\(\int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx\) [549]

Optimal result

Integrand size = 21, antiderivative size = 51 \[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{2 \sqrt {7} d} \] Output:

1/2*EllipticE(sin(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d-3/14*InverseJacob 
iAM(1/2*d*x+1/2*c,2/7*14^(1/2))*7^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\frac {7 E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )-3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{2 \sqrt {7} d} \] Input:

Integrate[Cos[c + d*x]/Sqrt[3 + 4*Cos[c + d*x]],x]
 

Output:

(7*EllipticE[(c + d*x)/2, 8/7] - 3*EllipticF[(c + d*x)/2, 8/7])/(2*Sqrt[7] 
*d)
 

Rubi [A] (verified)

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

\(\displaystyle \frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}-\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{2 \sqrt {7} d}\)

Input:

Int[Cos[c + d*x]/Sqrt[3 + 4*Cos[c + d*x]],x]
 

Output:

(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(2*d) - (3*EllipticF[(c + d*x)/2, 8/ 
7])/(2*Sqrt[7]*d)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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]
 

rule 3140

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]
 

rule 3231

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(154\) vs. \(2(46)=92\).

Time = 1.95 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.04

method	result	size
default	\(\frac {\sqrt {\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (3 \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )+\operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )\right )}{2 \sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(155\)
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}\, \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 )}}}\)	\(497\)

Input:

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

Output:

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
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.12 \[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\frac {i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) - i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 2 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 2 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{4 \, d} \] Input:

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

Output:

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
 

Sympy [F]

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

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

Output:

Integral(cos(c + d*x)/sqrt(4*cos(c + d*x) + 3), x)
 

Maxima [F]

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

Giac [F]

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

Mupad [B] (verification not implemented)

Time = 44.62 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.06 \[ \int \frac {\cos (c+d x)}{\sqrt {3+4 \cos (c+d x)}} \, dx=\frac {\sqrt {\frac {4\,\cos \left (c+d\,x\right )}{7}+\frac {3}{7}}\,\left (7\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {8}{7}\right )-3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |\frac {8}{7}\right )\right )}{2\,d\,\sqrt {4\,\cos \left (c+d\,x\right )+3}} \] Input:

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

Output:

(((4*cos(c + d*x))/7 + 3/7)^(1/2)*(7*ellipticE(c/2 + (d*x)/2, 8/7) - 3*ell 
ipticF(c/2 + (d*x)/2, 8/7)))/(2*d*(4*cos(c + d*x) + 3)^(1/2))
 

Reduce [F]

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