\(\int \cos (c+d x) \sqrt {3+4 \cos (c+d x)} \, dx\) [511]

Optimal result

Integrand size = 21, antiderivative size = 78 \[ \int \cos (c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}+\frac {\sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{6 d}+\frac {2 \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{3 d} \] Output:

1/2*EllipticE(sin(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d+1/6*InverseJacobi 
AM(1/2*d*x+1/2*c,2/7*14^(1/2))*7^(1/2)/d+2/3*(3+4*cos(d*x+c))^(1/2)*sin(d* 
x+c)/d
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int \cos (c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )+\sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )+4 \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{6 d} \] Input:

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

Output:

(3*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] + Sqrt[7]*EllipticF[(c + d*x)/2, 8/ 
7] + 4*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(6*d)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 3232, 27, 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.

\(\displaystyle \int \cos (c+d x) \sqrt {4 \cos (c+d x)+3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}dx\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {2}{3} \int \frac {3 \cos (c+d x)+4}{2 \sqrt {4 \cos (c+d x)+3}}dx+\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {3 \cos (c+d x)+4}{\sqrt {4 \cos (c+d x)+3}}dx+\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \int \frac {3 \sin \left (c+d x+\frac {\pi }{2}\right )+4}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {1}{3} \left (\frac {7}{4} \int \frac {1}{\sqrt {4 \cos (c+d x)+3}}dx+\frac {3}{4} \int \sqrt {4 \cos (c+d x)+3}dx\right )+\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \left (\frac {7}{4} \int \frac {1}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+\frac {3}{4} \int \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}dx\right )+\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{3} \left (\frac {7}{4} \int \frac {1}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}\right )+\frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 \sin (c+d x) \sqrt {4 \cos (c+d x)+3}}{3 d}+\frac {1}{3} \left (\frac {\sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{2 d}+\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}\right )\)

Input:

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

Output:

((3*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/(2*d) + (Sqrt[7]*EllipticF[(c + d 
*x)/2, 8/7])/(2*d))/3 + (2*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(3*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]
 

rule 3232

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( 
f*(m + 1))), x] + Simp[1/(m + 1)   Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* 
d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ 
[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 
 0] && IntegerQ[2*m]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(230\) vs. \(2(69)=138\).

Time = 2.58 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.96

method	result	size
default	\(-\frac {\sqrt {\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (64 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )\right )}{6 \sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(231\)

Input:

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

Output:

-1/6*((8*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(64*sin(1/2*d 
*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-56*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+ 
7*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(8*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*EllipticF( 
cos(1/2*d*x+1/2*c),2*2^(1/2))-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(8*sin(1/2*d* 
x+1/2*c)^2-7)^(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 = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.64 \[ \int \cos (c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {8 \, \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) - 5 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 5 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 6 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 6 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{12 \, d} \] Input:

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

Output:

1/12*(8*sqrt(4*cos(d*x + c) + 3)*sin(d*x + c) - 5*I*sqrt(2)*weierstrassPIn 
verse(-1, 1, cos(d*x + c) + I*sin(d*x + c) + 1/2) + 5*I*sqrt(2)*weierstras 
sPInverse(-1, 1, cos(d*x + c) - I*sin(d*x + c) + 1/2) + 6*I*sqrt(2)*weiers 
trassZeta(-1, 1, weierstrassPInverse(-1, 1, cos(d*x + c) + I*sin(d*x + c) 
+ 1/2)) - 6*I*sqrt(2)*weierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, co 
s(d*x + c) - I*sin(d*x + c) + 1/2)))/d
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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