\(\int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx\) [439]

Optimal result

Integrand size = 21, antiderivative size = 235 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {1}{16} \left (8 a^4+36 a^2 b^2+5 b^4\right ) x-\frac {a \left (4 a^4-121 a^2 b^2-128 b^4\right ) \sin (c+d x)}{60 b d}-\frac {\left (8 a^4-178 a^2 b^2-75 b^4\right ) \cos (c+d x) \sin (c+d x)}{240 d}-\frac {a \left (4 a^2-53 b^2\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{120 b d}-\frac {\left (4 a^2-25 b^2\right ) (a+b \cos (c+d x))^3 \sin (c+d x)}{120 b d}-\frac {a (a+b \cos (c+d x))^4 \sin (c+d x)}{30 b d}+\frac {(a+b \cos (c+d x))^5 \sin (c+d x)}{6 b d} \] Output:

1/16*(8*a^4+36*a^2*b^2+5*b^4)*x-1/60*a*(4*a^4-121*a^2*b^2-128*b^4)*sin(d*x 
+c)/b/d-1/240*(8*a^4-178*a^2*b^2-75*b^4)*cos(d*x+c)*sin(d*x+c)/d-1/120*a*( 
4*a^2-53*b^2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/b/d-1/120*(4*a^2-25*b^2)*(a+b* 
cos(d*x+c))^3*sin(d*x+c)/b/d-1/30*a*(a+b*cos(d*x+c))^4*sin(d*x+c)/b/d+1/6* 
(a+b*cos(d*x+c))^5*sin(d*x+c)/b/d
 

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.66 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {60 \left (8 a^4+36 a^2 b^2+5 b^4\right ) (c+d x)+480 a b \left (6 a^2+5 b^2\right ) \sin (c+d x)+15 \left (16 a^4+96 a^2 b^2+15 b^4\right ) \sin (2 (c+d x))+80 a b \left (4 a^2+5 b^2\right ) \sin (3 (c+d x))+45 b^2 \left (4 a^2+b^2\right ) \sin (4 (c+d x))+48 a b^3 \sin (5 (c+d x))+5 b^4 \sin (6 (c+d x))}{960 d} \] Input:

Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4,x]
 

Output:

(60*(8*a^4 + 36*a^2*b^2 + 5*b^4)*(c + d*x) + 480*a*b*(6*a^2 + 5*b^2)*Sin[c 
 + d*x] + 15*(16*a^4 + 96*a^2*b^2 + 15*b^4)*Sin[2*(c + d*x)] + 80*a*b*(4*a 
^2 + 5*b^2)*Sin[3*(c + d*x)] + 45*b^2*(4*a^2 + b^2)*Sin[4*(c + d*x)] + 48* 
a*b^3*Sin[5*(c + d*x)] + 5*b^4*Sin[6*(c + d*x)])/(960*d)
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 3270, 3042, 3232, 3042, 3232, 27, 3042, 3232, 3042, 3213}

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.

\(\displaystyle \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3270

\(\displaystyle \frac {\int (5 b-a \cos (c+d x)) (a+b \cos (c+d x))^4dx}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \left (5 b-a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4dx}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {1}{5} \int (a+b \cos (c+d x))^3 \left (21 a b-\left (4 a^2-25 b^2\right ) \cos (c+d x)\right )dx-\frac {a \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (21 a b+\left (25 b^2-4 a^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {a \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int 3 (a+b \cos (c+d x))^2 \left (b \left (24 a^2+25 b^2\right )-a \left (4 a^2-53 b^2\right ) \cos (c+d x)\right )dx-\frac {\left (4 a^2-25 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )-\frac {a \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int (a+b \cos (c+d x))^2 \left (b \left (24 a^2+25 b^2\right )-a \left (4 a^2-53 b^2\right ) \cos (c+d x)\right )dx-\frac {\left (4 a^2-25 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )-\frac {a \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b \left (24 a^2+25 b^2\right )-a \left (4 a^2-53 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {\left (4 a^2-25 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )-\frac {a \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3232

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (a b \left (64 a^2+181 b^2\right )-\left (8 a^4-178 b^2 a^2-75 b^4\right ) \cos (c+d x)\right )dx-\frac {a \left (4 a^2-53 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )-\frac {a \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (a b \left (64 a^2+181 b^2\right )+\left (-8 a^4+178 b^2 a^2+75 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {a \left (4 a^2-53 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )-\frac {a \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (-\frac {2 a \left (4 a^4-121 a^2 b^2-128 b^4\right ) \sin (c+d x)}{d}-\frac {b \left (8 a^4-178 a^2 b^2-75 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {15}{2} b x \left (8 a^4+36 a^2 b^2+5 b^4\right )\right )-\frac {a \left (4 a^2-53 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {\left (4 a^2-25 b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\right )-\frac {a \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}}{6 b}+\frac {\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}\)

