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

Optimal. Leaf size=23 \[ \frac{2 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.011538, antiderivative size = 23, 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 = {2653} \[ \frac{2 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

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

Rule 2653

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+d x)|\frac{8}{7}\right )}{d}\\ \end{align*}

Mathematica [A]  time = 0.0236297, size = 23, normalized size = 1. \[ \frac{2 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

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

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Maple [B]  time = 1.635, size = 137, normalized size = 6. \begin{align*} 2\,{\frac{\sqrt{ \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \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{-8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2\,\sqrt{2} \right ) }{\sqrt{-8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+7\, \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}-1}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)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-8*cos(1/2*d*x+1/2*c)^
2+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2*2^(1/2))/(-8*sin(1/2*d*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(
1/2*d*x+1/2*c)/(8*cos(1/2*d*x+1/2*c)^2-1)^(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{4 \cos{\left (c + d x \right )} + 3}\, 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(4*cos(c + d*x) + 3), 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)