3.6.51 \(\int \frac {\sec (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx\) [551]

Optimal. Leaf size=24 \[ \frac {2 \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 24, 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 = {2884} \begin {gather*} \frac {2 \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2884

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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+d x)|\frac {8}{7}\right )}{\sqrt {7} d}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 24, normalized size = 1.00 \begin {gather*} \frac {2 \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(48)=96\).
time = 0.13, size = 138, normalized size = 5.75

method	result	size
default	\(\frac {2 \sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, 2 \sqrt {2}\right )}{\sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(138\)

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,method=_RETURNVERBOSE)

[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)/(-8*sin(1/2*d*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),2,2*2^(1/2))/s
in(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

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(4*cos(c + d*x) + 3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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