\(\int \sqrt {3-4 \cos (c+d x)} \, dx\) [519]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.01 (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 = {2733} \[ \int \sqrt {3-4 \cos (c+d x)} \, dx=\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{d} \]

[In]

Int[Sqrt[3 - 4*Cos[c + d*x]],x]

[Out]

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

Rule 2733

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

Mathematica [A] (verified)

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

[In]

Integrate[Sqrt[3 - 4*Cos[c + d*x]],x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(47)=94\).

Time = 3.81 (sec) , antiderivative size = 138, normalized size of antiderivative = 5.75

method	result	size
default	\(-\frac {2 \sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{\sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d}\)	\(138\)
risch	\(-\frac {2 i \sqrt {-\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{-i \left (d x +c \right )}}}{d}+\frac {i \left (\frac {6 \left (-\frac {3}{4}+\frac {i \sqrt {7}}{4}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {3}{4}+\frac {i \sqrt {7}}{4}}{-\frac {3}{4}+\frac {i \sqrt {7}}{4}}}\, \sqrt {14}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{\frac {3}{4}-\frac {i \sqrt {7}}{4}}}\, F\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {3}{4}+\frac {i \sqrt {7}}{4}}{-\frac {3}{4}+\frac {i \sqrt {7}}{4}}}, \frac {\sqrt {14}\, \sqrt {i \left (\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}}{7}\right )}{7 \sqrt {-2 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}-2 \,{\mathrm e}^{i \left (d x +c \right )}}}-\frac {4 \left (-2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}-2\right )}{\sqrt {\left (-2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}-2\right ) {\mathrm e}^{i \left (d x +c \right )}}}-\frac {8 \left (-\frac {3}{4}+\frac {i \sqrt {7}}{4}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {3}{4}+\frac {i \sqrt {7}}{4}}{-\frac {3}{4}+\frac {i \sqrt {7}}{4}}}\, \sqrt {14}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{\frac {3}{4}-\frac {i \sqrt {7}}{4}}}\, \left (-\frac {i \sqrt {7}\, E\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {3}{4}+\frac {i \sqrt {7}}{4}}{-\frac {3}{4}+\frac {i \sqrt {7}}{4}}}, \frac {\sqrt {14}\, \sqrt {i \left (\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}}{7}\right )}{2}+\left (\frac {3}{4}+\frac {i \sqrt {7}}{4}\right ) F\left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}-\frac {3}{4}+\frac {i \sqrt {7}}{4}}{-\frac {3}{4}+\frac {i \sqrt {7}}{4}}}, \frac {\sqrt {14}\, \sqrt {i \left (\frac {3}{4}-\frac {i \sqrt {7}}{4}\right ) \sqrt {7}}}{7}\right )\right )}{7 \sqrt {-2 \,{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}-2 \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {-\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{-i \left (d x +c \right )}}\, \sqrt {-\left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right ) {\mathrm e}^{i \left (d x +c \right )}}}{d \left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}+2\right )}\)	\(662\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.42 \[ \int \sqrt {3-4 \cos (c+d x)} \, dx=-\frac {\sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 4 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) + 4 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{2 \, d} \]

[In]

integrate((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) + 4*sqrt(2)*weierstrassZeta(-1, -1, weierstrassPInverse(-1, -1, co
s(d*x + c) + I*sin(d*x + c) - 1/2)) + 4*sqrt(2)*weierstrassZeta(-1, -1, weierstrassPInverse(-1, -1, cos(d*x +
c) - I*sin(d*x + c) - 1/2)))/d

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(3 - 4*cos(c + d*x)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((3 - 4*cos(c + d*x))^(1/2),x)

[Out]

int((3 - 4*cos(c + d*x))^(1/2), x)