3.658 \(\int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{-3+2 \cos (c+d x)}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05566, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {2815} \[ -\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]),x]

[Out]

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

Rule 2815

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-2*
Sqrt[a^2]*Sqrt[-Cot[e + f*x]^2]*Rt[(a + b)/d, 2]*EllipticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/(Sqrt[d*Sin[e + f*x
]]*Rt[(a + b)/d, 2])], -((a + b)/(a - b))])/(a*f*Sqrt[a^2 - b^2]*Cot[e + f*x]), x] /; FreeQ[{a, b, d, e, f}, x
] && GtQ[a^2 - b^2, 0] && PosQ[(a + b)/d] && GtQ[a^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{-3+2 \cos (c+d x)}} \, dx &=-\frac{2 \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{-3+2 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right ) \sqrt{-\tan ^2(c+d x)}}{\sqrt{5} d}\\ \end{align*}

Mathematica [B]  time = 0.633724, size = 160, normalized size = 2.58 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{-\cot ^2\left (\frac{1}{2} (c+d x)\right )} \cot (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{-(2 \cos (c+d x)-3) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{2 \cos (c+d x)-3}{\cos (c+d x)-1}}}{\sqrt{3}}\right )|\frac{6}{5}\right )}{\sqrt{5} d (-\cos (c+d x))^{3/2} \sqrt{2 \cos (c+d x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]),x]

[Out]

(4*Sqrt[-Cot[(c + d*x)/2]^2]*Cot[c + d*x]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[-((-3 + 2*Cos[c + d*x]
)*Csc[(c + d*x)/2]^2)]*EllipticF[ArcSin[Sqrt[(-3 + 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]/Sqrt[3]], 6/5]*Sin[(c
+ d*x)/2]^4)/(Sqrt[5]*d*(-Cos[c + d*x])^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]])

________________________________________________________________________________________

Maple [A]  time = 0.256, size = 98, normalized size = 1.6 \begin{align*} 2\,{\frac{\sqrt{2}\sqrt{-3+2\,\cos \left ( dx+c \right ) }}{d\sqrt{-\cos \left ( dx+c \right ) }}{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},i\sqrt{5} \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(-cos(d*x+c))^(1/2)/(-3+2*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-\cos \left (d x + c\right )} \sqrt{2 \, \cos \left (d x + c\right ) - 3}}{2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(-cos(d*x+c))**(1/2)/(-3+2*cos(d*x+c))**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/(-cos(d*x+c))^(1/2)/(-3+2*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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