Integrand size = 21, antiderivative size = 51 \[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{2 \sqrt {7} d} \]
-3/14*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d* x+1/2*c),2/7*14^(1/2))/d*7^(1/2)+1/2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2* d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\frac {7 E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )-3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{2 \sqrt {7} d} \]
Time = 0.29 (sec) , antiderivative size = 51, 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, 3132, 3140}
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.
\(\displaystyle \int \frac {\cos (c+d x)}{\sqrt {4 \cos (c+d x)+3}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
\(\Big \downarrow \) 3231 |
\(\displaystyle \frac {1}{4} \int \sqrt {4 \cos (c+d x)+3}dx-\frac {3}{4} \int \frac {1}{\sqrt {4 \cos (c+d x)+3}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{4} \int \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}dx-\frac {3}{4} \int \frac {1}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle \frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {3}{4} \int \frac {1}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle \frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{2 \sqrt {7} d}\) |
3.6.49.3.1 Defintions of rubi rules used
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]
Time = 2.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.04
method | result | size |
default | \(\frac {\sqrt {\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (3 F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )+E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )\right )}{2 \sqrt {-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(155\) |
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}\, 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 )}}\, {\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 )}}}\) | \(497\) |
1/2*((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)*(3*EllipticF(cos(1/2*d*x+1 /2*c),2*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 higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.12 \[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\frac {i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) - i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 2 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 2 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{4 \, d} \]
1/4*(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) + 2*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, cos (d*x + c) + I*sin(d*x + c) + 1/2)) - 2*I*sqrt(2)*weierstrassZeta(-1, 1, we ierstrassPInverse(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2)))/d
\[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\int \frac {\cos {\left (c + d x \right )}}{\sqrt {4 \cos {\left (c + d x \right )} + 3}}\, dx \]
\[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
\[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )}{\sqrt {4 \, \cos \left (d x + c\right ) + 3}} \,d x } \]
Time = 14.50 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.06 \[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\frac {\sqrt {\frac {4\,\cos \left (c+d\,x\right )}{7}+\frac {3}{7}}\,\left (7\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {8}{7}\right )-3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {8}{7}\right )\right )}{2\,d\,\sqrt {4\,\cos \left (c+d\,x\right )+3}} \]