\(\int \cos (c+d x) (a+b \cos (c+d x))^3 (A+C \cos ^2(c+d x)) \, dx\) [540]

Optimal result

Integrand size = 31, antiderivative size = 264 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) x+\frac {a \left (5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (15 A b^2+\left (a^2+12 b^2\right ) C\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d} \]

[Out]

1/16*b*(6*a^2*(4*A+3*C)+b^2*(6*A+5*C))*x+1/15*a*(5*a^2*(3*A+2*C)+6*b^2*(5*A+4*C))*sin(d*x+c)/d+1/16*b*(6*a^2*(
4*A+3*C)+b^2*(6*A+5*C))*cos(d*x+c)*sin(d*x+c)/d+1/15*a*(15*A*b^2+(a^2+12*b^2)*C)*cos(d*x+c)^2*sin(d*x+c)/d+1/1
20*b*(6*a^2*C+5*b^2*(6*A+5*C))*cos(d*x+c)^3*sin(d*x+c)/d+1/10*a*C*cos(d*x+c)^2*(a+b*cos(d*x+c))^2*sin(d*x+c)/d
+1/6*C*cos(d*x+c)^2*(a+b*cos(d*x+c))^3*sin(d*x+c)/d

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3129, 3128, 3112, 3102, 2813} \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a \left (5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \sin (c+d x) \cos ^3(c+d x)}{120 d}+\frac {a \left (C \left (a^2+12 b^2\right )+15 A b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac {b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} b x \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac {a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d} \]

[In]

Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]

[Out]

(b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*x)/16 + (a*(5*a^2*(3*A + 2*C) + 6*b^2*(5*A + 4*C))*Sin[c + d*x])/(15*
d) + (b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(15*A*b^2 + (a^2 + 12*b^2
)*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*d) + (b*(6*a^2*C + 5*b^2*(6*A + 5*C))*Cos[c + d*x]^3*Sin[c + d*x])/(120*
d) + (a*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(10*d) + (C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^
3*Sin[c + d*x])/(6*d)

Rule 2813

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)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[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]

Rule 3112

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +
b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3128

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

Rule 3129

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

Rubi steps \begin{align*} \text {integral}& = \frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (2 a (3 A+C)+b (6 A+5 C) \cos (c+d x)+3 a C \cos ^2(c+d x)\right ) \, dx \\ & = \frac {a C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{30} \int \cos (c+d x) (a+b \cos (c+d x)) \left (2 a^2 (15 A+8 C)+a b (60 A+47 C) \cos (c+d x)+\left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{120} \int \cos (c+d x) \left (8 a^3 (15 A+8 C)+15 b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \cos (c+d x)+24 a \left (15 A b^2+\left (a^2+12 b^2\right ) C\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (15 A b^2+\left (a^2+12 b^2\right ) C\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{360} \int \cos (c+d x) \left (24 a \left (5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right )+45 b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \cos (c+d x)\right ) \, dx \\ & = \frac {1}{16} b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) x+\frac {a \left (5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (15 A b^2+\left (a^2+12 b^2\right ) C\right ) \cos ^2(c+d x) \sin (c+d x)}{15 d}+\frac {b \left (6 a^2 C+5 b^2 (6 A+5 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a C \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{10 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{6 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.03 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.95 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1440 a^2 A b c+360 A b^3 c+1080 a^2 b c C+300 b^3 c C+1440 a^2 A b d x+360 A b^3 d x+1080 a^2 b C d x+300 b^3 C d x+120 a \left (3 b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \sin (c+d x)+15 b \left (48 a^2 (A+C)+b^2 (16 A+15 C)\right ) \sin (2 (c+d x))+240 a A b^2 \sin (3 (c+d x))+80 a^3 C \sin (3 (c+d x))+300 a b^2 C \sin (3 (c+d x))+30 A b^3 \sin (4 (c+d x))+90 a^2 b C \sin (4 (c+d x))+45 b^3 C \sin (4 (c+d x))+36 a b^2 C \sin (5 (c+d x))+5 b^3 C \sin (6 (c+d x))}{960 d} \]

