Integrand size = 41, antiderivative size = 187 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {8 a^2 (63 A+57 B+47 C) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (63 A+57 B+47 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 d}+\frac {2 (63 A-18 B+22 C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{315 d}+\frac {2 C \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {2 (3 B+C) (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{21 a d} \] Output:
8/315*a^2*(63*A+57*B+47*C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/315*a*(63 *A+57*B+47*C)*(a+a*cos(d*x+c))^(1/2)*sin(d*x+c)/d+2/315*(63*A-18*B+22*C)*( a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d+2/9*C*cos(d*x+c)^2*(a+a*cos(d*x+c))^(3/ 2)*sin(d*x+c)/d+2/21*(3*B+C)*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/a/d
Time = 0.68 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.60 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a \sqrt {a (1+\cos (c+d x))} (3276 A+2964 B+2689 C+2 (756 A+759 B+799 C) \cos (c+d x)+4 (63 A+117 B+137 C) \cos (2 (c+d x))+90 B \cos (3 (c+d x))+170 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))) \tan \left (\frac {1}{2} (c+d x)\right )}{1260 d} \] Input:
Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C* Cos[c + d*x]^2),x]
Output:
(a*Sqrt[a*(1 + Cos[c + d*x])]*(3276*A + 2964*B + 2689*C + 2*(756*A + 759*B + 799*C)*Cos[c + d*x] + 4*(63*A + 117*B + 137*C)*Cos[2*(c + d*x)] + 90*B* Cos[3*(c + d*x)] + 170*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)
Time = 1.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.341, Rules used = {3042, 3524, 27, 3042, 3447, 3042, 3502, 27, 3042, 3230, 3042, 3126, 3042, 3125}
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.
| \(\displaystyle \int \cos (c+d x) (a \cos (c+d x)+a)^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos (c+d x) (\cos (c+d x) a+a)^{3/2} (a (9 A+4 C)+3 a (3 B+C) \cos (c+d x))dx}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos (c+d x) (\cos (c+d x) a+a)^{3/2} (a (9 A+4 C)+3 a (3 B+C) \cos (c+d x))dx}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (a (9 A+4 C)+3 a (3 B+C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^{3/2} \left (3 a (3 B+C) \cos ^2(c+d x)+a (9 A+4 C) \cos (c+d x)\right )dx}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (3 a (3 B+C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (9 A+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} (\cos (c+d x) a+a)^{3/2} \left (15 (3 B+C) a^2+(63 A-18 B+22 C) \cos (c+d x) a^2\right )dx}{7 a}+\frac {6 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int (\cos (c+d x) a+a)^{3/2} \left (15 (3 B+C) a^2+(63 A-18 B+22 C) \cos (c+d x) a^2\right )dx}{7 a}+\frac {6 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left (15 (3 B+C) a^2+(63 A-18 B+22 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx}{7 a}+\frac {6 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {\frac {3}{5} a^2 (63 A+57 B+47 C) \int (\cos (c+d x) a+a)^{3/2}dx+\frac {2 a^2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}}{7 a}+\frac {6 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} a^2 (63 A+57 B+47 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}dx+\frac {2 a^2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}}{7 a}+\frac {6 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3126 |
| \(\displaystyle \frac {\frac {\frac {3}{5} a^2 (63 A+57 B+47 C) \left (\frac {4}{3} a \int \sqrt {\cos (c+d x) a+a}dx+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a^2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}}{7 a}+\frac {6 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3}{5} a^2 (63 A+57 B+47 C) \left (\frac {4}{3} a \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )+\frac {2 a^2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}}{7 a}+\frac {6 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
| \(\Big \downarrow \) 3125 |
| \(\displaystyle \frac {\frac {\frac {2 a^2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac {3}{5} a^2 (63 A+57 B+47 C) \left (\frac {8 a^2 \sin (c+d x)}{3 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{3 d}\right )}{7 a}+\frac {6 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}\) |
Input:
Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(2*C*Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(9*d) + ((6*( 3*B + C)*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + ((2*a^2*(63*A - 18*B + 22*C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (3*a^2*(63*A + 57*B + 47*C)*((8*a^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Cos[c + d*x]]) + (2* a*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5)/(7*a))/(9*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos [c + d*x]/(d*Sqrt[a + b*Sin[c + d*x]])), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[a*((2*n - 1)/n) Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ a^2 - b^2, 0] && IGtQ[n - 1/2, 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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
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]
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] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 0.64 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.71
| method | result | size |
| default | \(\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (280 C \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-180 B -900 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (126 A +504 B +1134 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-315 A -525 B -735 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+315 A +315 B +315 C \right ) \sqrt {2}}{315 \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(132\) |
