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

Optimal. Leaf size=75 \[ \frac {3 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \Pi \left (-\frac {1}{2};\sin ^{-1}\left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{\sqrt {5} d} \]

[Out]

3/5*cot(d*x+c)*EllipticPi((3-2*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2),-1/2,1/5*I*5^(1/2))*(1-sec(d*x+c))^(1/2)*(1+
sec(d*x+c))^(1/2)/d*5^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 75, 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} \[ \frac {3 \cot (c+d x) \sqrt {1-\sec (c+d x)} \sqrt {\sec (c+d x)+1} \Pi \left (-\frac {1}{2};\sin ^{-1}\left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right )}{\sqrt {5} d} \]

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[-1/2, ArcSin[Sqrt[3 - 2*Cos[c + d*x]]/Sqrt[Cos[c + d*x]]], -1/5]*Sqrt[1 - Sec[c + d
*x]]*Sqrt[1 + Sec[c + d*x]])/(Sqrt[5]*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 {1}{2};\sin ^{-1}\left (\frac {\sqrt {3-2 \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\right )|-\frac {1}{5}\right ) \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}{\sqrt {5} d}\\ \end {align*}

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Mathematica [A]  time = 0.84, size = 117, normalized size = 1.56 \[ -\frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {3-2 \cos (c+d x)}{\cos (c+d x)+1}} \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)+1}} \left (F\left (\left .\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )\right |-5\right )-2 \Pi \left (-1;\left .\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )\right |-5\right )\right )}{d \sqrt {3-2 \cos (c+d x)} \sqrt {\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-2 \, \cos \left (d x + c\right ) + 3} \sqrt {\cos \left (d x + c\right )}}{2 \, \cos \left (d x + c\right ) - 3}, x\right ) \]

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(-2*cos(d*x + c) + 3)*sqrt(cos(d*x + c))/(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 = 153, normalized size = 2.04 \[ \frac {\sqrt {2}\, \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, i \sqrt {5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, i \sqrt {5}\right )\right ) \sqrt {-\frac {2 \left (-3+2 \cos \left (d x +c \right )\right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {3-2 \cos \left (d x +c \right )}\, \left (\sin ^{2}\left (d x +c \right )\right )}{d \left (2 \left (\cos ^{2}\left (d x +c \right )\right )-5 \cos \left (d x +c \right )+3\right ) \sqrt {\cos \left (d x +c \right )}} \]

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/d*2^(1/2)*(EllipticF((-1+cos(d*x+c))/sin(d*x+c),I*5^(1/2))-2*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,I*5^(1
/2)))*(-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/(2*cos(d*x+c)^2-5*cos(d*x+c)+3)/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 \[ \int \frac {\sqrt {\cos \left (c+d\,x\right )}}{\sqrt {3-2\,\cos \left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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(3 - 2*cos(c + d*x)), x)

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