3.519 \(\int \sqrt{3-4 \cos (c+d x)} \, dx\)

Optimal. Leaf size=24 \[ \frac{2 \sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{d} \]

[Out]

(2*Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/d

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Rubi [A]  time = 0.0113837, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2654} \[ \frac{2 \sqrt{7} E\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 - 4*Cos[c + d*x]],x]

[Out]

(2*Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/d

Rule 2654

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a - b]*EllipticE[(1*(c + Pi/2 + d*x)
)/2, (-2*b)/(a - b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a - b, 0]

Rubi steps

\begin{align*} \int \sqrt{3-4 \cos (c+d x)} \, dx &=\frac{2 \sqrt{7} E\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.0317568, size = 44, normalized size = 1.83 \[ -\frac{2 \sqrt{4 \cos (c+d x)-3} E\left (\left .\frac{1}{2} (c+d x)\right |8\right )}{d \sqrt{3-4 \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 - 4*Cos[c + d*x]],x]

[Out]

(-2*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8])/(d*Sqrt[3 - 4*Cos[c + d*x]])

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Maple [B]  time = 1.914, size = 138, normalized size = 5.8 \begin{align*} -2\,{\frac{\sqrt{- \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2/7\,\sqrt{14} \right ) }{\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+7}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3-4*cos(d*x+c))^(1/2),x)

[Out]

-2*(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c
)^2-7)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))/(8*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2)^(1/2)/si
n(1/2*d*x+1/2*c)/(-8*cos(1/2*d*x+1/2*c)^2+7)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-4 \, \cos \left (d x + c\right ) + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-4*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-4*cos(d*x + c) + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{-4 \, \cos \left (d x + c\right ) + 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-4*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(-4*cos(d*x + c) + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{3 - 4 \cos{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-4*cos(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(3 - 4*cos(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-4 \, \cos \left (d x + c\right ) + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-4*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-4*cos(d*x + c) + 3), x)