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

Optimal. Leaf size=97 \[ -\frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right )}{d \sqrt{-\cos (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.102807, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2810, 2808} \[ -\frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (\frac{5}{2};\left .\sin ^{-1}\left (\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right )}{d \sqrt{-\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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{\sqrt{\cos (c+d x)} \int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-3-2 \cos (c+d x)}} \, dx}{\sqrt{-\cos (c+d x)}}\\ &=-\frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (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 \sqrt{-\cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.669417, size = 115, normalized size = 1.19 \[ -\frac{2 \cos ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cos (c+d x) (2 \cos (c+d x)+3) \sec ^4\left (\frac{1}{2} (c+d x)\right )} \left (F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )+2 \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )\right )}{\sqrt{5} d \sqrt{-2 \cos (c+d x)-3} \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.436, size = 168, normalized size = 1.7 \begin{align*}{\frac{-{\frac{i}{5}}\sqrt{5}\sqrt{2}\sqrt{10} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d \left ( 2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\cos \left ( dx+c \right ) -3 \right ) } \left ({\it EllipticF} \left ({\frac{{\frac{i}{5}} \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},i\sqrt{5} \right ) -2\,{\it EllipticPi} \left ({\frac{i/5 \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},5,i\sqrt{5} \right ) \right ) \sqrt{-3-2\,\cos \left ( dx+c \right ) }\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{{\frac{3+2\,\cos \left ( dx+c \right ) }{1+\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*I/d*5^(1/2)*2^(1/2)*10^(1/2)*(EllipticF(1/5*I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),I*5^(1/2))-2*EllipticPi(
1/5*I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),5,I*5^(1/2)))*sin(d*x+c)^2*(-3-2*cos(d*x+c))^(1/2)*(cos(d*x+c)/(1+cos
(d*x+c)))^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)/(2*cos(d*x+c)^2+cos(d*x+c)-3)/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{-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 ) \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(-2*cos(d*x + c) - 3)*sqrt(cos(d*x + c))/(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)