[In]

Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]

[Out]

(1440*a^2*A*b*c + 360*A*b^3*c + 1080*a^2*b*c*C + 300*b^3*c*C + 1440*a^2*A*b*d*x + 360*A*b^3*d*x + 1080*a^2*b*C
*d*x + 300*b^3*C*d*x + 120*a*(3*b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Sin[c + d*x] + 15*b*(48*a^2*(A + C) + b^2*(
16*A + 15*C))*Sin[2*(c + d*x)] + 240*a*A*b^2*Sin[3*(c + d*x)] + 80*a^3*C*Sin[3*(c + d*x)] + 300*a*b^2*C*Sin[3*
(c + d*x)] + 30*A*b^3*Sin[4*(c + d*x)] + 90*a^2*b*C*Sin[4*(c + d*x)] + 45*b^3*C*Sin[4*(c + d*x)] + 36*a*b^2*C*
Sin[5*(c + d*x)] + 5*b^3*C*Sin[6*(c + d*x)])/(960*d)

Maple [A] (verified)

Time = 8.82 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.69

method	result	size
parallelrisch	\(\frac {720 \left (\left (\frac {A}{3}+\frac {5 C}{16}\right ) b^{2}+a^{2} \left (A +C \right )\right ) b \sin \left (2 d x +2 c \right )+240 \left (\left (A +\frac {5 C}{4}\right ) b^{2}+\frac {a^{2} C}{3}\right ) a \sin \left (3 d x +3 c \right )+30 \left (b^{2} \left (A +\frac {3 C}{2}\right )+3 a^{2} C \right ) b \sin \left (4 d x +4 c \right )+36 C a \,b^{2} \sin \left (5 d x +5 c \right )+5 C \,b^{3} \sin \left (6 d x +6 c \right )+960 \left (\frac {3 \left (3 A +\frac {5 C}{2}\right ) b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right )\right ) a \sin \left (d x +c \right )+1440 x b \left (\frac {\left (A +\frac {5 C}{6}\right ) b^{2}}{4}+a^{2} \left (A +\frac {3 C}{4}\right )\right ) d}{960 d}\)	\(181\)
parts	\(\frac {\left (A \,b^{3}+3 C \,a^{2} b \right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (3 A a \,b^{2}+C \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {a^{3} A \sin \left (d x +c \right )}{d}+\frac {C \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {3 C a \,b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\)	\(219\)
derivativedivides	\(\frac {C \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 C a \,b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C \,a^{2} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A a \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,a^{3} \sin \left (d x +c \right )}{d}\)	\(249\)
default	\(\frac {C \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 C a \,b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+A \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 C \,a^{2} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A a \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 A \,a^{2} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,a^{3} \sin \left (d x +c \right )}{d}\)	\(249\)
risch	\(\frac {3 A \,a^{2} b x}{2}+\frac {3 A \,b^{3} x}{8}+\frac {9 C \,a^{2} b x}{8}+\frac {5 b^{3} C x}{16}+\frac {a^{3} A \sin \left (d x +c \right )}{d}+\frac {9 \sin \left (d x +c \right ) A a \,b^{2}}{4 d}+\frac {3 a^{3} C \sin \left (d x +c \right )}{4 d}+\frac {15 \sin \left (d x +c \right ) C a \,b^{2}}{8 d}+\frac {C \,b^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 C a \,b^{2} \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (4 d x +4 c \right ) A \,b^{3}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) C \,a^{2} b}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) C \,b^{3}}{64 d}+\frac {\sin \left (3 d x +3 c \right ) A a \,b^{2}}{4 d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{3}}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) C a \,b^{2}}{16 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{2} b}{4 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{3}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) C \,a^{2} b}{4 d}+\frac {15 \sin \left (2 d x +2 c \right ) C \,b^{3}}{64 d}\)	\(315\)
norman	\(\frac {\left (\frac {3}{2} A \,a^{2} b +\frac {3}{8} A \,b^{3}+\frac {9}{8} C \,a^{2} b +\frac {5}{16} C \,b^{3}\right ) x +\left (9 A \,a^{2} b +\frac {9}{4} A \,b^{3}+\frac {27}{4} C \,a^{2} b +\frac {15}{8} C \,b^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (9 A \,a^{2} b +\frac {9}{4} A \,b^{3}+\frac {27}{4} C \,a^{2} b +\frac {15}{8} C \,b^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 A \,a^{2} b +\frac {15}{2} A \,b^{3}+\frac {45}{2} C \,a^{2} b +\frac {25}{4} C \,b^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {3}{2} A \,a^{2} b +\frac {3}{8} A \,b^{3}+\frac {9}{8} C \,a^{2} b +\frac {5}{16} C \,b^{3}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {45}{2} A \,a^{2} b +\frac {45}{8} A \,b^{3}+\frac {135}{8} C \,a^{2} b +\frac {75}{16} C \,b^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {45}{2} A \,a^{2} b +\frac {45}{8} A \,b^{3}+\frac {135}{8} C \,a^{2} b +\frac {75}{16} C \,b^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (16 A \,a^{3}-24 A \,a^{2} b +48 A a \,b^{2}-10 A \,b^{3}+16 C \,a^{3}-30 C \,a^{2} b +48 C a \,b^{2}-11 C \,b^{3}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {\left (16 A \,a^{3}+24 A \,a^{2} b +48 A a \,b^{2}+10 A \,b^{3}+16 C \,a^{3}+30 C \,a^{2} b +48 C a \,b^{2}+11 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (240 A \,a^{3}-216 A \,a^{2} b +528 A a \,b^{2}-42 A \,b^{3}+176 C \,a^{3}-126 C \,a^{2} b +336 C a \,b^{2}+5 C \,b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {\left (240 A \,a^{3}+216 A \,a^{2} b +528 A a \,b^{2}+42 A \,b^{3}+176 C \,a^{3}+126 C \,a^{2} b +336 C a \,b^{2}-5 C \,b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {\left (400 A \,a^{3}-120 A \,a^{2} b +720 A a \,b^{2}-10 A \,b^{3}+240 C \,a^{3}-30 C \,a^{2} b +624 C a \,b^{2}-75 C \,b^{3}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {\left (400 A \,a^{3}+120 A \,a^{2} b +720 A a \,b^{2}+10 A \,b^{3}+240 C \,a^{3}+30 C \,a^{2} b +624 C a \,b^{2}+75 C \,b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\)	\(699\)