| parts | \(\frac {4 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2\right ) \sqrt {2}}{5 \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}+\frac {4 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (60 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-12 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+19 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+38\right ) \sqrt {2}}{105 \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}+\frac {4 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (280 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-220 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+114 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+47 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+94\right ) \sqrt {2}}{315 \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(258\) |
Input:
int(cos(d*x+c)*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,me thod=_RETURNVERBOSE)
Output:
4/315*cos(1/2*d*x+1/2*c)*a^2*sin(1/2*d*x+1/2*c)*(280*C*sin(1/2*d*x+1/2*c)^ 8+(-180*B-900*C)*sin(1/2*d*x+1/2*c)^6+(126*A+504*B+1134*C)*sin(1/2*d*x+1/2 *c)^4+(-315*A-525*B-735*C)*sin(1/2*d*x+1/2*c)^2+315*A+315*B+315*C)*2^(1/2) /(a*cos(1/2*d*x+1/2*c)^2)^(1/2)/d
Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.61 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (35 \, C a \cos \left (d x + c\right )^{4} + 5 \, {\left (9 \, B + 17 \, C\right )} a \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 39 \, B + 34 \, C\right )} a \cos \left (d x + c\right )^{2} + {\left (189 \, A + 156 \, B + 136 \, C\right )} a \cos \left (d x + c\right ) + 2 \, {\left (189 \, A + 156 \, B + 136 \, C\right )} a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x, algorithm="fricas")
Output:
2/315*(35*C*a*cos(d*x + c)^4 + 5*(9*B + 17*C)*a*cos(d*x + c)^3 + 3*(21*A + 39*B + 34*C)*a*cos(d*x + c)^2 + (189*A + 156*B + 136*C)*a*cos(d*x + c) + 2*(189*A + 156*B + 136*C)*a)*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/(d*cos( d*x + c) + d)
Timed out. \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)* *2),x)
Output:
Timed out
Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.10 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {252 \, {\left (\sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 20 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} A \sqrt {a} + 6 \, {\left (15 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 63 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 175 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 735 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} B \sqrt {a} + {\left (35 \, \sqrt {2} a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, \sqrt {2} a \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 378 \, \sqrt {2} a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 1050 \, \sqrt {2} a \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3780 \, \sqrt {2} a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} C \sqrt {a}}{2520 \, d} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x, algorithm="maxima")
Output:
1/2520*(252*(sqrt(2)*a*sin(5/2*d*x + 5/2*c) + 5*sqrt(2)*a*sin(3/2*d*x + 3/ 2*c) + 20*sqrt(2)*a*sin(1/2*d*x + 1/2*c))*A*sqrt(a) + 6*(15*sqrt(2)*a*sin( 7/2*d*x + 7/2*c) + 63*sqrt(2)*a*sin(5/2*d*x + 5/2*c) + 175*sqrt(2)*a*sin(3 /2*d*x + 3/2*c) + 735*sqrt(2)*a*sin(1/2*d*x + 1/2*c))*B*sqrt(a) + (35*sqrt (2)*a*sin(9/2*d*x + 9/2*c) + 135*sqrt(2)*a*sin(7/2*d*x + 7/2*c) + 378*sqrt (2)*a*sin(5/2*d*x + 5/2*c) + 1050*sqrt(2)*a*sin(3/2*d*x + 3/2*c) + 3780*sq rt(2)*a*sin(1/2*d*x + 1/2*c))*C*sqrt(a))/d
Time = 0.93 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.27 \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (35 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 45 \, {\left (2 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 126 \, {\left (2 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 3 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 210 \, {\left (6 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 5 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 630 \, {\left (8 \, A a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 7 \, B a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 6 \, C a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{2520 \, d} \] Input:
integrate(cos(d*x+c)*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x, algorithm="giac")
Output:
1/2520*sqrt(2)*(35*C*a*sgn(cos(1/2*d*x + 1/2*c))*sin(9/2*d*x + 9/2*c) + 45 *(2*B*a*sgn(cos(1/2*d*x + 1/2*c)) + 3*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(7 /2*d*x + 7/2*c) + 126*(2*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 3*B*a*sgn(cos(1/2 *d*x + 1/2*c)) + 3*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(5/2*d*x + 5/2*c) + 2 10*(6*A*a*sgn(cos(1/2*d*x + 1/2*c)) + 5*B*a*sgn(cos(1/2*d*x + 1/2*c)) + 5* C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(3/2*d*x + 3/2*c) + 630*(8*A*a*sgn(cos(1 /2*d*x + 1/2*c)) + 7*B*a*sgn(cos(1/2*d*x + 1/2*c)) + 6*C*a*sgn(cos(1/2*d*x + 1/2*c)))*sin(1/2*d*x + 1/2*c))*sqrt(a)/d
Timed out. \[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int(cos(c + d*x)*(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
int(cos(c + d*x)*(a + a*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)
\[ \int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\sqrt {a}\, a \left (\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{3}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{2}d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{2}d x \right ) b \right ) \] Input:
int(cos(d*x+c)*(a+a*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
sqrt(a)*a*(int(sqrt(cos(c + d*x) + 1)*cos(c + d*x),x)*a + int(sqrt(cos(c + d*x) + 1)*cos(c + d*x)**4,x)*c + int(sqrt(cos(c + d*x) + 1)*cos(c + d*x)* *3,x)*b + int(sqrt(cos(c + d*x) + 1)*cos(c + d*x)**3,x)*c + int(sqrt(cos(c + d*x) + 1)*cos(c + d*x)**2,x)*a + int(sqrt(cos(c + d*x) + 1)*cos(c + d*x )**2,x)*b)