\(\int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx\) [667]

Optimal result

Integrand size = 25, antiderivative size = 97 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=-\frac {3 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \operatorname {EllipticPi}\left (\frac {5}{2},\arcsin \left (\frac {\sqrt {-3-2 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{d \sqrt {-\cos (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2889, 2887} \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=-\frac {3 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \operatorname {EllipticPi}\left (\frac {5}{2},\arcsin \left (\frac {\sqrt {-2 \cos (c+d x)-3}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),-5\right )}{d \sqrt {-\cos (c+d x)}} \]

[In]

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

[Out]

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

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 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \operatorname {EllipticPi}\left (\frac {5}{2},\arcsin \left (\frac {\sqrt {-3-2 \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)}}{d \sqrt {-\cos (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-3-2 \cos (c+d x)}} \, dx=\frac {2 i \cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {5}}\right ),-5\right )-2 \operatorname {EllipticPi}\left (5,i \text {arcsinh}\left (\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {5}}\right ),-5\right )\right ) \sqrt {\cos (c+d x) (3+2 \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )}}{d \sqrt {-3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 6.82 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.63

method	result	size
default	\(-\frac {i \sqrt {10}\, \sqrt {2}\, \left (F\left (\frac {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, i \sqrt {5}\right )-2 \Pi \left (\frac {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, 5, i \sqrt {5}\right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-3-2 \cos \left (d x +c \right )}\, \left (1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 d \sqrt {\cos \left (d x +c \right )}\, \left (3+2 \cos \left (d x +c \right )\right )}\)	\(158\)

[In]

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

[Out]

-1/5*I/d*10^(1/2)*2^(1/2)*(EllipticF(1/5*I*(csc(d*x+c)-cot(d*x+c))*5^(1/2),I*5^(1/2))-2*EllipticPi(1/5*I*(csc(
d*x+c)-cot(d*x+c))*5^(1/2),5,I*5^(1/2)))*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*((3+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))*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(-2*cos(d*x + c) - 3)*sqrt(cos(d*x + c))/(2*cos(d*x + c) + 3), x)

Sympy [F]

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

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

[In]

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

[Out]

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