[In]

int(cos(d*x+c)*(a+cos(d*x+c)*b)^3*(A+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/960*(720*((1/3*A+5/16*C)*b^2+a^2*(A+C))*b*sin(2*d*x+2*c)+240*((A+5/4*C)*b^2+1/3*a^2*C)*a*sin(3*d*x+3*c)+30*(
b^2*(A+3/2*C)+3*a^2*C)*b*sin(4*d*x+4*c)+36*C*a*b^2*sin(5*d*x+5*c)+5*C*b^3*sin(6*d*x+6*c)+960*(3/4*(3*A+5/2*C)*
b^2+a^2*(A+3/4*C))*a*sin(d*x+c)+1440*x*b*(1/4*(A+5/6*C)*b^2+a^2*(A+3/4*C))*d)/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.72 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} d x + {\left (40 \, C b^{3} \cos \left (d x + c\right )^{5} + 144 \, C a b^{2} \cos \left (d x + c\right )^{4} + 80 \, {\left (3 \, A + 2 \, C\right )} a^{3} + 96 \, {\left (5 \, A + 4 \, C\right )} a b^{2} + 10 \, {\left (18 \, C a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (5 \, C a^{3} + 3 \, {\left (5 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (6 \, {\left (4 \, A + 3 \, C\right )} a^{2} b + {\left (6 \, A + 5 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]

[In]

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2),x, algorithm="fricas")

[Out]

