3.6.16 \(\int \sqrt {3-4 \cos (c+d x)} \cos ^3(c+d x) \, dx\) [516]

3.6.16.1 Optimal result

Integrand size = 23, antiderivative size = 140 \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^3(c+d x) \, dx=-\frac {47 E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}-\frac {59 \operatorname {EllipticF}\left (\frac {1}{2} (c+\pi +d x),\frac {8}{7}\right )}{60 \sqrt {7} d}+\frac {59 \sqrt {3-4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3-4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}-\frac {(3-4 \cos (c+d x))^{3/2} \cos (c+d x) \sin (c+d x)}{14 d} \]

output

-3/70*(3-4*cos(d*x+c))^(3/2)*sin(d*x+c)/d-1/14*(3-4*cos(d*x+c))^(3/2)*cos( 
d*x+c)*sin(d*x+c)/d+47/140*(sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c) 
*EllipticE(cos(1/2*d*x+1/2*c),2/7*14^(1/2))/d*7^(1/2)+59/420*(sin(1/2*d*x+ 
1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)*EllipticF(cos(1/2*d*x+1/2*c),2/7*14^(1/ 
2))/d*7^(1/2)+59/105*sin(d*x+c)*(3-4*cos(d*x+c))^(1/2)/d
 

3.6.16.2 Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.81 \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^3(c+d x) \, dx=\frac {141 \sqrt {-3+4 \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |8\right )-413 \sqrt {-3+4 \cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),8\right )+654 \sin (c+d x)-511 \sin (2 (c+d x))+108 \sin (3 (c+d x))-60 \sin (4 (c+d x))}{420 d \sqrt {3-4 \cos (c+d x)}} \]

input

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

output

(141*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] - 413*Sqrt[-3 + 4 
*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8] + 654*Sin[c + d*x] - 511*Sin[2*(c 
 + d*x)] + 108*Sin[3*(c + d*x)] - 60*Sin[4*(c + d*x)])/(420*d*Sqrt[3 - 4*C 
os[c + d*x]])
 

3.6.16.3 Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 3272, 3042, 3502, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3272

\(\displaystyle -\frac {1}{14} \int \sqrt {3-4 \cos (c+d x)} \left (-6 \cos ^2(c+d x)-10 \cos (c+d x)+3\right )dx-\frac {\sin (c+d x) \cos (c+d x) (3-4 \cos (c+d x))^{3/2}}{14 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{14} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \left (-6 \sin \left (c+d x+\frac {\pi }{2}\right )^2-10 \sin \left (c+d x+\frac {\pi }{2}\right )+3\right )dx-\frac {\sin (c+d x) \cos (c+d x) (3-4 \cos (c+d x))^{3/2}}{14 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {1}{14} \left (\frac {1}{10} \int 2 \sqrt {3-4 \cos (c+d x)} (59 \cos (c+d x)+3)dx-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{5} \int \sqrt {3-4 \cos (c+d x)} (59 \cos (c+d x)+3)dx-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{14} \left (\frac {1}{5} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )} \left (59 \sin \left (c+d x+\frac {\pi }{2}\right )+3\right )dx-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {1}{14} \left (\frac {1}{5} \left (\frac {2}{3} \int -\frac {209-141 \cos (c+d x)}{2 \sqrt {3-4 \cos (c+d x)}}dx+\frac {118 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{5} \left (\frac {118 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}-\frac {1}{3} \int \frac {209-141 \cos (c+d x)}{\sqrt {3-4 \cos (c+d x)}}dx\right )-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{14} \left (\frac {1}{5} \left (\frac {118 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}-\frac {1}{3} \int \frac {209-141 \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {1}{14} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {413}{4} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}}dx-\frac {141}{4} \int \sqrt {3-4 \cos (c+d x)}dx\right )+\frac {118 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{14} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {413}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {141}{4} \int \sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {118 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

\(\Big \downarrow \) 3133

\(\displaystyle \frac {1}{14} \left (\frac {1}{5} \left (\frac {1}{3} \left (-\frac {413}{4} \int \frac {1}{\sqrt {3-4 \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {141 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}\right )+\frac {118 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}\right )-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

\(\Big \downarrow \) 3141

\(\displaystyle \frac {1}{14} \left (\frac {1}{5} \left (\frac {118 \sin (c+d x) \sqrt {3-4 \cos (c+d x)}}{3 d}+\frac {1}{3} \left (-\frac {59 \sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x+\pi ),\frac {8}{7}\right )}{2 d}-\frac {141 \sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{2 d}\right )\right )-\frac {3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{5 d}\right )-\frac {\sin (c+d x) (3-4 \cos (c+d x))^{3/2} \cos (c+d x)}{14 d}\)

input

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

output

-1/14*((3 - 4*Cos[c + d*x])^(3/2)*Cos[c + d*x]*Sin[c + d*x])/d + ((-3*(3 - 
 4*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((-141*Sqrt[7]*EllipticE[(c 
+ Pi + d*x)/2, 8/7])/(2*d) - (59*Sqrt[7]*EllipticF[(c + Pi + d*x)/2, 8/7]) 
/(2*d))/3 + (118*Sqrt[3 - 4*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/5)/14
 

3.6.16.3.1 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 3272

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m 
 + n))   Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d 
*(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 
 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m 
] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] 
&& NeQ[c, 0])))
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[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[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

3.6.16.4 Maple [A] (verified)

Time = 7.31 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.97

method	result	size
default	\(\frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (7680 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8064 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5432 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-568 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+59 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )+141 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )\right )}{420 \sqrt {8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d}\)	\(276\)

input

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

output

1/420*(-(8*cos(1/2*d*x+1/2*c)^2-7)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(7680*cos(1 
/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-8064*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2 
*c)^6+5432*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-568*sin(1/2*d*x+1/2*c)^ 
2*cos(1/2*d*x+1/2*c)+59*(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))+141*(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
 

3.6.16.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06 \[ \int \sqrt {3-4 \cos (c+d x)} \cos ^3(c+d x) \, dx=\frac {4 \, {\left (60 \, \cos \left (d x + c\right )^{2} - 9 \, \cos \left (d x + c\right ) + 91\right )} \sqrt {-4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) + 277 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 277 \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right ) + 282 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right ) + 282 \, \sqrt {2} {\rm weierstrassZeta}\left (-1, -1, {\rm weierstrassPInverse}\left (-1, -1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) - \frac {1}{2}\right )\right )}{840 \, d} \]

input

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

output

1/840*(4*(60*cos(d*x + c)^2 - 9*cos(d*x + c) + 91)*sqrt(-4*cos(d*x + c) + 
3)*sin(d*x + c) + 277*sqrt(2)*weierstrassPInverse(-1, -1, cos(d*x + c) + I 
*sin(d*x + c) - 1/2) + 277*sqrt(2)*weierstrassPInverse(-1, -1, cos(d*x + c 
) - I*sin(d*x + c) - 1/2) + 282*sqrt(2)*weierstrassZeta(-1, -1, weierstras 
sPInverse(-1, -1, cos(d*x + c) + I*sin(d*x + c) - 1/2)) + 282*sqrt(2)*weie 
rstrassZeta(-1, -1, weierstrassPInverse(-1, -1, cos(d*x + c) - I*sin(d*x + 
 c) - 1/2)))/d
 

3.6.16.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.6.16.7 Maxima [F]

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

input

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

output

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

3.6.16.8 Giac [F]

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

input

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

output

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

3.6.16.9 Mupad [F(-1)]

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

input

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

output

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