\(\int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx\) [520]

Optimal result

Integrand size = 21, antiderivative size = 50 \[ \int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx=-\frac {8 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{\sqrt {7} d}-\frac {6 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{\sqrt {7} d} \]

[Out]

8/7*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)*EllipticF(cos(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)+6/7*(
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)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2882, 2741, 2885} \[ \int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx=-\frac {8 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{\sqrt {7} d}-\frac {6 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{\sqrt {7} d} \]

[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 2741

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

Rule 2882

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 2885

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*} \text {integral}& = 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 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{\sqrt {7} d}-\frac {6 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{\sqrt {7} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.22 \[ \int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx=\frac {2 \sqrt {-3+4 \cos (c+d x)} \left (-4 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )+3 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),8\right )\right )}{d \sqrt {3-4 \cos (c+d x)}} \]

[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]])

Maple [A] (verified)

Time = 3.01 (sec) , antiderivative size = 159, normalized size of antiderivative = 3.18

method	result	size
default	\(\frac {2 \sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \left (4 F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )+3 \Pi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \frac {2 \sqrt {14}}{7}\right )\right )}{7 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d}\)	\(159\)

[In]

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

[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

Fricas [F]

\[ \int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right ) \,d x } \]

[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)

Sympy [F]

\[ \int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx=\int \sqrt {3 - 4 \cos {\left (c + d x \right )}} \sec {\left (c + d x \right )}\, dx \]

[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)

Maxima [F]

\[ \int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right ) \,d x } \]

[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)

Giac [F]

\[ \int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right ) \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {3-4 \cos (c+d x)} \sec (c+d x) \, dx=\int \frac {\sqrt {3-4\,\cos \left (c+d\,x\right )}}{\cos \left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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