Integrand size = 14, antiderivative size = 24 \[ \int \sqrt {3-4 \cos (c+d x)} \, dx=\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{d} \] Output:
2*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \sqrt {3-4 \cos (c+d x)} \, dx=-\frac {2 \sqrt {-3+4 \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )}{d \sqrt {3-4 \cos (c+d x)}} \] Input:
Integrate[Sqrt[3 - 4*Cos[c + d*x]],x]
Output:
(-2*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8])/(d*Sqrt[3 - 4*Cos [c + d*x]])
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3133}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {3-4 \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3133 |
| \(\displaystyle \frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{d}\) |
Input:
Int[Sqrt[3 - 4*Cos[c + d*x]],x]
Output:
(2*Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/d
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]
Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(23)=46\).
Time = 4.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 5.75
| method | result | size |
| default | \(-\frac {2 \sqrt {-\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{\sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+7}\, d}\) | \(138\) |
| 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}}}\, \operatorname {EllipticF}\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}\, \operatorname {EllipticE}\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 ) \operatorname {EllipticF}\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 )}\) | \(662\) |
Input:
int((3-4*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-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)/sin( 1/2*d*x+1/2*c)/(-8*cos(1/2*d*x+1/2*c)^2+7)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.42 \[ \int \sqrt {3-4 \cos (c+d x)} \, dx=-\frac {\sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 4 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) + 4 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{2 \, d} \] Input:
integrate((3-4*cos(d*x+c))^(1/2),x, algorithm="fricas")
Output:
-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, cos( d*x + c) + I*sin(d*x + c) - 1/2)) + 4*sqrt(2)*weierstrassZeta(-1, -1, weie rstrassPInverse(-1, -1, cos(d*x + c) - I*sin(d*x + c) - 1/2)))/d
\[ \int \sqrt {3-4 \cos (c+d x)} \, dx=\int \sqrt {3 - 4 \cos {\left (c + d x \right )}}\, dx \] Input:
integrate((3-4*cos(d*x+c))**(1/2),x)
Output:
Integral(sqrt(3 - 4*cos(c + d*x)), x)
\[ \int \sqrt {3-4 \cos (c+d x)} \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \,d x } \] Input:
integrate((3-4*cos(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-4*cos(d*x + c) + 3), x)
\[ \int \sqrt {3-4 \cos (c+d x)} \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \,d x } \] Input:
integrate((3-4*cos(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-4*cos(d*x + c) + 3), x)
Timed out. \[ \int \sqrt {3-4 \cos (c+d x)} \, dx=\int \sqrt {3-4\,\cos \left (c+d\,x\right )} \,d x \] Input:
int((3 - 4*cos(c + d*x))^(1/2),x)
Output:
int((3 - 4*cos(c + d*x))^(1/2), x)
\[ \int \sqrt {3-4 \cos (c+d x)} \, dx=\int \sqrt {-4 \cos \left (d x +c \right )+3}d x \] Input:
int((3-4*cos(d*x+c))^(1/2),x)
Output:
int(sqrt( - 4*cos(c + d*x) + 3),x)