\(\int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2+3 \cos (c+d x)}} \, dx\) [660]

Optimal result

Integrand size = 25, antiderivative size = 77 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2+3 \cos (c+d x)}} \, dx=-\frac {4 \cot (c+d x) \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {2+3 \cos (c+d x)}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),5\right ) \sqrt {-1-\sec (c+d x)} \sqrt {1-\sec (c+d x)}}{3 d} \]

[Out]

-4/3*cot(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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 77, 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 = {2888} \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2+3 \cos (c+d x)}} \, dx=-\frac {4 \cot (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {5}{3},\arcsin \left (\frac {\sqrt {3 \cos (c+d x)+2}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right ),5\right )}{3 d} \]

[In]

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

[Out]

(-4*Cot[c + d*x]*EllipticPi[5/3, ArcSin[Sqrt[2 + 3*Cos[c + d*x]]/(Sqrt[5]*Sqrt[Cos[c + d*x]])], 5]*Sqrt[-1 - S
ec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])/(3*d)

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {4 \cot (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} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(175\) vs. \(2(77)=154\).

Time = 12.63 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.27 \[ \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {2+3 \cos (c+d x)}} \, dx=\frac {2 \sqrt {\cos (c+d x)} \sqrt {2+3 \cos (c+d x)} \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \left (3 \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{2} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right )-5 \operatorname {EllipticPi}\left (-\frac {2}{3},\arcsin \left (\frac {1}{2} \sqrt {(2+3 \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right ),-4\right )\right )}{3 d \sqrt {\frac {-2-3 \cos (c+d x)}{-1+\cos (c+d x)}} \sqrt {\frac {\cos (c+d x)}{-1+\cos (c+d x)}}} \]

[In]

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

[Out]

(2*Sqrt[Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]*Sqrt[Cot[(c + d*x)/2]^2]*Csc[c + d*x]*(3*EllipticF[ArcSin[Sqrt[
(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4] - 5*EllipticPi[-2/3, ArcSin[Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c
+ d*x)/2]^2]/2], -4]))/(3*d*Sqrt[(-2 - 3*Cos[c + d*x])/(-1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*
x])])

Maple [A] (verified)

Time = 6.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.66

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {10}\, \left (F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \frac {\sqrt {5}}{5}\right )-2 \Pi \left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), -1, \frac {\sqrt {5}}{5}\right )\right ) \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )\right )}{5 d \sqrt {2+3 \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\)	\(128\)

[In]

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

[Out]

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

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(cos(d*x + c))/sqrt(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)