Integrand size = 31, antiderivative size = 201 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {1}{16} a^3 (23 A+26 B) x+\frac {a^3 (17 A+19 B) \sin (c+d x)}{5 d}+\frac {a^3 (23 A+26 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (21 A+22 B) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A+3 B) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (17 A+19 B) \sin ^3(c+d x)}{15 d} \] Output:
1/16*a^3*(23*A+26*B)*x+1/5*a^3*(17*A+19*B)*sin(d*x+c)/d+1/16*a^3*(23*A+26* B)*cos(d*x+c)*sin(d*x+c)/d+1/40*a^3*(21*A+22*B)*cos(d*x+c)^3*sin(d*x+c)/d+ 1/6*a*A*cos(d*x+c)^5*(a+a*sec(d*x+c))^2*sin(d*x+c)/d+1/15*(4*A+3*B)*cos(d* x+c)^4*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d-1/15*a^3*(17*A+19*B)*sin(d*x+c)^3 /d
Time = 0.36 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.67 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (1380 A c+1380 A d x+1560 B d x+120 (21 A+23 B) \sin (c+d x)+15 (63 A+64 B) \sin (2 (c+d x))+380 A \sin (3 (c+d x))+340 B \sin (3 (c+d x))+135 A \sin (4 (c+d x))+90 B \sin (4 (c+d x))+36 A \sin (5 (c+d x))+12 B \sin (5 (c+d x))+5 A \sin (6 (c+d x)))}{960 d} \] Input:
Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]
Output:
(a^3*(1380*A*c + 1380*A*d*x + 1560*B*d*x + 120*(21*A + 23*B)*Sin[c + d*x] + 15*(63*A + 64*B)*Sin[2*(c + d*x)] + 380*A*Sin[3*(c + d*x)] + 340*B*Sin[3 *(c + d*x)] + 135*A*Sin[4*(c + d*x)] + 90*B*Sin[4*(c + d*x)] + 36*A*Sin[5* (c + d*x)] + 12*B*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)
Time = 1.11 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.98, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {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.
| \(\displaystyle \int \cos ^6(c+d x) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {1}{6} \int \cos ^5(c+d x) (\sec (c+d x) a+a)^2 (2 a (4 A+3 B)+3 a (A+2 B) \sec (c+d x))dx+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (2 a (4 A+3 B)+3 a (A+2 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int 3 \cos ^4(c+d x) (\sec (c+d x) a+a) \left ((21 A+22 B) a^2+(13 A+16 B) \sec (c+d x) a^2\right )dx+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \int \cos ^4(c+d x) (\sec (c+d x) a+a) \left ((21 A+22 B) a^2+(13 A+16 B) \sec (c+d x) a^2\right )dx+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((21 A+22 B) a^2+(13 A+16 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) \left (8 (17 A+19 B) a^3+5 (23 A+26 B) \sec (c+d x) a^3\right )dx\right )+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) \left (8 (17 A+19 B) a^3+5 (23 A+26 B) \sec (c+d x) a^3\right )dx+\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \frac {8 (17 A+19 B) a^3+5 (23 A+26 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (8 a^3 (17 A+19 B) \int \cos ^3(c+d x)dx+5 a^3 (23 A+26 B) \int \cos ^2(c+d x)dx\right )+\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 a^3 (23 A+26 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+8 a^3 (17 A+19 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 a^3 (23 A+26 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 a^3 (17 A+19 B) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 a^3 (23 A+26 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {8 a^3 (17 A+19 B) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (5 a^3 (23 A+26 B) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {8 a^3 (17 A+19 B) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {3}{5} \left (\frac {a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (5 a^3 (23 A+26 B) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 a^3 (17 A+19 B) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )\right )+\frac {2 (4 A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\) |
Input:
Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]
Output:
(a*A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((2*(4*A + 3*B)*Cos[c + d*x]^4*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(5*d) + (3*(( a^3*(21*A + 22*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (5*a^3*(23*A + 26*B )*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (8*a^3*(17*A + 19*B)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d)/4))/5)/6
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]
Time = 1.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.56
| method | result | size |
| parallelrisch | \(\frac {9 \left (\left (7 A +\frac {64 B}{9}\right ) \sin \left (2 d x +2 c \right )+\frac {4 \left (19 A +17 B \right ) \sin \left (3 d x +3 c \right )}{27}+\left (A +\frac {2 B}{3}\right ) \sin \left (4 d x +4 c \right )+\frac {4 \left (A +\frac {B}{3}\right ) \sin \left (5 d x +5 c \right )}{15}+\frac {A \sin \left (6 d x +6 c \right )}{27}+\frac {8 \left (7 A +\frac {23 B}{3}\right ) \sin \left (d x +c \right )}{3}+\frac {92 d x \left (A +\frac {26 B}{23}\right )}{9}\right ) a^{3}}{64 d}\) | \(112\) |
| risch | \(\frac {23 a^{3} A x}{16}+\frac {13 a^{3} x B}{8}+\frac {21 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {23 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {a^{3} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 a^{3} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{3}}{80 d}+\frac {9 a^{3} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) B \,a^{3}}{32 d}+\frac {19 a^{3} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {17 \sin \left (3 d x +3 c \right ) B \,a^{3}}{48 d}+\frac {63 \sin \left (2 d x +2 c \right ) a^{3} A}{64 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}\) | \(207\) |
| derivativedivides | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {3 a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(266\) |
| default | \(\frac {\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {3 a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(266\) |
