\(\int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx\) [663]

Optimal result

Integrand size = 25, antiderivative size = 101 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {-2-3 \cos (c+d x)}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{3 d \sqrt {-\cos (c+d x)}} \]

[Out]

-4/3*cos(d*x+c)^(3/2)*csc(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/(-cos(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 101, 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, 2888} \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=-\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {-3 \cos (c+d x)-2}}{\sqrt {5} \sqrt {-\cos (c+d x)}}\right ),5\right )}{3 d \sqrt {-\cos (c+d x)}} \]

[In]

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

[Out]

(-4*Cos[c + d*x]^(3/2)*Csc[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*Sqrt[-Cos[c + d*x]])

Rule 2888

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))]*El
lipticPi[(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]

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 {-2-3 \cos (c+d x)}} \, dx}{\sqrt {-\cos (c+d x)}} \\ & = -\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (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 \sqrt {-\cos (c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.50 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {-2-3 \cos (c+d x)}} \, dx=-\frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {-\frac {(2+3 \cos (c+d x))^2}{(1+\cos (c+d x))^2}} \left (\operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {1}{5}\right )-2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {1}{5}\right )\right )}{\sqrt {5} d \sqrt {-2-3 \cos (c+d x)} \sqrt {\cos (c+d x)} \sqrt {-\frac {2+3 \cos (c+d x)}{1+\cos (c+d x)}}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 6.59 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {\left (F\left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, \sqrt {5}\right )-2 \Pi \left (\frac {\left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ) \sqrt {5}}{5}, -5, \sqrt {5}\right )\right ) \sqrt {2}\, \sqrt {10}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {-2-3 \cos \left (d x +c \right )}\, \left (1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 d \sqrt {\cos \left (d x +c \right )}\, \left (2+3 \cos \left (d x +c \right )\right )}\)	\(149\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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