Integrand size = 41, antiderivative size = 265 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} a^3 (21 A+23 B+26 C) x+\frac {a^3 (108 A+119 B+133 C) \sin (c+d x)}{35 d}+\frac {a^3 (21 A+23 B+26 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (129 A+147 B+154 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(3 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 a d}+\frac {(3 A+4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (108 A+119 B+133 C) \sin ^3(c+d x)}{105 d} \] Output:
1/16*a^3*(21*A+23*B+26*C)*x+1/35*a^3*(108*A+119*B+133*C)*sin(d*x+c)/d+1/16 *a^3*(21*A+23*B+26*C)*cos(d*x+c)*sin(d*x+c)/d+1/280*a^3*(129*A+147*B+154*C )*cos(d*x+c)^3*sin(d*x+c)/d+1/7*A*cos(d*x+c)^6*(a+a*sec(d*x+c))^3*sin(d*x+ c)/d+1/42*(3*A+7*B)*cos(d*x+c)^5*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/a/d+1/1 5*(3*A+4*B+3*C)*cos(d*x+c)^4*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d-1/105*a^3*( 108*A+119*B+133*C)*sin(d*x+c)^3/d
Time = 0.80 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.77 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (3360 A c+9660 B c+8820 A d x+9660 B d x+10920 C d x+105 (155 A+168 B+184 C) \sin (c+d x)+105 (61 A+63 B+64 C) \sin (2 (c+d x))+2835 A \sin (3 (c+d x))+2660 B \sin (3 (c+d x))+2380 C \sin (3 (c+d x))+1155 A \sin (4 (c+d x))+945 B \sin (4 (c+d x))+630 C \sin (4 (c+d x))+399 A \sin (5 (c+d x))+252 B \sin (5 (c+d x))+84 C \sin (5 (c+d x))+105 A \sin (6 (c+d x))+35 B \sin (6 (c+d x))+15 A \sin (7 (c+d x)))}{6720 d} \] Input:
Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
(a^3*(3360*A*c + 9660*B*c + 8820*A*d*x + 9660*B*d*x + 10920*C*d*x + 105*(1 55*A + 168*B + 184*C)*Sin[c + d*x] + 105*(61*A + 63*B + 64*C)*Sin[2*(c + d *x)] + 2835*A*Sin[3*(c + d*x)] + 2660*B*Sin[3*(c + d*x)] + 2380*C*Sin[3*(c + d*x)] + 1155*A*Sin[4*(c + d*x)] + 945*B*Sin[4*(c + d*x)] + 630*C*Sin[4* (c + d*x)] + 399*A*Sin[5*(c + d*x)] + 252*B*Sin[5*(c + d*x)] + 84*C*Sin[5* (c + d*x)] + 105*A*Sin[6*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 15*A*Sin[7*( c + d*x)]))/(6720*d)
Time = 1.59 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 4574, 3042, 4505, 3042, 4505, 27, 3042, 4484, 25, 3042, 4274, 3042, 3113, 2009, 3115, 24}
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.
| \(\displaystyle \int \cos ^7(c+d x) (a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int \cos ^6(c+d x) (\sec (c+d x) a+a)^3 (a (3 A+7 B)+a (3 A+7 C) \sec (c+d x))dx}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (3 A+7 B)+a (3 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{6} \int \cos ^5(c+d x) (\sec (c+d x) a+a)^2 \left (14 (3 A+4 B+3 C) a^2+3 (9 A+7 B+14 C) \sec (c+d x) a^2\right )dx+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (14 (3 A+4 B+3 C) a^2+3 (9 A+7 B+14 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{5} \int 3 \cos ^4(c+d x) (\sec (c+d x) a+a) \left ((129 A+147 B+154 C) a^3+(87 A+91 B+112 C) \sec (c+d x) a^3\right )dx+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \int \cos ^4(c+d x) (\sec (c+d x) a+a) \left ((129 A+147 B+154 C) a^3+(87 A+91 B+112 C) \sec (c+d x) a^3\right )dx+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((129 A+147 B+154 C) a^3+(87 A+91 B+112 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {a^4 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) \left (8 (108 A+119 B+133 C) a^4+35 (21 A+23 B+26 C) \sec (c+d x) a^4\right )dx\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) \left (8 (108 A+119 B+133 C) a^4+35 (21 A+23 B+26 C) \sec (c+d x) a^4\right )dx+\frac {a^4 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \frac {8 (108 A+119 B+133 C) a^4+35 (21 A+23 B+26 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^4 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (8 a^4 (108 A+119 B+133 C) \int \cos ^3(c+d x)dx+35 a^4 (21 A+23 B+26 C) \int \cos ^2(c+d x)dx\right )+\frac {a^4 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (21 A+23 B+26 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+8 a^4 (108 A+119 B+133 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^4 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (21 A+23 B+26 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 a^4 (108 A+119 B+133 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a^4 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (21 A+23 B+26 