1/240*(15*(6*(4*A + 3*C)*a^2*b + (6*A + 5*C)*b^3)*d*x + (40*C*b^3*cos(d*x + c)^5 + 144*C*a*b^2*cos(d*x + c)^4
+ 80*(3*A + 2*C)*a^3 + 96*(5*A + 4*C)*a*b^2 + 10*(18*C*a^2*b + (6*A + 5*C)*b^3)*cos(d*x + c)^3 + 16*(5*C*a^3 +
3*(5*A + 4*C)*a*b^2)*cos(d*x + c)^2 + 15*(6*(4*A + 3*C)*a^2*b + (6*A + 5*C)*b^3)*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. 668 vs. \(2 (245) = 490\).

Time = 0.40 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.53 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {A a^{3} \sin {\left (c + d x \right )}}{d} + \frac {3 A a^{2} b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{2} b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{2} b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 A a b^{2} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {3 A a b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A b^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 A b^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 A b^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 A b^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 C a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 C a^{2} b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a^{2} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {9 C a^{2} b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a^{2} b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {15 C a^{2} b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 C a b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {4 C a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a b^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C b^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 C b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 C b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 C b^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 C b^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 C b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 C b^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a + b \cos {\left (c \right )}\right )^{3} \cos {\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)*(a+b*cos(d*x+c))**3*(A+C*cos(d*x+c)**2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.92 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{2} - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a b^{2} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{3} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} - 960 \, A a^{3} \sin \left (d x + c\right )}{960 \, d} \]

[In]

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/960*(320*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 - 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^2*b - 90*(12*d*
x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^2*b + 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a*b^2 - 19
2*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a*b^2 - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*
sin(2*d*x + 2*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))*
C*b^3 - 960*A*a^3*sin(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.82 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C b^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, C a b^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{16} \, {\left (24 \, A a^{2} b + 18 \, C a^{2} b + 6 \, A b^{3} + 5 \, C b^{3}\right )} x + \frac {{\left (6 \, C a^{2} b + 2 \, A b^{3} + 3 \, C b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, C a^{3} + 12 \, A a b^{2} + 15 \, C a b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (48 \, A a^{2} b + 48 \, C a^{2} b + 16 \, A b^{3} + 15 \, C b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (8 \, A a^{3} + 6 \, C a^{3} + 18 \, A a b^{2} + 15 \, C a b^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \]

[In]

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2),x, algorithm="giac")

[Out]

1/192*C*b^3*sin(6*d*x + 6*c)/d + 3/80*C*a*b^2*sin(5*d*x + 5*c)/d + 1/16*(24*A*a^2*b + 18*C*a^2*b + 6*A*b^3 + 5
*C*b^3)*x + 1/64*(6*C*a^2*b + 2*A*b^3 + 3*C*b^3)*sin(4*d*x + 4*c)/d + 1/48*(4*C*a^3 + 12*A*a*b^2 + 15*C*a*b^2)
*sin(3*d*x + 3*c)/d + 1/64*(48*A*a^2*b + 48*C*a^2*b + 16*A*b^3 + 15*C*b^3)*sin(2*d*x + 2*c)/d + 1/8*(8*A*a^3 +
6*C*a^3 + 18*A*a*b^2 + 15*C*a*b^2)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 3.37 (sec) , antiderivative size = 617, normalized size of antiderivative = 2.34 \[ \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (2\,A\,a^3-\frac {5\,A\,b^3}{4}+2\,C\,a^3-\frac {11\,C\,b^3}{8}+6\,A\,a\,b^2-3\,A\,a^2\,b+6\,C\,a\,b^2-\frac {15\,C\,a^2\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (10\,A\,a^3-\frac {7\,A\,b^3}{4}+\frac {22\,C\,a^3}{3}+\frac {5\,C\,b^3}{24}+22\,A\,a\,b^2-9\,A\,a^2\,b+14\,C\,a\,b^2-\frac {21\,C\,a^2\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (20\,A\,a^3-\frac {A\,b^3}{2}+12\,C\,a^3-\frac {15\,C\,b^3}{4}+36\,A\,a\,b^2-6\,A\,a^2\,b+\frac {156\,C\,a\,b^2}{5}-\frac {3\,C\,a^2\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (20\,A\,a^3+\frac {A\,b^3}{2}+12\,C\,a^3+\frac {15\,C\,b^3}{4}+36\,A\,a\,b^2+6\,A\,a^2\,b+\frac {156\,C\,a\,b^2}{5}+\frac {3\,C\,a^2\,b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (10\,A\,a^3+\frac {7\,A\,b^3}{4}+\frac {22\,C\,a^3}{3}-\frac {5\,C\,b^3}{24}+22\,A\,a\,b^2+9\,A\,a^2\,b+14\,C\,a\,b^2+\frac {21\,C\,a^2\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^3+\frac {5\,A\,b^3}{4}+2\,C\,a^3+\frac {11\,C\,b^3}{8}+6\,A\,a\,b^2+3\,A\,a^2\,b+6\,C\,a\,b^2+\frac {15\,C\,a^2\,b}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {b\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (24\,A\,a^2+6\,A\,b^2+18\,C\,a^2+5\,C\,b^2\right )}{8\,d}+\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,A\,a^2+6\,A\,b^2+18\,C\,a^2+5\,C\,b^2\right )}{8\,\left (\frac {3\,A\,b^3}{4}+\frac {5\,C\,b^3}{8}+3\,A\,a^2\,b+\frac {9\,C\,a^2\,b}{4}\right )}\right )\,\left (24\,A\,a^2+6\,A\,b^2+18\,C\,a^2+5\,C\,b^2\right )}{8\,d} \]

