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

Optimal. Leaf size=25 \[ -\frac {2 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2806} \[ -\frac {2 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.06, size = 45, normalized size = 1.80 \[ \frac {2 \sqrt {4 \cos (c+d x)-3} \Pi \left (2;\left .\frac {1}{2} (c+d x)\right |8\right )}{d \sqrt {3-4 \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 2.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )}{4 \, \cos \left (d x + c\right ) - 3}, x\right ) \]

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)/(4*cos(d*x + c) - 3), x)

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

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

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maple [B]  time = 0.59, size = 139, normalized size = 5.56 \[ \frac {2 \sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \frac {2 \sqrt {14}}{7}\right )}{7 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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)/(8*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,2/7*14^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}}\,{d x} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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