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 a^4 (108 A+119 B+133 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^4 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^4 (21 A+23 B+26 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {8 a^4 (108 A+119 B+133 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^4 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {3}{5} \left (\frac {a^4 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (35 a^4 (21 A+23 B+26 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 a^4 (108 A+119 B+133 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )\right )+\frac {14 (3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{5 d}\right )+\frac {(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{6 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}\) |
Input:
Int[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + (((3*A + 7* B)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((14*(3 *A + 4*B + 3*C)*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(5*d ) + (3*((a^4*(129*A + 147*B + 154*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (35*a^4*(21*A + 23*B + 26*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - ( 8*a^4*(108*A + 119*B + 133*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d)/4))/5 )/6)/(7*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 5.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.55
| method | result | size |
| parallelrisch | \(\frac {27 \left (\frac {\left (\frac {61 A}{9}+7 B +\frac {64 C}{9}\right ) \sin \left (2 d x +2 c \right )}{3}+\left (A +\frac {76 B}{81}+\frac {68 C}{81}\right ) \sin \left (3 d x +3 c \right )+\frac {\left (\frac {11 A}{9}+B +\frac {2 C}{3}\right ) \sin \left (4 d x +4 c \right )}{3}+\frac {\left (\frac {19 A}{3}+4 B +\frac {4 C}{3}\right ) \sin \left (5 d x +5 c \right )}{45}+\frac {\left (A +\frac {B}{3}\right ) \sin \left (6 d x +6 c \right )}{27}+\frac {A \sin \left (7 d x +7 c \right )}{189}+\frac {\left (\frac {155 A}{3}+56 B +\frac {184 C}{3}\right ) \sin \left (d x +c \right )}{9}+\frac {28 x \left (A +\frac {23 B}{21}+\frac {26 C}{21}\right ) d}{9}\right ) a^{3}}{64 d}\) | \(147\) |
| risch | \(\frac {21 a^{3} A x}{16}+\frac {23 a^{3} B x}{16}+\frac {13 a^{3} x C}{8}+\frac {155 a^{3} A \sin \left (d x +c \right )}{64 d}+\frac {21 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {23 a^{3} C \sin \left (d x +c \right )}{8 d}+\frac {a^{3} A \sin \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} A \sin \left (6 d x +6 c \right )}{64 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{3}}{192 d}+\frac {19 a^{3} A \sin \left (5 d x +5 c \right )}{320 d}+\frac {3 \sin \left (5 d x +5 c \right ) B \,a^{3}}{80 d}+\frac {\sin \left (5 d x +5 c \right ) a^{3} C}{80 d}+\frac {11 a^{3} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {9 \sin \left (4 d x +4 c \right ) B \,a^{3}}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {27 a^{3} A \sin \left (3 d x +3 c \right )}{64 d}+\frac {19 \sin \left (3 d x +3 c \right ) B \,a^{3}}{48 d}+\frac {17 \sin \left (3 d x +3 c \right ) a^{3} C}{48 d}+\frac {61 \sin \left (2 d x +2 c \right ) a^{3} A}{64 d}+\frac {63 \sin \left (2 d x +2 c \right ) B \,a^{3}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{d}\) | \(337\) |
| derivativedivides | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {3 a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{3} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{3} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(427\) |
| default | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {3 a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{3} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{3} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(427\) |
Input:
int(cos(d*x+c)^7*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,meth od=_RETURNVERBOSE)
Output:
27/64*(1/3*(61/9*A+7*B+64/9*C)*sin(2*d*x+2*c)+(A+76/81*B+68/81*C)*sin(3*d* x+3*c)+1/3*(11/9*A+B+2/3*C)*sin(4*d*x+4*c)+1/45*(19/3*A+4*B+4/3*C)*sin(5*d *x+5*c)+1/27*(A+1/3*B)*sin(6*d*x+6*c)+1/189*A*sin(7*d*x+7*c)+1/9*(155/3*A+ 56*B+184/3*C)*sin(d*x+c)+28/9*x*(A+23/21*B+26/21*C)*d)*a^3/d
