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*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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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fricas [F]  time = 1.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-4 \, \cos \left (d x + c\right ) + 3}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-4 \, \cos \left (d x + c\right ) + 3}\,{d x} \]

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)

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maple [B]  time = 0.46, size = 138, normalized size = 5.75 \[ -\frac {2 \sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{\sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-4 \, \cos \left (d x + c\right ) + 3}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {3-4\,\cos \left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3 - 4*cos(c + d*x))^(1/2),x)

[Out]

int((3 - 4*cos(c + d*x))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {3 - 4 \cos {\left (c + d x \right )}}\, dx \]

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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