\(\int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx\) [517]

Optimal result

Integrand size = 23, antiderivative size = 107 \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\frac {21 \sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 d}-\frac {\sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{20 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{5 d}-\frac {(3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{10 d} \] Output:

21/20*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d-1/20*InverseJac 
obiAM(1/2*d*x+1/2*Pi+1/2*c,2/7*14^(1/2))*7^(1/2)/d+1/5*(3-4*cos(d*x+c))^(1 
/2)*sin(d*x+c)/d-1/10*(3-4*cos(d*x+c))^(3/2)*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.97 \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=-\frac {21 \sqrt {-3+4 \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )+7 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )+14 \sin (c+d x)-16 \sin (2 (c+d x))+8 \sin (3 (c+d x))}{20 d \sqrt {3-4 \cos (c+d x)}} \] Input:

Integrate[Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]^2,x]
 

Output:

-1/20*(21*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] + 7*Sqrt[-3 
+ 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8] + 14*Sin[c + d*x] - 16*Sin[2*( 
c + d*x)] + 8*Sin[3*(c + d*x)])/(d*Sqrt[3 - 4*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 3270, 27, 3042, 3232, 27, 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.

\(\displaystyle \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3270

\(\displaystyle -\frac {1}{10} \int -3 \sqrt {3-4 \cos (c+d x)} (\cos (c+d x)+2)dx-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{10} \int \sqrt {3-4 \cos (c+d x)} (\cos (c+d x)+2)dx-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{10} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right )+2\right )dx-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {3}{10} \left (\frac {2}{3} \int \frac {7 (2-3 \cos (c+d x))}{2 \sqrt {3-4 \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{10} \left (\frac {7}{3} \int \frac {2-3 \cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{10} \left (\frac {7}{3} \int \frac {2-3 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {3}{10} \left (\frac {7}{3} \left (\frac {3}{4} \int \sqrt {3-4 \cos (c+d x)}dx-\frac {1}{4} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3}{10} \left (\frac {7}{3} \left (\frac {3}{4} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {1}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

\(\Big \downarrow \) 3133

\(\displaystyle \frac {3}{10} \left (\frac {7}{3} \left (\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}-\frac {1}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

\(\Big \downarrow \) 3141

\(\displaystyle \frac {3}{10} \left (\frac {2 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}+\frac {7}{3} \left (\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}-\frac {\operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{2 \sqrt {7} d}\right )\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}\)

Input:

Int[Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]^2,x]
 

Output:

-1/10*((3 - 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/d + (3*((7*((3*Sqrt[7]*Ell 
ipticE[(c + Pi + d*x)/2, 8/7])/(2*d) - EllipticF[(c + Pi + d*x)/2, 8/7]/(2 
*Sqrt[7]*d)))/3 + (2*Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/10
 

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 3133

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 3141

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]
 

rule 3270

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(252\) vs. \(2(95)=190\).

Time = 6.08 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.36

method	result	size
default	\(\frac {\sqrt {-\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-256 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+128 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )\right )}{20 \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+7}\, d}\)	\(253\)

Input:

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

Output:

1/20*(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-256*cos(1/ 
2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+128*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2 
*c)-12*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+(sin(1/2*d*x+1/2*c)^2)^(1/2 
)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2/7*14^(1 
/2))-21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*Ell 
ipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2)))/(8*sin(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
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.29 \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\frac {4 \, {\left (4 \, \cos \left (d x + c\right ) - 1\right )} \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) - 7 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) - 7 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) - 42 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) - 42 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{40 \, d} \] Input:

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

Output:

1/40*(4*(4*cos(d*x + c) - 1)*sqrt(-4*cos(d*x + c) + 3)*sin(d*x + c) - 7*sq 
rt(2)*weierstrassPInverse(-1, -1, cos(d*x + c) + I*sin(d*x + c) - 1/2) - 7 
*sqrt(2)*weierstrassPInverse(-1, -1, cos(d*x + c) - I*sin(d*x + c) - 1/2) 
- 42*sqrt(2)*weierstrassZeta(-1, -1, weierstrassPInverse(-1, -1, cos(d*x + 
 c) + I*sin(d*x + c) - 1/2)) - 42*sqrt(2)*weierstrassZeta(-1, -1, weierstr 
assPInverse(-1, -1, cos(d*x + c) - I*sin(d*x + c) - 1/2)))/d
 

Sympy [F]

\[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\int \sqrt {3 - 4 \cos {\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )}\, dx \] Input:

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

Output:

Integral(sqrt(3 - 4*cos(c + d*x))*cos(c + d*x)**2, x)
 

Maxima [F]

\[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2} \,d x } \] Input:

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

Output:

integrate(sqrt(-4*cos(d*x + c) + 3)*cos(d*x + c)^2, x)
 

Giac [F]

\[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\int { \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \cos \left (d x + c\right )^{2} \,d x } \] Input:

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

Output:

integrate(sqrt(-4*cos(d*x + c) + 3)*cos(d*x + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\sqrt {3-4\,\cos \left (c+d\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx=\int \sqrt {-4 \cos \left (d x +c \right )+3}\, \cos \left (d x +c \right )^{2}d x \] Input:

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

Output:

int(sqrt( - 4*cos(c + d*x) + 3)*cos(c + d*x)**2,x)