\(\int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx\) [671]

Optimal result

Integrand size = 27, antiderivative size = 79 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=-\frac {4 \cot (c+d x) \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{3 d} \] Output:

-4/3*cot(d*x+c)*EllipticPi(1/5*(-2-3*cos(d*x+c))^(1/2)*5^(1/2)/(-cos(d*x+c 
))^(1/2),5/3,5^(1/2))*(-1-sec(d*x+c))^(1/2)*(1-sec(d*x+c))^(1/2)/d
 

Mathematica [A] (warning: unable to verify)

Time = 0.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {-\frac {(2+3 \cos (c+d x))^2}{(1+\cos (c+d x))^2}} \left (\operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {1}{5}\right )-2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {1}{5}\right )\right )}{\sqrt {5} d \sqrt {-2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)} \sqrt {\frac {-2-3 \cos (c+d x)}{1+\cos (c+d x)}}} \] Input:

Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]],x]
 

Output:

(4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[-((2 + 3* 
Cos[c + d*x])^2/(1 + Cos[c + d*x])^2)]*(EllipticF[ArcSin[Tan[(c + d*x)/2]] 
, 1/5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], 1/5]))/(Sqrt[5]*d*Sqrt 
[-2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]*Sqrt[(-2 - 3*Cos[c + d*x])/(1 + 
Cos[c + d*x])])
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {3042, 3288}

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.

\(\displaystyle \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-3 \cos (c+d x)-2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {-\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {-3 \sin \left (c+d x+\frac {\pi }{2}\right )-2}}dx\)

\(\Big \downarrow \) 3288

\(\displaystyle -\frac {4 \cot (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {-3 \cos (c+d x)-2}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),5\right )}{3 d}\)

Input:

Int[Sqrt[-Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]],x]
 

Output:

(-4*Cot[c + d*x]*EllipticPi[5/3, ArcSin[Sqrt[-2 - 3*Cos[c + d*x]]/(Sqrt[5] 
*Sqrt[-Cos[c + d*x]])], 5]*Sqrt[-1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]]) 
/(3*d)
 

Defintions of rubi rules used

rule 3042

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

rule 3288

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c 
*((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti 
cPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + 
 d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - 
 d^2, 0] && PosQ[(c + d)/b]
 

Maple [A] (verified)

Time = 8.60 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.67

method	result	size
default	\(-\frac {\sqrt {2}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {-\cos \left (d x +c \right )}\, \left (-\operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {\sqrt {5}}{5}\right )+2 \operatorname {EllipticPi}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), -1, \frac {\sqrt {5}}{5}\right )\right ) \left (1+\sec \left (d x +c \right )\right )}{5 d \sqrt {-2-3 \cos \left (d x +c \right )}}\)	\(132\)

Input:

int((-cos(d*x+c))^(1/2)/(-2-3*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/5/d/(-2-3*cos(d*x+c))^(1/2)*2^(1/2)*10^(1/2)*((2+3*cos(d*x+c))/(cos(d*x 
+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(-cos(d*x+c))^(1/2)*(-Elli 
pticF(cot(d*x+c)-csc(d*x+c),1/5*5^(1/2))+2*EllipticPi(cot(d*x+c)-csc(d*x+c 
),-1,1/5*5^(1/2)))*(1+sec(d*x+c))
 

Fricas [F]

\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2}} \,d x } \] Input:

integrate((-cos(d*x+c))^(1/2)/(-2-3*cos(d*x+c))^(1/2),x, algorithm="fricas 
")
 

Output:

integral(-sqrt(-cos(d*x + c))*sqrt(-3*cos(d*x + c) - 2)/(3*cos(d*x + c) + 
2), x)
 

Sympy [F]

\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int \frac {\sqrt {- \cos {\left (c + d x \right )}}}{\sqrt {- 3 \cos {\left (c + d x \right )} - 2}}\, dx \] Input:

integrate((-cos(d*x+c))**(1/2)/(-2-3*cos(d*x+c))**(1/2),x)
 

Output:

Integral(sqrt(-cos(c + d*x))/sqrt(-3*cos(c + d*x) - 2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2}} \,d x } \] Input:

integrate((-cos(d*x+c))^(1/2)/(-2-3*cos(d*x+c))^(1/2),x, algorithm="maxima 
")
 

Output:

integrate(sqrt(-cos(d*x + c))/sqrt(-3*cos(d*x + c) - 2), x)
 

Giac [F]

\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int { \frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {-3 \, \cos \left (d x + c\right ) - 2}} \,d x } \] Input:

integrate((-cos(d*x+c))^(1/2)/(-2-3*cos(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-cos(d*x + c))/sqrt(-3*cos(d*x + c) - 2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=\int \frac {\sqrt {-\cos \left (c+d\,x\right )}}{\sqrt {-3\,\cos \left (c+d\,x\right )-2}} \,d x \] Input:

int((-cos(c + d*x))^(1/2)/(- 3*cos(c + d*x) - 2)^(1/2),x)
 

Output:

int((-cos(c + d*x))^(1/2)/(- 3*cos(c + d*x) - 2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=-\left (\int \frac {\sqrt {-3 \cos \left (d x +c \right )-2}\, \sqrt {\cos \left (d x +c \right )}}{3 \cos \left (d x +c \right )+2}d x \right ) i \] Input:

int((-cos(d*x+c))^(1/2)/(-2-3*cos(d*x+c))^(1/2),x)
 

Output:

 - int((sqrt( - 3*cos(c + d*x) - 2)*sqrt(cos(c + d*x)))/(3*cos(c + d*x) + 
2),x)*i