\(\int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx\) [646]

Optimal result

Integrand size = 25, antiderivative size = 56 \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {\cos (c+d x)}} \] Output:

-2/5*(-cos(d*x+c))^(1/2)*EllipticF(sin(d*x+c)/(1-cos(d*x+c)),1/5*5^(1/2))* 
5^(1/2)/d/cos(d*x+c)^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(56)=112\).

Time = 2.95 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.55 \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\frac {4 \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(2-3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{d \sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \] Input:

Integrate[1/(Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]
 

Output:

(-4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[(2 - 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2] 
*Sqrt[Cos[c + d*x]*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[ 
Cos[c + d*x]*Csc[(c + d*x)/2]^2]/2], -4]*Sin[(c + d*x)/2]^4)/(d*Sqrt[2 - 3 
*Cos[c + d*x]]*Sqrt[Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3293, 3042, 3292}

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.

\(\displaystyle \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {2-3 \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 3293

\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {-\cos (c+d x)}}dx}{\sqrt {\cos (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {-\cos (c+d x)} \int \frac {1}{\sqrt {2-3 \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {-\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {\cos (c+d x)}}\)

\(\Big \downarrow \) 3292

\(\displaystyle -\frac {2 \sqrt {-\cos (c+d x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sin (c+d x)}{1-\cos (c+d x)}\right ),\frac {1}{5}\right )}{\sqrt {5} d \sqrt {\cos (c+d x)}}\)

Input:

Int[1/(Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]
 

Output:

(-2*Sqrt[-Cos[c + d*x]]*EllipticF[ArcSin[Sin[c + d*x]/(1 - Cos[c + d*x])], 
 1/5])/(Sqrt[5]*d*Sqrt[Cos[c + d*x]])
 

Defintions of rubi rules used

rule 3042

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

rule 3292

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[-2*(d/(f*Sqrt[a + b*d]))*EllipticF[ArcSin[Co 
s[e + f*x]/(1 + d*Sin[e + f*x])], -(a - b*d)/(a + b*d)], x] /; FreeQ[{a, b, 
 d, e, f}, x] && LtQ[a^2 - b^2, 0] && EqQ[d^2, 1] && GtQ[b*d, 0]
 

rule 3293

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[Sqrt[Sign[b]*Sin[e + f*x]]/Sqrt[d*Sin[e + f* 
x]]   Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[Sign[b]*Sin[e + f*x]]), x], x] / 
; FreeQ[{a, b, d, e, f}, x] && LtQ[a^2 - b^2, 0] && GtQ[b^2, 0] &&  !(EqQ[d 
^2, 1] && GtQ[b*d, 0])
 

Maple [A] (verified)

Time = 9.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.70

method	result	size
default	\(-\frac {2 \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos \left (d x +c \right )+1\right ) \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {5}\right )}{d \sqrt {2-3 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\)	\(95\)

Input:

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

Output:

-2/d/(2-3*cos(d*x+c))^(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)*(cos(d*x+c)+1)*EllipticF(cot 
(d*x+c)-csc(d*x+c),5^(1/2))
 

Fricas [F]

\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {2 - 3 \cos {\left (c + d x \right )}} \sqrt {\cos {\left (c + d x \right )}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {2-3\,\cos \left (c+d\,x\right )}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)}} \, dx=-\left (\int \frac {\sqrt {-3 \cos \left (d x +c \right )+2}\, \sqrt {\cos \left (d x +c \right )}}{3 \cos \left (d x +c \right )^{2}-2 \cos \left (d x +c \right )}d x \right ) \] Input:

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