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

Optimal. Leaf size=78 \[ \frac{\sqrt{7} F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{6 d}+\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{2 d}+\frac{2 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{3 d} \]

[Out]

(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(2*d) + (Sqrt[7]*EllipticF[(c + d*x)/2, 8/7])/(6*d) + (2*Sqrt[3 + 4*Cos[
c + d*x]]*Sin[c + d*x])/(3*d)

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Rubi [A]  time = 0.0836391, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2753, 2752, 2661, 2653} \[ \frac{\sqrt{7} F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{6 d}+\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{2 d}+\frac{2 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(2*d) + (Sqrt[7]*EllipticF[(c + d*x)/2, 8/7])/(6*d) + (2*Sqrt[3 + 4*Cos[
c + d*x]]*Sin[c + d*x])/(3*d)

Rule 2753

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[
b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2661

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

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 \cos (c+d x) \sqrt{3+4 \cos (c+d x)} \, dx &=\frac{2 \sqrt{3+4 \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2}{3} \int \frac{2+\frac{3}{2} \cos (c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx\\ &=\frac{2 \sqrt{3+4 \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{1}{4} \int \sqrt{3+4 \cos (c+d x)} \, dx+\frac{7}{12} \int \frac{1}{\sqrt{3+4 \cos (c+d x)}} \, dx\\ &=\frac{\sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{2 d}+\frac{\sqrt{7} F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )}{6 d}+\frac{2 \sqrt{3+4 \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0604662, size = 69, normalized size = 0.88 \[ \frac{\sqrt{7} F\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )+3 \sqrt{7} E\left (\frac{1}{2} (c+d x)|\frac{8}{7}\right )+4 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*Sqrt[3 + 4*Cos[c + d*x]],x]

[Out]

(3*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] + Sqrt[7]*EllipticF[(c + d*x)/2, 8/7] + 4*Sqrt[3 + 4*Cos[c + d*x]]*Sin[
c + d*x])/(6*d)

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Maple [A]  time = 2.113, size = 231, normalized size = 3. \begin{align*} -{\frac{1}{6\,d}\sqrt{ \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 64\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +7\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2\,\sqrt{2} \right ) -3\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2\,\sqrt{2} \right ) -56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\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}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*((8*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(64*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+7*(si
n(1/2*d*x+1/2*c)^2)^(1/2)*(8*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2*2^(1/2))-3*(sin(1/2*
d*x+1/2*c)^2)^(1/2)*(8*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2*2^(1/2))-56*sin(1/2*d*x+1/
2*c)^2*cos(1/2*d*x+1/2*c))/(-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} \cos \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(4*cos(d*x + c) + 3)*cos(d*x + c), 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} \cos \left (d x + c\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(4*cos(d*x + c) + 3)*cos(d*x + c), 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} \cos{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(4*cos(c + d*x) + 3)*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} \cos \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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