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

Optimal. Leaf size=58 \[ \frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0529605, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2815} \[ \frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right )}{d} \]

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[5]*Sqrt[Cos[c + d*x]])], -5]*Sqrt[-Tan[c + d*x
]^2])/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 (\left .\sin ^{-1}\left (\frac{\sqrt{3+2 \cos (c+d x)}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right )\right |-5\right ) \sqrt{-\tan ^2(c+d x)}}{d}\\ \end{align*}

Mathematica [B]  time = 1.08296, size = 140, normalized size = 2.41 \[ \frac{4 \sqrt{\cos (c+d x)} \sqrt{2 \cos (c+d x)+3} \sqrt{-\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{(2 \cos (c+d x)+3) \csc ^2\left (\frac{1}{2} (c+d x)\right )}}{\sqrt{6}}\right )\right |6\right )}{d \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{(2 \cos (c+d x)+3) \csc ^2\left (\frac{1}{2} (c+d x)\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.458, size = 116, normalized size = 2. \begin{align*} -{\frac{\sqrt{2}\sqrt{10} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{5\,d \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{3+2\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},{\frac{i}{5}}\sqrt{5} \right ){\frac{1}{\sqrt{3+2\,\cos \left ( dx+c \right ) }}} \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \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]

-1/5/d*2^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(3/2)/(3+2*cos(d*x+c))^(1/2)*10^(1/2)*((3+2*cos(d*x+c))/(1+cos(d*x+
c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*I*5^(1/2))*sin(d*x+c)^4/cos(d*x+c)^(3/2)/(-1+cos(d*x+c))^2

________________________________________________________________________________________

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

________________________________________________________________________________________

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 )^{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(2*cos(d*x + c) + 3)*sqrt(cos(d*x + c))/(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{2 \cos{\left (c + d x \right )} + 3} \sqrt{\cos{\left (c + d x \right )}}}\, 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(2*cos(c + d*x) + 3)*sqrt(cos(c + d*x))), x)

________________________________________________________________________________________

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