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

Optimal. Leaf size=32 \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )|\frac{1}{5}\right )}{\sqrt{5} d} \]

[Out]

(2*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 1/5])/(Sqrt[5]*d)

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Rubi [A]  time = 0.0588819, antiderivative size = 32, 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 = {2813} \[ \frac{2 F\left (\sin ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)+1}\right )|\frac{1}{5}\right )}{\sqrt{5} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticF[ArcSin[Sin[c + d*x]/(1 + Cos[c + d*x])], 1/5])/(Sqrt[5]*d)

Rule 2813

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

Rubi steps

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

Mathematica [B]  time = 3.63154, size = 131, normalized size = 4.09 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{3 \cos (c+d x)+2} \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left (\frac{1}{2} (c+d x)\right )}\right )\right |-4\right )}{d \sqrt{\frac{-3 \cos (c+d x)-2}{\cos (c+d x)-1}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.462, size = 115, normalized size = 3.6 \begin{align*} -{\frac{\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}\sqrt{10}}{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{2+3\,\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{\sqrt{5}}{5}} \right ){\frac{1}{\sqrt{2+3\,\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)/(2+3*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)/(2+3*cos(d*x+c))^(1/2)*sin(d*x+c)^4*10^(1/2)*((2+3*cos(d*x+c)
)/(1+cos(d*x+c)))^(1/2)*EllipticF((-1+cos(d*x+c))/sin(d*x+c),1/5*5^(1/2))/cos(d*x+c)^(3/2)/(-1+cos(d*x+c))^2

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{3 \, \cos \left (d x + c\right ) + 2} \sqrt{\cos \left (d x + c\right )}}{3 \, \cos \left (d x + c\right )^{2} + 2 \, \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)/(2+3*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(3*cos(d*x + c) + 2)*sqrt(cos(d*x + c))/(3*cos(d*x + c)^2 + 2*cos(d*x + c)), x)

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

[Out]

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

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

[Out]

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