Integrand size = 33, antiderivative size = 166 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} a^3 (20 A+15 B+13 C) x+\frac {a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac {3 a^3 (20 A+15 B+13 C) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {(5 B-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}-\frac {a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d} \] Output:
1/8*a^3*(20*A+15*B+13*C)*x+1/5*a^3*(20*A+15*B+13*C)*sin(d*x+c)/d+3/40*a^3* (20*A+15*B+13*C)*cos(d*x+c)*sin(d*x+c)/d+1/20*(5*B-C)*(a+a*cos(d*x+c))^3*s in(d*x+c)/d+1/5*C*(a+a*cos(d*x+c))^4*sin(d*x+c)/a/d-1/60*a^3*(20*A+15*B+13 *C)*sin(d*x+c)^3/d
Time = 0.93 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^3 \sin (c+d x) \left (30 (20 A+15 B+13 C) \arcsin \left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )+\left (8 (55 A+45 B+38 C)+15 (12 A+15 B+13 C) \cos (c+d x)+8 (5 A+15 B+19 C) \cos ^2(c+d x)+30 (B+3 C) \cos ^3(c+d x)+24 C \cos ^4(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{120 d \sqrt {\sin ^2(c+d x)}} \] Input:
Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x ]
Output:
(a^3*Sin[c + d*x]*(30*(20*A + 15*B + 13*C)*ArcSin[Sqrt[Sin[(c + d*x)/2]^2] ] + (8*(55*A + 45*B + 38*C) + 15*(12*A + 15*B + 13*C)*Cos[c + d*x] + 8*(5* A + 15*B + 19*C)*Cos[c + d*x]^2 + 30*(B + 3*C)*Cos[c + d*x]^3 + 24*C*Cos[c + d*x]^4)*Sqrt[Sin[c + d*x]^2]))/(120*d*Sqrt[Sin[c + d*x]^2])
Time = 0.54 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3042, 3502, 3042, 3230, 3042, 3124, 2009}
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.
| \(\displaystyle \int (a \cos (c+d x)+a)^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^3 (a (5 A+4 C)+a (5 B-C) \cos (c+d x))dx}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (5 A+4 C)+a (5 B-C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {1}{4} a (20 A+15 B+13 C) \int (\cos (c+d x) a+a)^3dx+\frac {a (5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} a (20 A+15 B+13 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3dx+\frac {a (5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {\frac {1}{4} a (20 A+15 B+13 C) \int \left (\cos ^3(c+d x) a^3+3 \cos ^2(c+d x) a^3+3 \cos (c+d x) a^3+a^3\right )dx+\frac {a (5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{4} a (20 A+15 B+13 C) \left (-\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {5 a^3 x}{2}\right )+\frac {a (5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
Input:
Int[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) + ((a*(5*B - C)*(a + a*Cos [c + d*x])^3*Sin[c + d*x])/(4*d) + (a*(20*A + 15*B + 13*C)*((5*a^3*x)/2 + (4*a^3*Sin[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^3*Si n[c + d*x]^3)/(3*d)))/4)/(5*a)
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTri g[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
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[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
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]
Time = 156.18 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.64
| method | result | size |
| parallelrisch | \(\frac {\left (3 \left (3 A +4 B +4 C \right ) \sin \left (2 d x +2 c \right )+\left (A +3 B +\frac {17 C}{4}\right ) \sin \left (3 d x +3 c \right )+\frac {3 \left (B +3 C \right ) \sin \left (4 d x +4 c \right )}{8}+\frac {3 C \sin \left (5 d x +5 c \right )}{20}+3 \left (15 A +13 B +\frac {23 C}{2}\right ) \sin \left (d x +c \right )+30 x d \left (A +\frac {3 B}{4}+\frac {13 C}{20}\right )\right ) a^{3}}{12 d}\) | \(107\) |
| parts | \(a^{3} x A +\frac {\left (3 a^{3} A +B \,a^{3}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (a^{3} A +3 B \,a^{3}+3 C \,a^{3}\right ) \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (3 a^{3} A +3 B \,a^{3}+C \,a^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(197\) |
| risch | \(\frac {5 a^{3} x A}{2}+\frac {15 a^{3} x B}{8}+\frac {13 a^{3} C x}{8}+\frac {15 a^{3} A \sin \left (d x +c \right )}{4 d}+\frac {13 a^{3} B \sin \left (d x +c \right )}{4 d}+\frac {23 a^{3} C \sin \left (d x +c \right )}{8 d}+\frac {C \,a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{3}}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) C \,a^{3}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} A}{12 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{4 d}+\frac {17 \sin \left (3 d x +3 c \right ) C \,a^{3}}{48 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{3} A}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}+\frac {\sin \left (2 d x +2 c \right ) C \,a^{3}}{d}\) | \(228\) |