Input:

Int[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4,x]
 

Output:

((a + b*Cos[c + d*x])^5*Sin[c + d*x])/(6*b*d) + (-1/5*(a*(a + b*Cos[c + d* 
x])^4*Sin[c + d*x])/d + (-1/4*((4*a^2 - 25*b^2)*(a + b*Cos[c + d*x])^3*Sin 
[c + d*x])/d + (3*(-1/3*(a*(4*a^2 - 53*b^2)*(a + b*Cos[c + d*x])^2*Sin[c + 
 d*x])/d + ((15*b*(8*a^4 + 36*a^2*b^2 + 5*b^4)*x)/2 - (2*a*(4*a^4 - 121*a^ 
2*b^2 - 128*b^4)*Sin[c + d*x])/d - (b*(8*a^4 - 178*a^2*b^2 - 75*b^4)*Cos[c 
 + d*x]*Sin[c + d*x])/(2*d))/3))/4)/5)/(6*b)
 

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 3213

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co 
s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free 
Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 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 [A] (verified)

Time = 112.54 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.65

method	result	size
parallelrisch	\(\frac {\left (240 a^{4}+1440 a^{2} b^{2}+225 b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (320 a^{3} b +400 a \,b^{3}\right ) \sin \left (3 d x +3 c \right )+\left (180 a^{2} b^{2}+45 b^{4}\right ) \sin \left (4 d x +4 c \right )+48 a \,b^{3} \sin \left (5 d x +5 c \right )+5 b^{4} \sin \left (6 d x +6 c \right )+\left (2880 a^{3} b +2400 a \,b^{3}\right ) \sin \left (d x +c \right )+480 \left (a^{4}+\frac {9}{2} a^{2} b^{2}+\frac {5}{8} b^{4}\right ) d x}{960 d}\)	\(153\)
derivativedivides	\(\frac {a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{3} b \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+6 a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a \,b^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(174\)
default	\(\frac {a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{3} b \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3}+6 a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a \,b^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(174\)
parts	\(\frac {a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {6 a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {4 a^{3} b \left (\cos \left (d x +c \right )^{2}+2\right ) \sin \left (d x +c \right )}{3 d}\)	\(185\)
risch	\(\frac {9 a^{2} b^{2} x}{4}+\frac {5 b^{4} x}{16}+\frac {a^{4} x}{2}+\frac {3 \sin \left (d x +c \right ) a^{3} b}{d}+\frac {5 \sin \left (d x +c \right ) a \,b^{3}}{2 d}+\frac {b^{4} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a \,b^{3} \sin \left (5 d x +5 c \right )}{20 d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{2} b^{2}}{16 d}+\frac {3 \sin \left (4 d x +4 c \right ) b^{4}}{64 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} b}{3 d}+\frac {5 \sin \left (3 d x +3 c \right ) a \,b^{3}}{12 d}+\frac {a^{4} \sin \left (2 d x +2 c \right )}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b^{2}}{2 d}+\frac {15 \sin \left (2 d x +2 c \right ) b^{4}}{64 d}\)	\(215\)
norman	\(\frac {\left (\frac {9}{4} a^{2} b^{2}+\frac {5}{16} b^{4}+\frac {1}{2} a^{4}\right ) x +\left (45 a^{2} b^{2}+\frac {25}{4} b^{4}+10 a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {9}{4} a^{2} b^{2}+\frac {5}{16} b^{4}+\frac {1}{2} a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {27}{2} a^{2} b^{2}+\frac {15}{8} b^{4}+3 a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {27}{2} a^{2} b^{2}+\frac {15}{8} b^{4}+3 a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {135}{4} a^{2} b^{2}+\frac {75}{16} b^{4}+\frac {15}{2} a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {135}{4} a^{2} b^{2}+\frac {75}{16} b^{4}+\frac {15}{2} a^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {\left (8 a^{4}-64 a^{3} b +60 a^{2} b^{2}-64 a \,b^{3}+11 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (8 a^{4}+64 a^{3} b +60 a^{2} b^{2}+64 a \,b^{3}+11 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {\left (40 a^{4}-960 a^{3} b +60 a^{2} b^{2}-832 a \,b^{3}+75 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}+\frac {\left (40 a^{4}+960 a^{3} b +60 a^{2} b^{2}+832 a \,b^{3}+75 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {\left (72 a^{4}-704 a^{3} b +252 a^{2} b^{2}-448 a \,b^{3}-5 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {\left (72 a^{4}+704 a^{3} b +252 a^{2} b^{2}+448 a \,b^{3}-5 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\)	\(511\)
orering	\(\text {Expression too large to display}\)	\(3492\)

