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

Optimal. Leaf size=48 \[ \frac {8 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d}+\frac {6 \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d} \]

[Out]

8/7*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)+6/7*(
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 = 0.09, antiderivative size = 48, normalized size of antiderivative = 1.00, 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, 2661, 2805} \[ \frac {8 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d}+\frac {6 \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 2805

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+d x)|\frac {8}{7}\right )}{\sqrt {7} d}+\frac {6 \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 = 0.05, size = 41, normalized size = 0.85 \[ \frac {8 F\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )+6 \Pi \left (2;\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{\sqrt {7} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 2.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right ), x\right ) \]

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), x)

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

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

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maple [A]  time = 0.55, size = 158, normalized size = 3.29 \[ -\frac {2 \sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (4 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )-3 \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, 2 \sqrt {2}\right )\right )}{\sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]

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

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {4\,\cos \left (c+d\,x\right )+3}}{\cos \left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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