\(\int \frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx\) [605]

Optimal result

Integrand size = 25, antiderivative size = 135 \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {\frac {a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt {\frac {a (1+\cos (c+d x))}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \csc (c+d x) \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {a+b} \sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}\right ),-\frac {a-b}{a+b}\right )}{\sqrt {a+b} d} \] Output:

-2*(a*(1-cos(d*x+c))/(a+b*cos(d*x+c)))^(1/2)*(a*(1+cos(d*x+c))/(a+b*cos(d* 
x+c)))^(1/2)*(a+b*cos(d*x+c))*csc(d*x+c)*EllipticPi((a+b)^(1/2)*cos(d*x+c) 
^(1/2)/(a+b*cos(d*x+c))^(1/2),b/(a+b),(-(a-b)/(a+b))^(1/2))/(a+b)^(1/2)/d
 

Mathematica [A] (verified)

Time = 2.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \left ((a-b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+2 b \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )\right )}{d \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {a+b \cos (c+d x)}} \] Input:

Integrate[Sqrt[a + b*Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]
 

Output:

(2*Sqrt[Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x] 
))]*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*E 
llipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]))/(d*Sqrt[Cos[c 
+ d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3042, 3290}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3290

\(\displaystyle -\frac {2 \csc (c+d x) \sqrt {\frac {a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt {\frac {a (\cos (c+d x)+1)}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {a+b} \sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}\right ),-\frac {a-b}{a+b}\right )}{d \sqrt {a+b}}\)

Input:

Int[Sqrt[a + b*Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]
 

Output:

(-2*Sqrt[(a*(1 - Cos[c + d*x]))/(a + b*Cos[c + d*x])]*Sqrt[(a*(1 + Cos[c + 
 d*x]))/(a + b*Cos[c + d*x])]*(a + b*Cos[c + d*x])*Csc[c + d*x]*EllipticPi 
[b/(a + b), ArcSin[(Sqrt[a + b]*Sqrt[Cos[c + d*x]])/Sqrt[a + b*Cos[c + d*x 
]]], -((a - b)/(a + b))])/(Sqrt[a + b]*d)
 

Defintions of rubi rules used

rule 3042

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

rule 3290

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

Maple [A] (verified)

Time = 8.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.34

method	result	size
default	\(-\frac {2 \sqrt {\frac {a +\cos \left (d x +c \right ) b}{\left (\cos \left (d x +c \right )+1\right ) \left (a +b \right )}}\, \left (\operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) a -\operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) b +2 b \operatorname {EllipticPi}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), -1, \sqrt {-\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+1\right )}{d \sqrt {a +\cos \left (d x +c \right ) b}\, \sqrt {\cos \left (d x +c \right )}}\)	\(181\)

Input:

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

Output:

-2/d*((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*(EllipticF(cot(d*x+c)-c 
sc(d*x+c),(-(a-b)/(a+b))^(1/2))*a-EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/ 
(a+b))^(1/2))*b+2*b*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,(-(a-b)/(a+b))^(1/ 
2)))/(a+cos(d*x+c)*b)^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+ 
1)/cos(d*x+c)^(1/2)
 

Fricas [F]

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

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

Output:

integral(sqrt(b*cos(d*x + c) + a)/sqrt(cos(d*x + c)), x)
 

Sympy [F]

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

integrate((a+b*cos(d*x+c))**(1/2)/cos(d*x+c)**(1/2),x)
 

Output:

Integral(sqrt(a + b*cos(c + d*x))/sqrt(cos(c + d*x)), x)
 

Maxima [F]

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

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

Output:

integrate(sqrt(b*cos(d*x + c) + a)/sqrt(cos(d*x + c)), x)
 

Giac [F]

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

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

Output:

integrate(sqrt(b*cos(d*x + c) + a)/sqrt(cos(d*x + c)), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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