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

Optimal. Leaf size=50 \[ -\frac{8 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d}-\frac{6 \Pi \left (2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d} \]

[Out]

(-8*EllipticF[(c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d) - (6*EllipticPi[2, (c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d)

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Rubi [A]  time = 0.0875074, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2803, 2662, 2806} \[ -\frac{8 F\left (\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d}-\frac{6 \Pi \left (2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right )}{\sqrt{7} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*EllipticF[(c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d) - (6*EllipticPi[2, (c + Pi + d*x)/2, 8/7])/(Sqrt[7]*d)

Rule 2803

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

Rule 2662

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 2806

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

Rubi steps

\begin{align*} \int \sqrt{3-4 \cos (c+d x)} \sec (c+d x) \, dx &=3 \int \frac{\sec (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx-4 \int \frac{1}{\sqrt{3-4 \cos (c+d x)}} \, dx\\ &=-\frac{8 F\left (\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{\sqrt{7} d}-\frac{6 \Pi \left (2;\frac{1}{2} (c+\pi +d x)|\frac{8}{7}\right )}{\sqrt{7} d}\\ \end{align*}

Mathematica [A]  time = 0.0595939, size = 61, normalized size = 1.22 \[ \frac{2 \sqrt{4 \cos (c+d x)-3} \left (3 \Pi \left (2;\left .\frac{1}{2} (c+d x)\right |8\right )-4 F\left (\left .\frac{1}{2} (c+d x)\right |8\right )\right )}{d \sqrt{3-4 \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-3 + 4*Cos[c + d*x]]*(-4*EllipticF[(c + d*x)/2, 8] + 3*EllipticPi[2, (c + d*x)/2, 8]))/(d*Sqrt[3 - 4*C
os[c + d*x]])

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Maple [A]  time = 2.533, size = 159, normalized size = 3.2 \begin{align*}{\frac{2}{7\,d}\sqrt{- \left ( 8\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}\sqrt{56\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-7} \left ( 4\,{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2/7\,\sqrt{14} \right ) +3\,{\it EllipticPi} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,2,2/7\,\sqrt{14} \right ) \right ){\frac{1}{\sqrt{8\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{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}+7}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(-(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)*(4*EllipticF(cos(1/2*d*x+1/2*c),2/7*14^(1/2))+3*EllipticPi(cos(1/2*d*x+1/2*c),2,2/7*14^(1/2)))/(
8*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(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} \sec \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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