Input:
int(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),x,method=_RETURNVERBO SE)
Output:
9/64*((7*A+64/9*B)*sin(2*d*x+2*c)+4/27*(19*A+17*B)*sin(3*d*x+3*c)+(A+2/3*B )*sin(4*d*x+4*c)+4/15*(A+1/3*B)*sin(5*d*x+5*c)+1/27*A*sin(6*d*x+6*c)+8/3*( 7*A+23/3*B)*sin(d*x+c)+92/9*d*x*(A+26/23*B))*a^3/d
Time = 0.09 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (23 \, A + 26 \, B\right )} a^{3} d x + {\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{4} + 10 \, {\left (23 \, A + 18 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (17 \, A + 19 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (23 \, A + 26 \, B\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (17 \, A + 19 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorithm="f ricas")
Output:
1/240*(15*(23*A + 26*B)*a^3*d*x + (40*A*a^3*cos(d*x + c)^5 + 48*(3*A + B)* a^3*cos(d*x + c)^4 + 10*(23*A + 18*B)*a^3*cos(d*x + c)^3 + 16*(17*A + 19*B )*a^3*cos(d*x + c)^2 + 15*(23*A + 26*B)*a^3*cos(d*x + c) + 32*(17*A + 19*B )*a^3)*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6*(a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.30 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3}}{960 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorithm="m axima")
Output:
1/960*(192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^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))*A*a^3 - 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 + 90*(12*d *x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 + 64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^3 - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 + 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d* x + 2*c))*B*a^3 + 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^3)/d
Time = 0.18 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.21 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (23 \, A a^{3} + 26 \, B a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (345 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 390 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2210 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5148 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5814 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5988 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3165 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4190 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1530 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),x, algorithm="g iac")
Output:
1/240*(15*(23*A*a^3 + 26*B*a^3)*(d*x + c) + 2*(345*A*a^3*tan(1/2*d*x + 1/2 *c)^11 + 390*B*a^3*tan(1/2*d*x + 1/2*c)^11 + 1955*A*a^3*tan(1/2*d*x + 1/2* c)^9 + 2210*B*a^3*tan(1/2*d*x + 1/2*c)^9 + 4554*A*a^3*tan(1/2*d*x + 1/2*c) ^7 + 5148*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 5814*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 5988*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 3165*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 4190*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 1575*A*a^3*tan(1/2*d*x + 1/2*c) + 153 0*B*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d
Time = 13.75 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.42 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\left (\frac {23\,A\,a^3}{8}+\frac {13\,B\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {391\,A\,a^3}{24}+\frac {221\,B\,a^3}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {759\,A\,a^3}{20}+\frac {429\,B\,a^3}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {969\,A\,a^3}{20}+\frac {499\,B\,a^3}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {211\,A\,a^3}{8}+\frac {419\,B\,a^3}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {105\,A\,a^3}{8}+\frac {51\,B\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,A+26\,B\right )}{8\,\left (\frac {23\,A\,a^3}{8}+\frac {13\,B\,a^3}{4}\right )}\right )\,\left (23\,A+26\,B\right )}{8\,d} \] Input:
int(cos(c + d*x)^6*(A + B/cos(c + d*x))*(a + a/cos(c + d*x))^3,x)
Output:
(tan(c/2 + (d*x)/2)*((105*A*a^3)/8 + (51*B*a^3)/4) + tan(c/2 + (d*x)/2)^11 *((23*A*a^3)/8 + (13*B*a^3)/4) + tan(c/2 + (d*x)/2)^3*((211*A*a^3)/8 + (41 9*B*a^3)/12) + tan(c/2 + (d*x)/2)^9*((391*A*a^3)/24 + (221*B*a^3)/12) + ta n(c/2 + (d*x)/2)^7*((759*A*a^3)/20 + (429*B*a^3)/10) + tan(c/2 + (d*x)/2)^ 5*((969*A*a^3)/20 + (499*B*a^3)/10))/(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*ta n(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(23*A + 26*B))/(8*((23*A*a^3)/8 + (13*B*a^3)/4)))*(23*A + 26*B ))/(8*d)
Time = 0.17 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.81 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^{3} \left (40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a -310 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -180 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +615 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +570 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +144 \sin \left (d x +c \right )^{5} a +48 \sin \left (d x +c \right )^{5} b -560 \sin \left (d x +c \right )^{3} a -400 \sin \left (d x +c \right )^{3} b +960 \sin \left (d x +c \right ) a +960 \sin \left (d x +c \right ) b +345 a d x +390 b d x \right )}{240 d} \] Input:
int(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)),x)
Output:
(a**3*(40*cos(c + d*x)*sin(c + d*x)**5*a - 310*cos(c + d*x)*sin(c + d*x)** 3*a - 180*cos(c + d*x)*sin(c + d*x)**3*b + 615*cos(c + d*x)*sin(c + d*x)*a + 570*cos(c + d*x)*sin(c + d*x)*b + 144*sin(c + d*x)**5*a + 48*sin(c + d* x)**5*b - 560*sin(c + d*x)**3*a - 400*sin(c + d*x)**3*b + 960*sin(c + d*x) *a + 960*sin(c + d*x)*b + 345*a*d*x + 390*b*d*x))/(240*d)