Input:

int(cos(d*x+c)^2*(a+cos(d*x+c)*b)^4,x,method=_RETURNVERBOSE)
 

Output:

1/960*((240*a^4+1440*a^2*b^2+225*b^4)*sin(2*d*x+2*c)+(320*a^3*b+400*a*b^3) 
*sin(3*d*x+3*c)+(180*a^2*b^2+45*b^4)*sin(4*d*x+4*c)+48*a*b^3*sin(5*d*x+5*c 
)+5*b^4*sin(6*d*x+6*c)+(2880*a^3*b+2400*a*b^3)*sin(d*x+c)+480*(a^4+9/2*a^2 
*b^2+5/8*b^4)*d*x)/d
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.64 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {15 \, {\left (8 \, a^{4} + 36 \, a^{2} b^{2} + 5 \, b^{4}\right )} d x + {\left (40 \, b^{4} \cos \left (d x + c\right )^{5} + 192 \, a b^{3} \cos \left (d x + c\right )^{4} + 640 \, a^{3} b + 512 \, a b^{3} + 10 \, {\left (36 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + 64 \, {\left (5 \, a^{3} b + 4 \, a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (8 \, a^{4} + 36 \, a^{2} b^{2} + 5 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^4,x, algorithm="fricas")
 

Output:

1/240*(15*(8*a^4 + 36*a^2*b^2 + 5*b^4)*d*x + (40*b^4*cos(d*x + c)^5 + 192* 
a*b^3*cos(d*x + c)^4 + 640*a^3*b + 512*a*b^3 + 10*(36*a^2*b^2 + 5*b^4)*cos 
(d*x + c)^3 + 64*(5*a^3*b + 4*a*b^3)*cos(d*x + c)^2 + 15*(8*a^4 + 36*a^2*b 
^2 + 5*b^4)*cos(d*x + c))*sin(d*x + c))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (211) = 422\).

Time = 0.37 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.95 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx=\begin {cases} \frac {a^{4} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {a^{4} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {8 a^{3} b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {4 a^{3} b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 a^{2} b^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {9 a^{2} b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {9 a^{2} b^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {9 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {15 a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {32 a b^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {16 a b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {4 a b^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 b^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 b^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 b^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 b^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 b^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right )^{4} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:

integrate(cos(d*x+c)**2*(a+b*cos(d*x+c))**4,x)
 

Output:

Piecewise((a**4*x*sin(c + d*x)**2/2 + a**4*x*cos(c + d*x)**2/2 + a**4*sin( 
c + d*x)*cos(c + d*x)/(2*d) + 8*a**3*b*sin(c + d*x)**3/(3*d) + 4*a**3*b*si 
n(c + d*x)*cos(c + d*x)**2/d + 9*a**2*b**2*x*sin(c + d*x)**4/4 + 9*a**2*b* 
*2*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + 9*a**2*b**2*x*cos(c + d*x)**4/4 + 
 9*a**2*b**2*sin(c + d*x)**3*cos(c + d*x)/(4*d) + 15*a**2*b**2*sin(c + d*x 
)*cos(c + d*x)**3/(4*d) + 32*a*b**3*sin(c + d*x)**5/(15*d) + 16*a*b**3*sin 
(c + d*x)**3*cos(c + d*x)**2/(3*d) + 4*a*b**3*sin(c + d*x)*cos(c + d*x)**4 
/d + 5*b**4*x*sin(c + d*x)**6/16 + 15*b**4*x*sin(c + d*x)**4*cos(c + d*x)* 
*2/16 + 15*b**4*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*b**4*x*cos(c + d* 
x)**6/16 + 5*b**4*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*b**4*sin(c + d*x 
)**3*cos(c + d*x)**3/(6*d) + 11*b**4*sin(c + d*x)*cos(c + d*x)**5/(16*d), 
Ne(d, 0)), (x*(a + b*cos(c))**4*cos(c)**2, True))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.72 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} b + 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b^{2} + 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a b^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{4}}{960 \, d} \] Input:

