\(\int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx\) [557]

Optimal result

Integrand size = 14, antiderivative size = 24 \[ \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{\sqrt {7} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2741} \[ \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{\sqrt {7} d} \]

[In]

Int[1/Sqrt[3 - 4*Cos[c + d*x]],x]

[Out]

(2*EllipticF[(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]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx=\frac {2 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )}{d \sqrt {3-4 \cos (c+d x)}} \]

[In]

Integrate[1/Sqrt[3 - 4*Cos[c + d*x]],x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.52 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25

method	result	size
default	\(\frac {2 \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \operatorname {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}| 2 \sqrt {2}\right )}{d \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}}\)	\(54\)

[In]

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

[Out]

2/d/(-8*cos(1/2*d*x+1/2*c)^2+7)^(1/2)*(8*cos(1/2*d*x+1/2*c)^2-7)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2*2^(1/2)
)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx=-\frac {\sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )}{2 \, d} \]

[In]

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

[Out]

-1/2*(sqrt(2)*weierstrassPInverse(-1, -1, cos(d*x + c) + I*sin(d*x + c) - 1/2) + sqrt(2)*weierstrassPInverse(-
1, -1, cos(d*x + c) - I*sin(d*x + c) - 1/2))/d

Sympy [F]

\[ \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {3 - 4 \cos {\left (c + d x \right )}}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(3 - 4*cos(c + d*x)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx=\frac {2\,\sqrt {4\,\cos \left (c+d\,x\right )-3}\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |8\right )}{d\,\sqrt {3-4\,\cos \left (c+d\,x\right )}} \]

[In]

int(1/(3 - 4*cos(c + d*x))^(1/2),x)

[Out]

(2*(4*cos(c + d*x) - 3)^(1/2)*ellipticF(c/2 + (d*x)/2, 8))/(d*(3 - 4*cos(c + d*x))^(1/2))