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(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

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Rubi [A]  time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, 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*}

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Mathematica [A]  time = 1.44, size = 115, normalized size = 1.58 \[ -\frac {2 \sqrt {\cos (c+d x)} \sqrt {2 \cos (c+d x)+3} \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|-\frac {1}{5}\right )-2 \Pi \left (-1;\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|-\frac {1}{5}\right )\right )}{\sqrt {5} d \sqrt {(3 \cos (c+d x)+\cos (2 (c+d x))+1) \sec ^4\left (\frac {1}{2} (c+d x)\right )}} \]

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)])*S
ec[(c + d*x)/2]^4])

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fricas [F]  time = 1.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {2 \, \cos \left (d x + c\right ) + 3}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {2 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]

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)

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maple [B]  time = 0.18, size = 144, normalized size = 1.97 \[ -\frac {\sqrt {10}\, \sqrt {2}\, \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, \frac {i \sqrt {5}}{5}\right )\right ) \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sin ^{2}\left (d x +c \right )\right )}{5 d \sqrt {3+2 \cos \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\cos \left (d x + c\right )}}{\sqrt {2 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\cos \left (c+d\,x\right )}}{\sqrt {2\,\cos \left (c+d\,x\right )+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\cos {\left (c + d x \right )}}}{\sqrt {2 \cos {\left (c + d x \right )} + 3}}\, dx \]

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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