\(\int \frac {\cos ^2(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\) [555]

Optimal result

Integrand size = 23, antiderivative size = 80 \[ \int \frac {\cos ^2(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=-\frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{4 d}+\frac {17 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{12 \sqrt {7} d}-\frac {\sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{6 d} \] Output:

-1/4*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d+17/84*InverseJac 
obiAM(1/2*d*x+1/2*Pi+1/2*c,2/7*14^(1/2))*7^(1/2)/d-1/6*(3-4*cos(d*x+c))^(1 
/2)*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^2(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\frac {3 \sqrt {-3+4 \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )+17 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )-6 \sin (c+d x)+4 \sin (2 (c+d x))}{12 d \sqrt {3-4 \cos (c+d x)}} \] Input:

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

Output:

(3*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] + 17*Sqrt[-3 + 4*Co 
s[c + d*x]]*EllipticF[(c + d*x)/2, 8] - 6*Sin[c + d*x] + 4*Sin[2*(c + d*x) 
])/(12*d*Sqrt[3 - 4*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 3270, 25, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3270

\(\displaystyle -\frac {1}{6} \int -\frac {3 \cos (c+d x)+2}{\sqrt {3-4 \cos (c+d x)}}dx-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{6 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \int \frac {3 \cos (c+d x)+2}{\sqrt {3-4 \cos (c+d x)}}dx-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \int \frac {3 \sin \left (c+d x+\frac {\pi }{2}\right )+2}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{6 d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {1}{6} \left (\frac {17}{4} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}}dx-\frac {3}{4} \int \sqrt {3-4 \cos (c+d x)}dx\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {17}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3}{4} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{6 d}\)

\(\Big \downarrow \) 3133

\(\displaystyle \frac {1}{6} \left (\frac {17}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{6 d}\)

\(\Big \downarrow \) 3141

\(\displaystyle \frac {1}{6} \left (\frac {17 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{2 \sqrt {7} d}-\frac {3 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}\right )-\frac {\sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{6 d}\)

Input:

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

Output:

((-3*Sqrt[7]*EllipticE[(c + Pi + d*x)/2, 8/7])/(2*d) + (17*EllipticF[(c + 
Pi + d*x)/2, 8/7])/(2*Sqrt[7]*d))/6 - (Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d* 
x])/(6*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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. \(231\) vs. \(2(72)=144\).

Time = 2.72 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.90

method	result	size
default	\(-\frac {\sqrt {-\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (224 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-28 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+17 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )\right )}{84 \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+7}\, d}\)	\(232\)

Input:

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

Output:

-1/84*(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(224*sin(1/ 
2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-28*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2* 
c)+17*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(56*sin(1/2*d*x+1/2*c)^2-7)^(1/2)*Ellip 
ticF(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)*EllipticE(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 = 128, normalized size of antiderivative = 1.60 \[ \int \frac {\cos ^2(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=-\frac {4 \, \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) + 7 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 7 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) - 6 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) - 6 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{24 \, d} \] Input:

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

Output:

-1/24*(4*sqrt(-4*cos(d*x + c) + 3)*sin(d*x + c) + 7*sqrt(2)*weierstrassPIn 
verse(-1, -1, cos(d*x + c) + I*sin(d*x + c) - 1/2) + 7*sqrt(2)*weierstrass 
PInverse(-1, -1, cos(d*x + c) - I*sin(d*x + c) - 1/2) - 6*sqrt(2)*weierstr 
assZeta(-1, -1, weierstrassPInverse(-1, -1, cos(d*x + c) + I*sin(d*x + c) 
- 1/2)) - 6*sqrt(2)*weierstrassZeta(-1, -1, weierstrassPInverse(-1, -1, co 
s(d*x + c) - I*sin(d*x + c) - 1/2)))/d
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^2(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx=\frac {\sqrt {4\,\cos \left (c+d\,x\right )-3}\,\left (6\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |8\right )+34\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |8\right )\right )}{24\,d\,\sqrt {3-4\,\cos \left (c+d\,x\right )}}-\frac {\sin \left (c+d\,x\right )\,\sqrt {3-4\,\cos \left (c+d\,x\right )}}{6\,d} \] Input:

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

Output:

((4*cos(c + d*x) - 3)^(1/2)*(6*ellipticE(c/2 + (d*x)/2, 8) + 34*ellipticF( 
c/2 + (d*x)/2, 8)))/(24*d*(3 - 4*cos(c + d*x))^(1/2)) - (sin(c + d*x)*(3 - 
 4*cos(c + d*x))^(1/2))/(6*d)
 

Reduce [F]

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

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

Output:

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