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

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

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2810, 2809} \[ -\frac {4 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {-\sec (c+d x)-1} \sqrt {1-\sec (c+d x)} \Pi \left (\frac {5}{3};\left .\sin ^{-1}\left (\frac {\sqrt {3 \cos (c+d x)+2}}{\sqrt {5} \sqrt {\cos (c+d x)}}\right )\right |5\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 \sqrt {-\cos (c+d x)} \sqrt {\cos (c+d x)} \csc (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*}

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Mathematica [A]  time = 0.77, size = 194, normalized size = 1.96 \[ \frac {4 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\cot ^2\left (\frac {1}{2} (c+d x)\right )} \csc (c+d x) \sqrt {-\cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {(3 \cos (c+d x)+2) \csc ^2\left (\frac {1}{2} (c+d x)\right )} \left (3 F\left (\left .\sin ^{-1}\left (\frac {1}{2} \sqrt {(3 \cos (c+d x)+2) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right |-4\right )-5 \Pi \left (-\frac {2}{3};\left .\sin ^{-1}\left (\frac {1}{2} \sqrt {(3 \cos (c+d x)+2) \csc ^2\left (\frac {1}{2} (c+d x)\right )}\right )\right |-4\right )\right )}{3 d \sqrt {-\cos (c+d x)} \sqrt {3 \cos (c+d x)+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(2 + 3*Cos[c + d*x])*Csc[(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])*Sin[(c + d*x)/2]^4)/(3*d*Sqrt[-Cos[c + d*x]]*
Sqrt[2 + 3*Cos[c + d*x]])

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fricas [F]  time = 1.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-\cos \left (d x + c\right )}}{\sqrt {3 \, \cos \left (d x + c\right ) + 2}}, x\right ) \]

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(-cos(d*x + c))/sqrt(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 = 0.17, size = 159, normalized size = 1.61 \[ -\frac {\sqrt {2}\, \sqrt {10}\, \left (\EllipticF \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, \sqrt {5}\right )-2 \EllipticPi \left (\frac {\sqrt {5}\, \left (-1+\cos \left (d x +c \right )\right )}{5 \sin \left (d x +c \right )}, -5, \sqrt {5}\right )\right ) \sqrt {-\cos \left (d x +c \right )}\, \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {5}}{5 d \sqrt {2+3 \cos \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right )} \]

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]

-1/5/d*2^(1/2)*10^(1/2)*(EllipticF(1/5*5^(1/2)*(-1+cos(d*x+c))/sin(d*x+c),5^(1/2))-2*EllipticPi(1/5*5^(1/2)*(-
1+cos(d*x+c))/sin(d*x+c),-5,5^(1/2)))/(2+3*cos(d*x+c))^(1/2)*(-cos(d*x+c))^(1/2)*sin(d*x+c)^2*(cos(d*x+c)/(1+c
os(d*x+c)))^(1/2)*((2+3*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)/(-1+cos(d*x+c))/cos(d*x+c)*5^(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 \[ \int \frac {\sqrt {-\cos \left (c+d\,x\right )}}{\sqrt {3\,\cos \left (c+d\,x\right )+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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(3*cos(c + d*x) + 2), x)

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