[In]

int(cos(c + d*x)*(A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3,x)

[Out]

(tan(c/2 + (d*x)/2)*(2*A*a^3 + (5*A*b^3)/4 + 2*C*a^3 + (11*C*b^3)/8 + 6*A*a*b^2 + 3*A*a^2*b + 6*C*a*b^2 + (15*
C*a^2*b)/4) + tan(c/2 + (d*x)/2)^11*(2*A*a^3 - (5*A*b^3)/4 + 2*C*a^3 - (11*C*b^3)/8 + 6*A*a*b^2 - 3*A*a^2*b +
6*C*a*b^2 - (15*C*a^2*b)/4) + tan(c/2 + (d*x)/2)^3*(10*A*a^3 + (7*A*b^3)/4 + (22*C*a^3)/3 - (5*C*b^3)/24 + 22*
A*a*b^2 + 9*A*a^2*b + 14*C*a*b^2 + (21*C*a^2*b)/4) + tan(c/2 + (d*x)/2)^9*(10*A*a^3 - (7*A*b^3)/4 + (22*C*a^3)
/3 + (5*C*b^3)/24 + 22*A*a*b^2 - 9*A*a^2*b + 14*C*a*b^2 - (21*C*a^2*b)/4) + tan(c/2 + (d*x)/2)^5*(20*A*a^3 + (
A*b^3)/2 + 12*C*a^3 + (15*C*b^3)/4 + 36*A*a*b^2 + 6*A*a^2*b + (156*C*a*b^2)/5 + (3*C*a^2*b)/2) + tan(c/2 + (d*
x)/2)^7*(20*A*a^3 - (A*b^3)/2 + 12*C*a^3 - (15*C*b^3)/4 + 36*A*a*b^2 - 6*A*a^2*b + (156*C*a*b^2)/5 - (3*C*a^2*
b)/2))/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^
8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) - (b*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(24*A*a^2
+ 6*A*b^2 + 18*C*a^2 + 5*C*b^2))/(8*d) + (b*atan((b*tan(c/2 + (d*x)/2)*(24*A*a^2 + 6*A*b^2 + 18*C*a^2 + 5*C*b^
2))/(8*((3*A*b^3)/4 + (5*C*b^3)/8 + 3*A*a^2*b + (9*C*a^2*b)/4)))*(24*A*a^2 + 6*A*b^2 + 18*C*a^2 + 5*C*b^2))/(8
*d)