\(\int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx\) [459]

Optimal result

Integrand size = 35, antiderivative size = 150 \[ \int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=-\frac {2 (A b-a B) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 A E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \] Output:

-2*(A*b-B*a)*((b+a*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c, 
2^(1/2)*(a/(a+b))^(1/2))*sec(d*x+c)^(1/2)/a/d/(a+b*sec(d*x+c))^(1/2)+2*A*E 
llipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2 
)/a/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 3.77 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.69 \[ \int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\frac {2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \left (A (a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+(-A b+a B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )\right ) \sqrt {\sec (c+d x)}}{a d \sqrt {a+b \sec (c+d x)}} \] Input:

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

Output:

(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(A*(a + b)*EllipticE[(c + d*x)/2, (2 
*a)/(a + b)] + (-(A*b) + a*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)])*Sqrt[ 
Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]])
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}

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.

\(\displaystyle \int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4523

\(\displaystyle \frac {A \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}-\frac {(A b-a B) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {(A b-a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}\)

\(\Big \downarrow \) 4343

\(\displaystyle \frac {A \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {(A b-a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}-\frac {(A b-a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {A \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {(A b-a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {(A b-a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2 A \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {(A b-a B) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}\)

\(\Big \downarrow \) 4345

\(\displaystyle \frac {2 A \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {(A b-a B) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 A \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {(A b-a B) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {2 A \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {(A b-a B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 A \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {(A b-a B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 A \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}-\frac {2 (A b-a B) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}\)

Input:

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

Output:

(-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, 
(2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*A*E 
llipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[( 
b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 4343

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] 
*(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S 
qrt[b + a*Sin[e + f*x]])   Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a 
, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4345

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) 
+ (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S 
qrt[a + b*Csc[e + f*x]])   Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ 
{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4523

Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d 
_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a   I 
nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) 
/(a*d)   Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ 
[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(546\) vs. \(2(147)=294\).

Time = 14.27 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.65

method	result	size
default	\(\frac {2 \left (\left (-\cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )-1\right ) A \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, a \operatorname {EllipticE}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) A \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, b \operatorname {EllipticE}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )+\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) A \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, a \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )+\left (-\cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )-1\right ) B \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, a \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )+A \sqrt {\frac {a -b}{a +b}}\, a \cos \left (d x +c \right ) \sin \left (d x +c \right )+A \sqrt {\frac {a -b}{a +b}}\, b \sin \left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{d \sqrt {\frac {a -b}{a +b}}\, a \left (\cos \left (d x +c \right )^{2} a +a \cos \left (d x +c \right )+b \cos \left (d x +c \right )+b \right ) \sqrt {\sec \left (d x +c \right )}}\)	\(547\)
parts	\(\frac {2 B \sqrt {\sec \left (d x +c \right )}\, \sqrt {a +b \sec \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )\right )}{d \sqrt {\frac {a -b}{a +b}}\, \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 A \left (\left (\cos \left (d x +c \right )^{2}+2 \cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, a \operatorname {EllipticE}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )+\left (-\cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )-1\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, b \operatorname {EllipticE}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )+\left (-\cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )-1\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, a \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )+\sqrt {\frac {a -b}{a +b}}\, a \cos \left (d x +c \right ) \sin \left (d x +c \right )+\sqrt {\frac {a -b}{a +b}}\, b \sin \left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{d \sqrt {\frac {a -b}{a +b}}\, a \left (\cos \left (d x +c \right )^{2} a +a \cos \left (d x +c \right )+b \cos \left (d x +c \right )+b \right ) \sqrt {\sec \left (d x +c \right )}}\)	\(589\)
risch	\(\text {Expression too large to display}\)	\(1140\)

Input:

int((A+B*sec(d*x+c))/sec(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

2/d/((a-b)/(a+b))^(1/2)/a*((-cos(d*x+c)^2-2*cos(d*x+c)-1)*A*(1/(a+b)*(b+a* 
cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a*EllipticE(((a 
-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(cos(d*x+c 
)^2+2*cos(d*x+c)+1)*A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/( 
1+cos(d*x+c)))^(1/2)*b*EllipticE(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+ 
c)),(-(a+b)/(a-b))^(1/2))+(cos(d*x+c)^2+2*cos(d*x+c)+1)*A*(1/(a+b)*(b+a*co 
s(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1+cos(d*x+c)))^(1/2)*a*EllipticF(((a-b 
)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))+(-cos(d*x+c) 
^2-2*cos(d*x+c)-1)*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(1/(1 
+cos(d*x+c)))^(1/2)*a*EllipticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c 
)),(-(a+b)/(a-b))^(1/2))+A*((a-b)/(a+b))^(1/2)*a*cos(d*x+c)*sin(d*x+c)+A*( 
(a-b)/(a+b))^(1/2)*b*sin(d*x+c))*(a+b*sec(d*x+c))^(1/2)/(cos(d*x+c)^2*a+a* 
cos(d*x+c)+b*cos(d*x+c)+b)/sec(d*x+c)^(1/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.47 \[ \int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\frac {3 i \, \sqrt {2} A a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 i \, \sqrt {2} A a^{\frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \sqrt {2} {\left (-3 i \, B a + 2 i \, A b\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (3 i \, B a - 2 i \, A b\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )}{3 \, a^{2} d} \] Input:

integrate((A+B*sec(d*x+c))/sec(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2),x, algo 
rithm="fricas")
 

Output:

1/3*(3*I*sqrt(2)*A*a^(3/2)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27* 
(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27* 
(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a 
)) - 3*I*sqrt(2)*A*a^(3/2)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27* 
(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27* 
(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a 
)) + sqrt(2)*(-3*I*B*a + 2*I*A*b)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 
- 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*si 
n(d*x + c) + 2*b)/a) + sqrt(2)*(3*I*B*a - 2*I*A*b)*sqrt(a)*weierstrassPInv 
erse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d* 
x + c) - 3*I*a*sin(d*x + c) + 2*b)/a))/(a^2*d)
 

Sympy [F]

\[ \int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {A + B \sec {\left (c + d x \right )}}{\sqrt {a + b \sec {\left (c + d x \right )}} \sqrt {\sec {\left (c + d x \right )}}}\, dx \] Input:

integrate((A+B*sec(d*x+c))/sec(d*x+c)**(1/2)/(a+b*sec(d*x+c))**(1/2),x)
 

Output:

Integral((A + B*sec(c + d*x))/(sqrt(a + b*sec(c + d*x))*sqrt(sec(c + d*x)) 
), x)
 

Maxima [F]

\[ \int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{\sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:

integrate((A+B*sec(d*x+c))/sec(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2),x, algo 
rithm="maxima")
 

Output:

integrate((B*sec(d*x + c) + A)/(sqrt(b*sec(d*x + c) + a)*sqrt(sec(d*x + c) 
)), x)
 

Giac [F]

\[ \int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\int { \frac {B \sec \left (d x + c\right ) + A}{\sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:

integrate((A+B*sec(d*x+c))/sec(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2),x, algo 
rithm="giac")
 

Output:

integrate((B*sec(d*x + c) + A)/(sqrt(b*sec(d*x + c) + a)*sqrt(sec(d*x + c) 
)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )}d x \] Input:

int((A+B*sec(d*x+c))/sec(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2),x)
 

Output:

int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a))/sec(c + d*x),x)