Integrand size = 21, antiderivative size = 53 \[ \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=-\frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{2 d}+\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{2 \sqrt {7} d} \] Output:
-1/2*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d+3/14*InverseJaco biAM(1/2*d*x+1/2*Pi+1/2*c,2/7*14^(1/2))*7^(1/2)/d
Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.13 \[ \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\frac {\sqrt {-3+4 \cos (c+d x)} \left (E\left (\left .\frac {1}{2} (c+d x)\right |8\right )+3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )\right )}{2 d \sqrt {3-4 \cos (c+d x)}} \] Input:
Integrate[Cos[c + d*x]/Sqrt[3 - 4*Cos[c + d*x]],x]
Output:
(Sqrt[-3 + 4*Cos[c + d*x]]*(EllipticE[(c + d*x)/2, 8] + 3*EllipticF[(c + d *x)/2, 8]))/(2*d*Sqrt[3 - 4*Cos[c + d*x]])
Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3231, 3042, 3133, 3141}
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.
| \(\displaystyle \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {3}{4} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}}dx-\frac {1}{4} \int \sqrt {3-4 \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {1}{4} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3133 |
| \(\displaystyle \frac {3}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}\) |
| \(\Big \downarrow \) 3141 |
| \(\displaystyle \frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{2 \sqrt {7} d}-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}\) |
Input:
Int[Cos[c + d*x]/Sqrt[3 - 4*Cos[c + d*x]],x]
Output:
-1/2*(Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/d + (3*EllipticF[(c + Pi + d*x)/2, 8/7])/(2*Sqrt[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]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a - b]))*EllipticF[(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]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*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]
Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(49)=98\).
Time = 1.89 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.98
| method | result | size |
| default | \(-\frac {\sqrt {-\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \left (3 \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-7 \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )\right )}{14 \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+7}\, d}\) | \(158\) |
| risch | \(-\frac {i \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 )}}{2 d \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 {i \left (\frac {-2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}-2}{\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 {2 \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 )}}\, {\mathrm e}^{-i \left (d x +c \right )}}{d \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 )}}}\) | \(499\) |
Input:
int(cos(d*x+c)/(3-4*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/14*(-(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)*(3*EllipticF(cos(1/2*d *x+1/2*c),2/7*14^(1/2))-7*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2)))/(8*s in(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 = 0.09 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.00 \[ \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=-\frac {\sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) - 2 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) - 2 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{4 \, d} \] Input:
integrate(cos(d*x+c)/(3-4*cos(d*x+c))^(1/2),x, algorithm="fricas")
Output:
-1/4*(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) - 2*sqrt(2)*weierstrassZeta(-1, -1, weierstrassPInverse(-1, -1, cos( d*x + c) + I*sin(d*x + c) - 1/2)) - 2*sqrt(2)*weierstrassZeta(-1, -1, weie rstrassPInverse(-1, -1, cos(d*x + c) - I*sin(d*x + c) - 1/2)))/d
\[ \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\int \frac {\cos {\left (c + d x \right )}}{\sqrt {3 - 4 \cos {\left (c + d x \right )}}}\, dx \] Input:
integrate(cos(d*x+c)/(3-4*cos(d*x+c))**(1/2),x)
Output:
Integral(cos(c + d*x)/sqrt(3 - 4*cos(c + d*x)), x)
\[ \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}} \,d x } \] Input:
integrate(cos(d*x+c)/(3-4*cos(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(cos(d*x + c)/sqrt(-4*cos(d*x + c) + 3), x)
\[ \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )}{\sqrt {-4 \, \cos \left (d x + c\right ) + 3}} \,d x } \] Input:
integrate(cos(d*x+c)/(3-4*cos(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(cos(d*x + c)/sqrt(-4*cos(d*x + c) + 3), x)
Time = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\frac {\sqrt {4\,\cos \left (c+d\,x\right )-3}\,\left (\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |8\right )+3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |8\right )\right )}{2\,d\,\sqrt {3-4\,\cos \left (c+d\,x\right )}} \] Input:
int(cos(c + d*x)/(3 - 4*cos(c + d*x))^(1/2),x)
Output:
((4*cos(c + d*x) - 3)^(1/2)*(ellipticE(c/2 + (d*x)/2, 8) + 3*ellipticF(c/2 + (d*x)/2, 8)))/(2*d*(3 - 4*cos(c + d*x))^(1/2))
\[ \int \frac {\cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=-\left (\int \frac {\sqrt {-4 \cos \left (d x +c \right )+3}\, \cos \left (d x +c \right )}{4 \cos \left (d x +c \right )-3}d x \right ) \] Input:
int(cos(d*x+c)/(3-4*cos(d*x+c))^(1/2),x)
Output:
- int((sqrt( - 4*cos(c + d*x) + 3)*cos(c + d*x))/(4*cos(c + d*x) - 3),x)