\(\int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx\) [665]

Optimal result

Integrand size = 25, antiderivative size = 75 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\frac {3 \cot (c+d x) \operatorname {EllipticPi}\left (-\frac {1}{2},\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{\sqrt {5} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2887} \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\frac {3 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \operatorname {EllipticPi}\left (-\frac {1}{2},\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right )}{\sqrt {5} d} \]

[In]

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

[Out]

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

Rule 2887

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

Rubi steps \begin{align*} \text {integral}& = \frac {3 \cot (c+d x) \operatorname {EllipticPi}\left (-\frac {1}{2},\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{\sqrt {5} d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.75 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\frac {2 i \sqrt {\cos (c+d x)} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right ),-\frac {1}{5}\right )-2 \operatorname {EllipticPi}\left (\frac {1}{5},i \text {arcsinh}\left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right ),-\frac {1}{5}\right )\right ) \sqrt {1+5 \tan ^2\left (\frac {1}{2} (c+d x)\right )}}{d \sqrt {30-20 \cos (c+d x)} \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}} \]

[In]

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

[Out]

((2*I)*Sqrt[Cos[c + d*x]]*(EllipticF[I*ArcSinh[Sqrt[5]*Tan[(c + d*x)/2]], -1/5] - 2*EllipticPi[1/5, I*ArcSinh[
Sqrt[5]*Tan[(c + d*x)/2]], -1/5])*Sqrt[1 + 5*Tan[(c + d*x)/2]^2])/(d*Sqrt[30 - 20*Cos[c + d*x]]*Sqrt[Cos[c + d
*x]/(1 + Cos[c + d*x])])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (65 ) = 130\).

Time = 5.62 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.87

method	result	size
default	\(-\frac {\sqrt {2}\, \left (F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), i \sqrt {5}\right )-2 \Pi \left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), -1, i \sqrt {5}\right )\right ) \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {3-2 \cos \left (d x +c \right )}\, \left (1+\cos \left (d x +c \right )\right )}{d \sqrt {\cos \left (d x +c \right )}\, \left (-3+2 \cos \left (d x +c \right )\right )}\)	\(140\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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