\(\int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx\) [673]

Optimal result

Integrand size = 27, antiderivative size = 97 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\frac {3 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {1}{2},\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{\sqrt {5} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 97, 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 = {2889, 2887} \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\frac {3 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \operatorname {EllipticPi}\left (-\frac {1}{2},\arcsin \left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right ),-\frac {1}{5}\right )}{\sqrt {5} d} \]

[In]

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

[Out]

(3*Sqrt[-Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Csc[c + d*x]*EllipticPi[-1/2, ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[C
os[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]

Rule 2889

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*S
in[e + f*x]]/Sqrt[(-b)*Sin[e + f*x]], Int[Sqrt[(-b)*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]], x], x] /; FreeQ[{b
, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && NegQ[(c + d)/b]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-\cos (c+d x)} \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx}{\sqrt {\cos (c+d x)}} \\ & = \frac {3 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (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 = 0.24 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {-\cos (c+d x)}}{\sqrt {3-2 \cos (c+d x)}} \, dx=\frac {2 i \sqrt {-\cos (c+d x)} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right ),-\frac {1}{5}\right )-2 \operatorname {EllipticPi}\left (\frac {1}{5},i \text {arcsinh}\left (\sqrt {5} \tan \left (\frac {1}{2} (c+d x)\right )\right ),-\frac {1}{5}\right )\right ) \sqrt {1+5 \tan ^2\left (\frac {1}{2} (c+d x)\right )}}{d \sqrt {30-20 \cos (c+d x)} \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}}} \]

[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 [A] (verified)

Time = 7.03 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.65

method	result	size
default	\(\frac {i \left (2 \Pi \left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}, \frac {1}{5}, \frac {i \sqrt {5}}{5}\right )-F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}, \frac {i \sqrt {5}}{5}\right )\right ) \sqrt {-\cos \left (d x +c \right )}\, \sqrt {3-2 \cos \left (d x +c \right )}\, \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \left (1+\sec \left (d x +c \right )\right ) \sqrt {5}}{5 d \left (-3+2 \cos \left (d x +c \right )\right )}\)	\(160\)

[In]

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

[Out]

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

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