\(\int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx\) [509]

Optimal result

Integrand size = 23, antiderivative size = 138 \[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {47 E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{20 \sqrt {7} d}+\frac {59 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{60 \sqrt {7} d}+\frac {59 \sqrt {3+4 \cos (c+d x)} \sin (c+d x)}{105 d}-\frac {3 (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{70 d}+\frac {\cos (c+d x) (3+4 \cos (c+d x))^{3/2} \sin (c+d x)}{14 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.67 \[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {141 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )+59 \sqrt {7} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )+\sqrt {3+4 \cos (c+d x)} (212 \sin (c+d x)+9 \sin (2 (c+d x))+30 \sin (3 (c+d x)))}{420 d} \] Input:

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

Output:

(141*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] + 59*Sqrt[7]*EllipticF[(c + d*x)/ 
2, 8/7] + Sqrt[3 + 4*Cos[c + d*x]]*(212*Sin[c + d*x] + 9*Sin[2*(c + d*x)] 
+ 30*Sin[3*(c + d*x)]))/(420*d)
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 153, 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, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3272

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3502

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3232

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3140

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

Input:

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

Output:

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

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]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(274\) vs. \(2(121)=242\).

Time = 6.65 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99

method	result	size
default	\(-\frac {\sqrt {\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (7680 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-14976 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+12344 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-4480 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+413 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )-141 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )\right )}{420 \sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(275\)

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)^2-1)*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)^8-14976*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/ 
2*c)^6+12344*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-4480*sin(1/2*d*x+1/2* 
c)^2*cos(1/2*d*x+1/2*c)+413*(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))-141*(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 = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.07 \[ \int \cos ^3(c+d x) \sqrt {3+4 \cos (c+d x)} \, dx=\frac {4 \, {\left (60 \, \cos \left (d x + c\right )^{2} + 9 \, \cos \left (d x + c\right ) + 91\right )} \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sin \left (d x + c\right ) - 277 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 277 i \, \sqrt {2} {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + 282 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right ) - 282 i \, \sqrt {2} {\rm weierstrassZeta}\left (-1, 1, {\rm weierstrassPInverse}\left (-1, 1, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {1}{2}\right )\right )}{840 \, d} \] 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*I*sqrt(2)*weierstrassPInverse(-1, 1, cos(d*x + c) + I 
*sin(d*x + c) + 1/2) + 277*I*sqrt(2)*weierstrassPInverse(-1, 1, cos(d*x + 
c) - I*sin(d*x + c) + 1/2) + 282*I*sqrt(2)*weierstrassZeta(-1, 1, weierstr 
assPInverse(-1, 1, cos(d*x + c) + I*sin(d*x + c) + 1/2)) - 282*I*sqrt(2)*w 
eierstrassZeta(-1, 1, weierstrassPInverse(-1, 1, cos(d*x + c) - I*sin(d*x 
+ c) + 1/2)))/d
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

Giac [F]

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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