| derivativedivides | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 C \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{3} A \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+a^{3} A \left (d x +c \right )+B \,a^{3} \sin \left (d x +c \right )+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(295\) |
| default | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 C \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{3} A \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+a^{3} A \left (d x +c \right )+B \,a^{3} \sin \left (d x +c \right )+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(295\) |
| norman | \(\frac {\frac {a^{3} \left (20 A +15 B +13 C \right ) x}{8}+\frac {32 a^{3} \left (20 A +15 B +13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {7 a^{3} \left (20 A +15 B +13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {a^{3} \left (20 A +15 B +13 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}+\frac {5 a^{3} \left (20 A +15 B +13 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {5 a^{3} \left (20 A +15 B +13 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {5 a^{3} \left (20 A +15 B +13 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {5 a^{3} \left (20 A +15 B +13 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}+\frac {a^{3} \left (20 A +15 B +13 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {a^{3} \left (44 A +49 B +51 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (212 A +183 B +133 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}\) | \(312\) |
| orering | \(\text {Expression too large to display}\) | \(10006\) |
Input:
int((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETURNVER BOSE)
Output:
1/12*(3*(3*A+4*B+4*C)*sin(2*d*x+2*c)+(A+3*B+17/4*C)*sin(3*d*x+3*c)+3/8*(B+ 3*C)*sin(4*d*x+4*c)+3/20*C*sin(5*d*x+5*c)+3*(15*A+13*B+23/2*C)*sin(d*x+c)+ 30*x*d*(A+3/4*B+13/20*C))*a^3/d
Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.73 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (20 \, A + 15 \, B + 13 \, C\right )} a^{3} d x + {\left (24 \, C a^{3} \cos \left (d x + c\right )^{4} + 30 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 15 \, B + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (12 \, A + 15 \, B + 13 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (55 \, A + 45 \, B + 38 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \] Input:
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm= "fricas")
Output:
1/120*(15*(20*A + 15*B + 13*C)*a^3*d*x + (24*C*a^3*cos(d*x + c)^4 + 30*(B + 3*C)*a^3*cos(d*x + c)^3 + 8*(5*A + 15*B + 19*C)*a^3*cos(d*x + c)^2 + 15* (12*A + 15*B + 13*C)*a^3*cos(d*x + c) + 8*(55*A + 45*B + 38*C)*a^3)*sin(d* x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (150) = 300\).
Time = 0.30 (sec) , antiderivative size = 658, normalized size of antiderivative = 3.96 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate((a+a*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)
Output:
Piecewise((3*A*a**3*x*sin(c + d*x)**2/2 + 3*A*a**3*x*cos(c + d*x)**2/2 + A *a**3*x + 2*A*a**3*sin(c + d*x)**3/(3*d) + A*a**3*sin(c + d*x)*cos(c + d*x )**2/d + 3*A*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + 3*A*a**3*sin(c + d*x)/ d + 3*B*a**3*x*sin(c + d*x)**4/8 + 3*B*a**3*x*sin(c + d*x)**2*cos(c + d*x) **2/4 + 3*B*a**3*x*sin(c + d*x)**2/2 + 3*B*a**3*x*cos(c + d*x)**4/8 + 3*B* a**3*x*cos(c + d*x)**2/2 + 3*B*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 2 *B*a**3*sin(c + d*x)**3/d + 5*B*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 3*B*a**3*sin(c + d*x)*cos(c + d*x)**2/d + 3*B*a**3*sin(c + d*x)*cos(c + d* x)/(2*d) + B*a**3*sin(c + d*x)/d + 9*C*a**3*x*sin(c + d*x)**4/8 + 9*C*a**3 *x*sin(c + d*x)**2*cos(c + d*x)**2/4 + C*a**3*x*sin(c + d*x)**2/2 + 9*C*a* *3*x*cos(c + d*x)**4/8 + C*a**3*x*cos(c + d*x)**2/2 + 8*C*a**3*sin(c + d*x )**5/(15*d) + 4*C*a**3*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*C*a**3*si n(c + d*x)**3*cos(c + d*x)/(8*d) + 2*C*a**3*sin(c + d*x)**3/d + C*a**3*sin (c + d*x)*cos(c + d*x)**4/d + 15*C*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 3*C*a**3*sin(c + d*x)*cos(c + d*x)**2/d + C*a**3*sin(c + d*x)*cos(c + d *x)/(2*d), Ne(d, 0)), (x*(a*cos(c) + a)**3*(A + B*cos(c) + C*cos(c)**2), T rue))