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^4,x, algorithm="maxima")
 

Output:

1/960*(240*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^4 - 1280*(sin(d*x + c)^3 - 3 
*sin(d*x + c))*a^3*b + 180*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x 
 + 2*c))*a^2*b^2 + 256*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x 
+ c))*a*b^3 - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) 
 - 48*sin(2*d*x + 2*c))*b^4)/d
 

Giac [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.71 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {b^{4} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {a b^{3} \sin \left (5 \, d x + 5 \, c\right )}{20 \, d} + \frac {1}{16} \, {\left (8 \, a^{4} + 36 \, a^{2} b^{2} + 5 \, b^{4}\right )} x + \frac {3 \, {\left (4 \, a^{2} b^{2} + b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (16 \, a^{4} + 96 \, a^{2} b^{2} + 15 \, b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (6 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )}{2 \, d} \] Input:

integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^4,x, algorithm="giac")
 

Output:

1/192*b^4*sin(6*d*x + 6*c)/d + 1/20*a*b^3*sin(5*d*x + 5*c)/d + 1/16*(8*a^4 
 + 36*a^2*b^2 + 5*b^4)*x + 3/64*(4*a^2*b^2 + b^4)*sin(4*d*x + 4*c)/d + 1/1 
2*(4*a^3*b + 5*a*b^3)*sin(3*d*x + 3*c)/d + 1/64*(16*a^4 + 96*a^2*b^2 + 15* 
b^4)*sin(2*d*x + 2*c)/d + 1/2*(6*a^3*b + 5*a*b^3)*sin(d*x + c)/d
 

Mupad [B] (verification not implemented)

Time = 40.43 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.91 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {a^4\,x}{2}+\frac {5\,b^4\,x}{16}+\frac {9\,a^2\,b^2\,x}{4}+\frac {a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {15\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,b^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {b^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {5\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {a\,b^3\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {3\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {5\,a\,b^3\,\sin \left (c+d\,x\right )}{2\,d}+\frac {3\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \] Input:

int(cos(c + d*x)^2*(a + b*cos(c + d*x))^4,x)
 

Output:

(a^4*x)/2 + (5*b^4*x)/16 + (9*a^2*b^2*x)/4 + (a^4*sin(2*c + 2*d*x))/(4*d) 
+ (15*b^4*sin(2*c + 2*d*x))/(64*d) + (3*b^4*sin(4*c + 4*d*x))/(64*d) + (b^ 
4*sin(6*c + 6*d*x))/(192*d) + (5*a*b^3*sin(3*c + 3*d*x))/(12*d) + (a^3*b*s 
in(3*c + 3*d*x))/(3*d) + (a*b^3*sin(5*c + 5*d*x))/(20*d) + (3*a^2*b^2*sin( 
2*c + 2*d*x))/(2*d) + (3*a^2*b^2*sin(4*c + 4*d*x))/(16*d) + (5*a*b^3*sin(c 
 + d*x))/(2*d) + (3*a^3*b*sin(c + d*x))/d
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.89 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{4}-360 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{2}-130 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{4}+120 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4}+900 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{2}+165 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{4}+192 \sin \left (d x +c \right )^{5} a \,b^{3}-320 \sin \left (d x +c \right )^{3} a^{3} b -640 \sin \left (d x +c \right )^{3} a \,b^{3}+960 \sin \left (d x +c \right ) a^{3} b +960 \sin \left (d x +c \right ) a \,b^{3}+120 a^{4} d x +540 a^{2} b^{2} d x +75 b^{4} d x}{240 d} \] Input:

int(cos(d*x+c)^2*(a+b*cos(d*x+c))^4,x)
 

Output:

(40*cos(c + d*x)*sin(c + d*x)**5*b**4 - 360*cos(c + d*x)*sin(c + d*x)**3*a 
**2*b**2 - 130*cos(c + d*x)*sin(c + d*x)**3*b**4 + 120*cos(c + d*x)*sin(c 
+ d*x)*a**4 + 900*cos(c + d*x)*sin(c + d*x)*a**2*b**2 + 165*cos(c + d*x)*s 
in(c + d*x)*b**4 + 192*sin(c + d*x)**5*a*b**3 - 320*sin(c + d*x)**3*a**3*b 
 - 640*sin(c + d*x)**3*a*b**3 + 960*sin(c + d*x)*a**3*b + 960*sin(c + d*x) 
*a*b**3 + 120*a**4*d*x + 540*a**2*b**2*d*x + 75*b**4*d*x)/(240*d)