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

Optimal. Leaf size=99 \[ -\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} \Pi \left (\frac {1}{3};\sin ^{-1}\left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|\frac {1}{5}\right )}{3 \sqrt {5} d \sqrt {-\cos (c+d x)}} \]

[Out]

-4/15*cos(d*x+c)^(3/2)*csc(d*x+c)*EllipticPi((2-3*cos(d*x+c))^(1/2)/(-cos(d*x+c))^(1/2),1/3,1/5*5^(1/2))*(-1+s
ec(d*x+c))^(1/2)*(1+sec(d*x+c))^(1/2)/d*5^(1/2)/(-cos(d*x+c))^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2810, 2809} \[ -\frac {4 \cos ^{\frac {3}{2}}(c+d x) \csc (c+d x) \sqrt {\sec (c+d x)-1} \sqrt {\sec (c+d x)+1} \Pi \left (\frac {1}{3};\sin ^{-1}\left (\frac {\sqrt {2-3 \cos (c+d x)}}{\sqrt {-\cos (c+d x)}}\right )|\frac {1}{5}\right )}{3 \sqrt {5} d \sqrt {-\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 2810

Int[Sqrt[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*S
in[e + f*x]]/Sqrt[-(b*Sin[e + f*x])], Int[Sqrt[-(b*Sin[e + f*x])]/Sqrt[c + d*Sin[e + f*x]], x], x] /; FreeQ[{b
, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && NegQ[(c + d)/b]

Rubi steps

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

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Mathematica [A]  time = 1.77, size = 145, normalized size = 1.46 \[ -\frac {4 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {-\frac {(2-3 \cos (c+d x))^2}{(\cos (c+d x)+1)^2}} \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)+1}} \left (F\left (\left .\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )\right |5\right )-2 \Pi \left (-1;\left .\sin ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )\right |5\right )\right )}{d \sqrt {2-3 \cos (c+d x)} \sqrt {\cos (c+d x)} \sqrt {\frac {2-3 \cos (c+d x)}{\cos (c+d x)+1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-3 \, \cos \left (d x + c\right ) + 2} \sqrt {\cos \left (d x + c\right )}}{3 \, \cos \left (d x + c\right ) - 2}, x\right ) \]

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(-3*cos(d*x + c) + 2)*sqrt(cos(d*x + c))/(3*cos(d*x + c) - 2), x)

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

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)

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maple [A]  time = 0.19, size = 144, normalized size = 1.45 \[ \frac {2 \left (\EllipticF \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, \sqrt {5}\right )-2 \EllipticPi \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}, -1, \sqrt {5}\right )\right ) \sqrt {\frac {-2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2-3 \cos \left (d x +c \right )}\, \left (\sin ^{2}\left (d x +c \right )\right )}{d \left (3 \left (\cos ^{2}\left (d x +c \right )\right )-5 \cos \left (d x +c \right )+2\right ) \sqrt {\cos \left (d x +c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(cos(c + d*x))/sqrt(2 - 3*cos(c + d*x)), x)

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