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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.100456, antiderivative size = 99, 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} \[ \frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left (-\frac{1}{2};\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d \sqrt{-\cos (c+d x)}} \]

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[-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*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{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 \sqrt{-\cos (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 0.961351, size = 135, normalized size = 1.36 \[ -\frac{2 i \sqrt{2 \cos (c+d x)-3} \sqrt{\frac{\cos (c+d x)}{5 \cos (c+d x)+5}} \left (F\left (i \sinh ^{-1}\left (\sqrt{5} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )-2 \Pi \left (\frac{1}{5};i \sinh ^{-1}\left (\sqrt{5} \tan \left (\frac{1}{2} (c+d x)\right )\right )|-\frac{1}{5}\right )\right )}{d \sqrt{\cos (c+d x)} \sqrt{\frac{3-2 \cos (c+d x)}{\cos (c+d x)+1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*Sqrt[-3 + 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]/(5 + 5*Cos[c + d*x])]*(EllipticF[I*ArcSinh[Sqrt[5]*Tan[(c
+ d*x)/2]], -1/5] - 2*EllipticPi[1/5, I*ArcSinh[Sqrt[5]*Tan[(c + d*x)/2]], -1/5]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[
(3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x])])

________________________________________________________________________________________

Maple [A]  time = 0.435, size = 158, normalized size = 1.6 \begin{align*}{\frac{-{\frac{i}{5}}\sqrt{5}\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d \left ( -1+\cos \left ( dx+c \right ) \right ) } \left ( 2\,{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},1/5,i/5\sqrt{5} \right ) -{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \right ) \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{-3+2\,\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)*(2*EllipticPi(I*(-1+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5,1/5*I*5^(1/2))-EllipticF(I*(-1
+cos(d*x+c))*5^(1/2)/sin(d*x+c),1/5*I*5^(1/2)))/(-3+2*cos(d*x+c))^(1/2)*sin(d*x+c)^2*(cos(d*x+c)/(1+cos(d*x+c)
))^(1/2)*(-2*(-3+2*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)/(-1+cos(d*x+c))/cos(d*x+c)^(1/2)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)