Time = 0.05 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.70 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 480 \, {\left (d x + c\right )} A a^{3} + 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} + 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 1440 \, A a^{3} \sin \left (d x + c\right ) - 480 \, B a^{3} \sin \left (d x + c\right )}{480 \, d} \] Input:
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm= "maxima")
Output:
-1/480*(160*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 - 360*(2*d*x + 2*c + s in(2*d*x + 2*c))*A*a^3 - 480*(d*x + c)*A*a^3 + 480*(sin(d*x + c)^3 - 3*sin (d*x + c))*B*a^3 - 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2* c))*B*a^3 - 360*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^3 - 32*(3*sin(d*x + c )^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^3 + 480*(sin(d*x + c)^3 - 3 *sin(d*x + c))*C*a^3 - 45*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3 - 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^3 - 1440*A*a^3*si n(d*x + c) - 480*B*a^3*sin(d*x + c))/d
Time = 0.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.98 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{8} \, {\left (20 \, A a^{3} + 15 \, B a^{3} + 13 \, C a^{3}\right )} x + \frac {{\left (B a^{3} + 3 \, C a^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (4 \, A a^{3} + 12 \, B a^{3} + 17 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (3 \, A a^{3} + 4 \, B a^{3} + 4 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (30 \, A a^{3} + 26 \, B a^{3} + 23 \, C a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm= "giac")
Output:
1/80*C*a^3*sin(5*d*x + 5*c)/d + 1/8*(20*A*a^3 + 15*B*a^3 + 13*C*a^3)*x + 1 /32*(B*a^3 + 3*C*a^3)*sin(4*d*x + 4*c)/d + 1/48*(4*A*a^3 + 12*B*a^3 + 17*C *a^3)*sin(3*d*x + 3*c)/d + 1/4*(3*A*a^3 + 4*B*a^3 + 4*C*a^3)*sin(2*d*x + 2 *c)/d + 1/8*(30*A*a^3 + 26*B*a^3 + 23*C*a^3)*sin(d*x + c)/d
Time = 1.17 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.94 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (5\,A\,a^3+\frac {15\,B\,a^3}{4}+\frac {13\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {70\,A\,a^3}{3}+\frac {35\,B\,a^3}{2}+\frac {91\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {128\,A\,a^3}{3}+32\,B\,a^3+\frac {416\,C\,a^3}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {106\,A\,a^3}{3}+\frac {61\,B\,a^3}{2}+\frac {133\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A\,a^3+\frac {49\,B\,a^3}{4}+\frac {51\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (20\,A+15\,B+13\,C\right )}{4\,\left (5\,A\,a^3+\frac {15\,B\,a^3}{4}+\frac {13\,C\,a^3}{4}\right )}\right )\,\left (20\,A+15\,B+13\,C\right )}{4\,d}-\frac {a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (20\,A+15\,B+13\,C\right )}{4\,d} \] Input:
int((a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
(tan(c/2 + (d*x)/2)^9*(5*A*a^3 + (15*B*a^3)/4 + (13*C*a^3)/4) + tan(c/2 + (d*x)/2)^7*((70*A*a^3)/3 + (35*B*a^3)/2 + (91*C*a^3)/6) + tan(c/2 + (d*x)/ 2)^3*((106*A*a^3)/3 + (61*B*a^3)/2 + (133*C*a^3)/6) + tan(c/2 + (d*x)/2)^5 *((128*A*a^3)/3 + 32*B*a^3 + (416*C*a^3)/15) + tan(c/2 + (d*x)/2)*(11*A*a^ 3 + (49*B*a^3)/4 + (51*C*a^3)/4))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(20*A + 15*B + 13* C))/(4*(5*A*a^3 + (15*B*a^3)/4 + (13*C*a^3)/4)))*(20*A + 15*B + 13*C))/(4* d) - (a^3*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(20*A + 15*B + 13*C))/(4*d)
Time = 0.16 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.05 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^{3} \left (-30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b -90 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +180 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +255 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +285 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +24 \sin \left (d x +c \right )^{5} c -40 \sin \left (d x +c \right )^{3} a -120 \sin \left (d x +c \right )^{3} b -200 \sin \left (d x +c \right )^{3} c +480 \sin \left (d x +c \right ) a +480 \sin \left (d x +c \right ) b +480 \sin \left (d x +c \right ) c +300 a d x +225 b d x +195 c d x \right )}{120 d} \] Input:
int((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
(a**3*( - 30*cos(c + d*x)*sin(c + d*x)**3*b - 90*cos(c + d*x)*sin(c + d*x) **3*c + 180*cos(c + d*x)*sin(c + d*x)*a + 255*cos(c + d*x)*sin(c + d*x)*b + 285*cos(c + d*x)*sin(c + d*x)*c + 24*sin(c + d*x)**5*c - 40*sin(c + d*x) **3*a - 120*sin(c + d*x)**3*b - 200*sin(c + d*x)**3*c + 480*sin(c + d*x)*a + 480*sin(c + d*x)*b + 480*sin(c + d*x)*c + 300*a*d*x + 225*b*d*x + 195*c *d*x))/(120*d)