 4455 |
| \(\displaystyle \frac {\frac {2 \sqrt {a} (c-d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{f}-\frac {\sqrt {2} \sqrt {a} c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{f}}{a c (c-d)}-\frac {2 d^2 \int \frac {1}{\frac {a^2 d \tan ^2(e+f x)}{\sec (e+f x) a+a}+a (c+d)}d\left (-\frac {a \tan (e+f x)}{\sqrt {\sec (e+f x) a+a}}\right )}{c f (c-d)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 d^{3/2} \arctan \left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a}}\right )}{\sqrt {a} c f (c-d) \sqrt {c+d}}+\frac {\frac {2 \sqrt {a} (c-d) \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{f}-\frac {\sqrt {2} \sqrt {a} c \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{f}}{a c (c-d)}\) |
((2*Sqrt[a]*(c - d)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]] ])/f - (Sqrt[2]*Sqrt[a]*c*ArcTan[(Sqrt[a]*Tan[e + f*x])/(Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/f)/(a*c*(c - d)) + (2*d^(3/2)*ArcTan[(Sqrt[a]*Sqrt[d]*T an[e + f*x])/(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])])/(Sqrt[a]*c*(c - d)*S qrt[c + d]*f)
3.2.69.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c/a Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[(b*c - a*d)/a Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; F reeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Int[1/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]* (d_.) + (c_))), x_Symbol] :> Simp[1/(c*(b*c - a*d)) Int[(b*c - a*d - b*d* Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[d^2/(c*(b*c - a*d)) Int[Csc[e + f*x]*(Sqrt[a + b*Csc[e + f*x]]/(c + d*Csc[e + f*x])), x], x] / ; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && (EqQ[a^2 - b^2, 0] | | EqQ[c^2 - d^2, 0])
Int[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/(c sc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Simp[-2*(b/f) Subst[In t[1/(b*c + a*d + d*x^2), x], x, b*(Cot[e + f*x]/Sqrt[a + b*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]
Leaf count of result is larger than twice the leaf count of optimal. \(658\) vs. \(2(137)=274\).
Time = 15.29 (sec) , antiderivative size = 659, normalized size of antiderivative = 3.97
| method | result | size |
| default | \(\frac {\left (2 \sqrt {\frac {d}{c -d}}\, \sqrt {2}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right ) c -2 \sqrt {\frac {d}{c -d}}\, \sqrt {2}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\right ) d -2 \sqrt {\frac {d}{c -d}}\, \sqrt {\left (c +d \right ) \left (c -d \right )}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right ) c +\sqrt {2}\, \ln \left (-\frac {2 \left (\sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, c -\sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, d +\sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-c +d \right )}{-c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right ) d^{2}-\sqrt {2}\, \ln \left (\frac {2 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, c -2 \sqrt {2}\, \sqrt {\frac {d}{c -d}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, d -2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-2 c +2 d}{c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right ) d^{2}\right ) \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\, \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}}{2 f \sqrt {\frac {d}{c -d}}\, \left (c -d \right ) c \sqrt {\left (c +d \right ) \left (c -d \right )}\, a}\) | \(659\) |
1/2/f/(d/(c-d))^(1/2)/(c-d)/c/((c+d)*(c-d))^(1/2)/a*(2*(d/(c-d))^(1/2)*2^( 1/2)*((c+d)*(c-d))^(1/2)*arctanh(2^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1) ^(1/2)*(-cot(f*x+e)+csc(f*x+e)))*c-2*(d/(c-d))^(1/2)*2^(1/2)*((c+d)*(c-d)) ^(1/2)*arctanh(2^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-cot(f*x+e )+csc(f*x+e)))*d-2*(d/(c-d))^(1/2)*((c+d)*(c-d))^(1/2)*ln(csc(f*x+e)-cot(f *x+e)+((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2))*c+2^(1/2)*ln(-2*(((1-cos(f* x+e))^2*csc(f*x+e)^2-1)^(1/2)*2^(1/2)*(d/(c-d))^(1/2)*c-2^(1/2)*(d/(c-d))^ (1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*d+((c+d)*(c-d))^(1/2)*(-cot( f*x+e)+csc(f*x+e))-c+d)/(-c*(-cot(f*x+e)+csc(f*x+e))+(-cot(f*x+e)+csc(f*x+ e))*d+((c+d)*(c-d))^(1/2)))*d^2-2^(1/2)*ln(2*(((1-cos(f*x+e))^2*csc(f*x+e) ^2-1)^(1/2)*2^(1/2)*(d/(c-d))^(1/2)*c-2^(1/2)*(d/(c-d))^(1/2)*((1-cos(f*x+ e))^2*csc(f*x+e)^2-1)^(1/2)*d-((c+d)*(c-d))^(1/2)*(-cot(f*x+e)+csc(f*x+e)) -c+d)/(c*(-cot(f*x+e)+csc(f*x+e))-(-cot(f*x+e)+csc(f*x+e))*d+((c+d)*(c-d)) ^(1/2)))*d^2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^(1/2)*(-2*a/((1-cos(f*x+e) )^2*csc(f*x+e)^2-1))^(1/2)
Time = 10.60 (sec) , antiderivative size = 1050, normalized size of antiderivative = 6.33 \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Too large to display} \]
[-1/2*(sqrt(2)*a*c*sqrt(-1/a)*log(-(2*sqrt(2)*sqrt((a*cos(f*x + e) + a)/co s(f*x + e))*sqrt(-1/a)*cos(f*x + e)*sin(f*x + e) - 3*cos(f*x + e)^2 - 2*co s(f*x + e) + 1)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + 2*a*d*sqrt(-d/(a* c + a*d))*log((2*(c + d)*sqrt(-d/(a*c + a*d))*sqrt((a*cos(f*x + e) + a)/co s(f*x + e))*cos(f*x + e)*sin(f*x + e) + (c + 2*d)*cos(f*x + e)^2 + (c + d) *cos(f*x + e) - d)/(c*cos(f*x + e)^2 + (c + d)*cos(f*x + e) + d)) + 2*sqrt (-a)*(c - d)*log((2*a*cos(f*x + e)^2 + 2*sqrt(-a)*sqrt((a*cos(f*x + e) + a )/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos(f*x + e) + 1)))/((a*c^2 - a*c*d)*f), -1/2*(sqrt(2)*a*c*sqrt(-1/a)*log(-(2*sqrt( 2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt(-1/a)*cos(f*x + e)*sin(f*x + e) - 3*cos(f*x + e)^2 - 2*cos(f*x + e) + 1)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + 4*a*d*sqrt(d/(a*c + a*d))*arctan((c + d)*sqrt(d/(a*c + a*d)) *sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(d*sin(f*x + e))) + 2*sqrt(-a)*(c - d)*log((2*a*cos(f*x + e)^2 + 2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos (f*x + e) + 1)))/((a*c^2 - a*c*d)*f), -(a*d*sqrt(-d/(a*c + a*d))*log((2*(c + d)*sqrt(-d/(a*c + a*d))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + (c + 2*d)*cos(f*x + e)^2 + (c + d)*cos(f*x + e) - d)/ (c*cos(f*x + e)^2 + (c + d)*cos(f*x + e) + d)) - sqrt(2)*sqrt(a)*c*arctan( sqrt(2)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*s...
\[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int \frac {1}{\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (c + d \sec {\left (e + f x \right )}\right )}\, dx \]
\[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int { \frac {1}{\sqrt {a \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}} \,d x } \]
Exception generated. \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {1}{\sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \, dx=\int \frac {1}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )} \,d x \]