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

Optimal. Leaf size=77 \[ -\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left (\frac{5}{3};\left .\sin ^{-1}\left (\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |5\right )}{3 d} \]

[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)

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Rubi [A]  time = 0.0488477, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2809} \[ -\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left (\frac{5}{3};\left .\sin ^{-1}\left (\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |5\right )}{3 d} \]

Antiderivative was successfully verified.

[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 2809

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

Rubi steps

\begin{align*} \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{2+3 \cos (c+d x)}} \, dx &=-\frac{4 \cot (c+d x) \Pi \left (\frac{5}{3};\left .\sin ^{-1}\left (\frac{\sqrt{2+3 \cos (c+d x)}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |5\right ) \sqrt{-1-\sec (c+d x)} \sqrt{1-\sec (c+d x)}}{3 d}\\ \end{align*}

Mathematica [B]  time = 2.84075, size = 175, normalized size = 2.27 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{3 \cos (c+d x)+2} \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \left (3 F\left (\left .\sin ^{-1}\left (\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )}\right )\right |-4\right )-5 \Pi \left (-\frac{2}{3};\left .\sin ^{-1}\left (\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )}\right )\right |-4\right )\right )}{3 d \sqrt{\frac{-3 \cos (c+d x)-2}{\cos (c+d x)-1}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}} \]

Antiderivative was successfully verified.

[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])])

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Maple [B]  time = 0.388, size = 142, normalized size = 1.8 \begin{align*} -{\frac{\sqrt{2}\sqrt{10} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{5\,d \left ( -1+\cos \left ( dx+c \right ) \right ) } \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},{\frac{\sqrt{5}}{5}} \right ) -2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},-1,1/5\,\sqrt{5} \right ) \right ) \sqrt{{\frac{2+3\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{2+3\,\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\cos \left (d x + c\right )}}{\sqrt{3 \, \cos \left (d x + c\right ) + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\cos \left (d x + c\right )}}{\sqrt{3 \, \cos \left (d x + c\right ) + 2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\cos{\left (c + d x \right )}}}{\sqrt{3 \cos{\left (c + d x \right )} + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\cos \left (d x + c\right )}}{\sqrt{3 \, \cos \left (d x + c\right ) + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)