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

Optimal. Leaf size=73 \[ -\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{d} \]

[Out]

(-3*Cot[c + d*x]*EllipticPi[5/2, ArcSin[Sqrt[3 + 2*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]])/d

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Rubi [A]  time = 0.0466321, antiderivative size = 73, 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 = {2808} \[ -\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Cot[c + d*x]*EllipticPi[5/2, ArcSin[Sqrt[3 + 2*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]])/d

Rule 2808

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]]*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))])/(d*f*Sqrt[c^2 - d^2]), x] /; F
reeQ[{b, c, d, e, f}, x] && GtQ[c^2 - d^2, 0] && PosQ[(c + d)/b] && GtQ[c^2, 0]

Rubi steps

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

Mathematica [A]  time = 1.36494, size = 117, normalized size = 1.6 \[ -\frac{2 \sqrt{\cos (c+d x)} \sqrt{2 \cos (c+d x)+3} \sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )+2 \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )\right )}{\sqrt{5} d \sqrt{(3 \cos (c+d x)+\cos (2 (c+d x))+1) \sec ^4\left (\frac{1}{2} (c+d x)\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.374, size = 144, normalized size = 2. \begin{align*} -{\frac{\sqrt{10}\sqrt{2} \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{i}{5}}\sqrt{5} \right ) -2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},-1,i/5\sqrt{5} \right ) \right ) \sqrt{{\frac{3+2\,\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{3+2\,\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)/(3+2*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*I*5^(1/2))-2*EllipticPi((-1+cos(d*x+c))/sin(
d*x+c),-1,1/5*I*5^(1/2)))*sin(d*x+c)^2/(3+2*cos(d*x+c))^(1/2)*((3+2*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{2 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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{2 \, \cos \left (d x + c\right ) + 3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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), 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{2 \cos{\left (c + d x \right )} + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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{2 \, \cos \left (d x + c\right ) + 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)