Integrand size = 14, antiderivative size = 23 \[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d} \] Output:
2*EllipticE(sin(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d} \] Input:
Integrate[Sqrt[3 + 4*Cos[c + d*x]],x]
Output:
(2*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/d
Time = 0.18 (sec) , antiderivative size = 23, 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, 3132}
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.
\(\displaystyle \int \sqrt {4 \cos (c+d x)+3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}dx\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}\) |
Input:
Int[Sqrt[3 + 4*Cos[c + d*x]],x]
Output:
(2*Sqrt[7]*EllipticE[(c + 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. \(136\) vs. \(2(23)=46\).
Time = 2.91 (sec) , antiderivative size = 137, normalized size of antiderivative = 5.96
method | result | size |
default | \(\frac {2 \sqrt {\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )}{\sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(137\) |
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 )}\) | \(659\) |
Input:
int((3+4*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
2*((8*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2 *c)^2)^(1/2)*(1-8*cos(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c) ,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
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.70 \[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\frac {-i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 4 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 4 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{2 \, d} \] Input:
integrate((3+4*cos(d*x+c))^(1/2),x, algorithm="fricas")
Output:
1/2*(-I*sqrt(2)*weierstrassPInverse(-1, 1, cos(d*x + c) + I*sin(d*x + c) + 1/2) + I*sqrt(2)*weierstrassPInverse(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2) + 4*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, co s(d*x + c) + I*sin(d*x + c) + 1/2)) - 4*I*sqrt(2)*weierstrassZeta(-1, 1, w eierstrassPInverse(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2)))/d
\[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\int \sqrt {4 \cos {\left (c + d x \right )} + 3}\, dx \] Input:
integrate((3+4*cos(d*x+c))**(1/2),x)
Output:
Integral(sqrt(4*cos(c + d*x) + 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="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. \[ \int \sqrt {3+4 \cos (c+d x)} \, dx=\int \sqrt {4\,\cos \left (c+d\,x\right )+3} \,d x \] Input:
int((4*cos(c + d*x) + 3)^(1/2),x)
Output:
int((4*cos(c + d*x) + 3)^(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)