Time = 0.08 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.63 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (21 \, A + 23 \, B + 26 \, C\right )} a^{3} d x + {\left (240 \, A a^{3} \cos \left (d x + c\right )^{6} + 280 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (27 \, A + 21 \, B + 7 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 70 \, {\left (21 \, A + 23 \, B + 18 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (108 \, A + 119 \, B + 133 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 105 \, {\left (21 \, A + 23 \, B + 26 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (108 \, A + 119 \, B + 133 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{1680 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
Output:
1/1680*(105*(21*A + 23*B + 26*C)*a^3*d*x + (240*A*a^3*cos(d*x + c)^6 + 280 *(3*A + B)*a^3*cos(d*x + c)^5 + 48*(27*A + 21*B + 7*C)*a^3*cos(d*x + c)^4 + 70*(21*A + 23*B + 18*C)*a^3*cos(d*x + c)^3 + 16*(108*A + 119*B + 133*C)* a^3*cos(d*x + c)^2 + 105*(21*A + 23*B + 26*C)*a^3*cos(d*x + c) + 32*(108*A + 119*B + 133*C)*a^3)*sin(d*x + c))/d
Timed out. \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**7*(a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)** 2),x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.60 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} A a^{3} - 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} + 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 1344 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} - 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 448 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} + 6720 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 630 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3}}{6720 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
Output:
-1/6720*(192*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 3 5*sin(d*x + c))*A*a^3 - 1344*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*si n(d*x + c))*A*a^3 + 105*(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))*A*a^3 - 210*(12*d*x + 12*c + sin(4*d*x + 4 *c) + 8*sin(2*d*x + 2*c))*A*a^3 - 1344*(3*sin(d*x + c)^5 - 10*sin(d*x + c) ^3 + 15*sin(d*x + c))*B*a^3 + 35*(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*a^3 + 2240*(sin(d*x + c)^3 - 3* sin(d*x + c))*B*a^3 - 630*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3 - 448*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c ))*C*a^3 + 6720*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 - 630*(12*d*x + 12 *c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3 - 1680*(2*d*x + 2*c + si n(2*d*x + 2*c))*C*a^3)/d
Time = 0.30 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.51 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (21 \, A a^{3} + 23 \, B a^{3} + 26 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (2205 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2415 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 2730 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 14700 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 16100 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 18200 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 41601 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 45563 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 51506 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 62592 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72576 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 77952 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63231 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 62853 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 71246 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25620 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33180 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40040 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11235 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11025 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10710 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}}{1680 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
Output:
1/1680*(105*(21*A*a^3 + 23*B*a^3 + 26*C*a^3)*(d*x + c) + 2*(2205*A*a^3*tan (1/2*d*x + 1/2*c)^13 + 2415*B*a^3*tan(1/2*d*x + 1/2*c)^13 + 2730*C*a^3*tan (1/2*d*x + 1/2*c)^13 + 14700*A*a^3*tan(1/2*d*x + 1/2*c)^11 + 16100*B*a^3*t an(1/2*d*x + 1/2*c)^11 + 18200*C*a^3*tan(1/2*d*x + 1/2*c)^11 + 41601*A*a^3 *tan(1/2*d*x + 1/2*c)^9 + 45563*B*a^3*tan(1/2*d*x + 1/2*c)^9 + 51506*C*a^3 *tan(1/2*d*x + 1/2*c)^9 + 62592*A*a^3*tan(1/2*d*x + 1/2*c)^7 + 72576*B*a^3 *tan(1/2*d*x + 1/2*c)^7 + 77952*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 63231*A*a^3 *tan(1/2*d*x + 1/2*c)^5 + 62853*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 71246*C*a^3 *tan(1/2*d*x + 1/2*c)^5 + 25620*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 33180*B*a^3 *tan(1/2*d*x + 1/2*c)^3 + 40040*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 11235*A*a^3 *tan(1/2*d*x + 1/2*c) + 11025*B*a^3*tan(1/2*d*x + 1/2*c) + 10710*C*a^3*tan (1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^7)/d
Time = 16.89 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.42 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {21\,A\,a^3}{8}+\frac {23\,B\,a^3}{8}+\frac {13\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {35\,A\,a^3}{2}+\frac {115\,B\,a^3}{6}+\frac {65\,C\,a^3}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^3}{40}+\frac {6509\,B\,a^3}{120}+\frac {3679\,C\,a^3}{60}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {2608\,A\,a^3}{35}+\frac {432\,B\,a^3}{5}+\frac {464\,C\,a^3}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3011\,A\,a^3}{40}+\frac {2993\,B\,a^3}{40}+\frac {5089\,C\,a^3}{60}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {61\,A\,a^3}{2}+\frac {79\,B\,a^3}{2}+\frac {143\,C\,a^3}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {107\,A\,a^3}{8}+\frac {105\,B\,a^3}{8}+\frac {51\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (21\,A+23\,B+26\,C\right )}{8\,\left (\frac {21\,A\,a^3}{8}+\frac {23\,B\,a^3}{8}+\frac {13\,C\,a^3}{4}\right )}\right )\,\left (21\,A+23\,B+26\,C\right )}{8\,d} \] Input:
int(cos(c + d*x)^7*(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)^13*((21*A*a^3)/8 + (23*B*a^3)/8 + (13*C*a^3)/4) + tan( c/2 + (d*x)/2)^11*((35*A*a^3)/2 + (115*B*a^3)/6 + (65*C*a^3)/3) + tan(c/2 + (d*x)/2)^3*((61*A*a^3)/2 + (79*B*a^3)/2 + (143*C*a^3)/3) + tan(c/2 + (d* x)/2)^7*((2608*A*a^3)/35 + (432*B*a^3)/5 + (464*C*a^3)/5) + tan(c/2 + (d*x )/2)^5*((3011*A*a^3)/40 + (2993*B*a^3)/40 + (5089*C*a^3)/60) + tan(c/2 + ( d*x)/2)^9*((1981*A*a^3)/40 + (6509*B*a^3)/120 + (3679*C*a^3)/60) + tan(c/2 + (d*x)/2)*((107*A*a^3)/8 + (105*B*a^3)/8 + (51*C*a^3)/4))/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan( c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^12 + ta n(c/2 + (d*x)/2)^14 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(21*A + 23*B + 26*C))/(8*((21*A*a^3)/8 + (23*B*a^3)/8 + (13*C*a^3)/4)))*(21*A + 23*B + 26*C))/(8*d)
Time = 0.18 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.97 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{3} \left (840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a +280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b -3150 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -2170 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b -1260 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +4515 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +4305 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +3990 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c -240 \sin \left (d x +c \right )^{7} a +2016 \sin \left (d x +c \right )^{5} a +1008 \sin \left (d x +c \right )^{5} b +336 \sin \left (d x +c \right )^{5} c -5040 \sin \left (d x +c \right )^{3} a -3920 \sin \left (d x +c \right )^{3} b -2800 \sin \left (d x +c \right )^{3} c +6720 \sin \left (d x +c \right ) a +6720 \sin \left (d x +c \right ) b +6720 \sin \left (d x +c \right ) c +2205 a d x +2415 b d x +2730 c d x \right )}{1680 d} \] Input:
int(cos(d*x+c)^7*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
(a**3*(840*cos(c + d*x)*sin(c + d*x)**5*a + 280*cos(c + d*x)*sin(c + d*x)* *5*b - 3150*cos(c + d*x)*sin(c + d*x)**3*a - 2170*cos(c + d*x)*sin(c + d*x )**3*b - 1260*cos(c + d*x)*sin(c + d*x)**3*c + 4515*cos(c + d*x)*sin(c + d *x)*a + 4305*cos(c + d*x)*sin(c + d*x)*b + 3990*cos(c + d*x)*sin(c + d*x)* c - 240*sin(c + d*x)**7*a + 2016*sin(c + d*x)**5*a + 1008*sin(c + d*x)**5* b + 336*sin(c + d*x)**5*c - 5040*sin(c + d*x)**3*a - 3920*sin(c + d*x)**3* b - 2800*sin(c + d*x)**3*c + 6720*sin(c + d*x)*a + 6720*sin(c + d*x)*b + 6 720*sin(c + d*x)*c + 2205*a*d*x + 2415*b*d*x + 2730*c*d*x))/(1680*d)