\(\int \frac {\sqrt {\sec (c+d x)} (A+B \sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx\) [458]

Optimal result

Integrand size = 35, antiderivative size = 138 \[ \int \frac {\sqrt {\sec (c+d x)} (A+B \sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx=\frac {2 A \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 B \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{d \sqrt {a+b \sec (c+d x)}} \] Output:

2*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)/d/(a+b*sec(d*x+c))^(1/2)+2*B*((b+a*cos(d* 
x+c))/(a+b))^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(a/(a+b))^(1/2) 
)*sec(d*x+c)^(1/2)/d/(a+b*sec(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {\sec (c+d x)} (A+B \sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx=\frac {2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \left (A \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )+B \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )\right ) \sqrt {\sec (c+d x)}}{d \sqrt {a+b \sec (c+d x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 138, 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, 4524, 3042, 4345, 3042, 3142, 3042, 3140, 4346, 3042, 3286, 3042, 3284}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4524

\(\displaystyle A \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+B \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle A \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+B \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4345

\(\displaystyle \frac {A \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+B \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+B \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {A \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+B \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+B \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 3140

\(\displaystyle B \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}\)

\(\Big \downarrow \) 4346

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3286

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3284

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

Input:

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

Output:

(2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + 
b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*B*Sqrt[(b + a*Co 
s[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c 
+ d*x]])/(d*Sqrt[a + b*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 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 3284

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 
2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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] && GtQ[c + d, 0]
 

rule 3286

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt 
[c + d*Sin[e + f*x]]   Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d/(c + 
 d))*Sin[e + f*x]]), x], 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] &&  !GtQ[c + d, 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 4346

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_. 
) + (a_)], x_Symbol] :> Simp[d*Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x 
]]/Sqrt[a + b*Csc[e + f*x]])   Int[1/(Sin[e + f*x]*Sqrt[b + a*Sin[e + f*x]] 
), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4524

Int[(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + ( 
A_)))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[A   Int 
[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[B/d   Int[(d* 
Csc[e + f*x])^(3/2)/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 [C] (verified)

Result contains complex when optimal does not.

Time = 11.02 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.82

method	result	size
default	\(-\frac {2 \left (A \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )-B \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )+2 B \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), \frac {a +b}{a -b}, \frac {i}{\sqrt {\frac {a -b}{a +b}}}\right )\right ) \sqrt {a +b \sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right )}\, \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, \cos \left (d x +c \right )}{d \sqrt {\frac {a -b}{a +b}}\, \left (b +a \cos \left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}}\)	\(251\)
parts	\(-\frac {2 B \left (\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \frac {a +b}{a -b}, \frac {i}{\sqrt {\frac {a -b}{a +b}}}\right )\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, \sec \left (d x +c \right )^{\frac {3}{2}} \sqrt {a +b \sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{d \sqrt {\frac {a -b}{a +b}}\, \left (b +a \cos \left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}}+\frac {2 A \sqrt {\sec \left (d x +c \right )}\, \sqrt {a +b \sec \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right ) \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (1+\cos \left (d x +c \right )\right )}}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )\right )}{d \sqrt {\frac {a -b}{a +b}}\, \left (b +a \cos \left (d x +c \right )\right )}\)	\(355\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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