Integrand size = 33, antiderivative size = 216 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} a^3 (23 A+30 C) x+\frac {a^3 (34 A+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (23 A+30 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (73 A+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac {(31 A+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d} \]
1/16*a^3*(23*A+30*C)*x+1/15*a^3*(34*A+45*C)*sin(d*x+c)/d+1/16*a^3*(23*A+30 *C)*cos(d*x+c)*sin(d*x+c)/d+1/120*a^3*(73*A+90*C)*cos(d*x+c)^2*sin(d*x+c)/ d+1/6*A*cos(d*x+c)^5*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+1/10*A*cos(d*x+c)^4*( a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/a/d+1/120*(31*A+30*C)*cos(d*x+c)^3*(a^3+a ^3*sec(d*x+c))*sin(d*x+c)/d
Time = 0.69 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.57 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (900 A c+1380 A d x+1800 C d x+120 (21 A+26 C) \sin (c+d x)+15 (63 A+64 C) \sin (2 (c+d x))+380 A \sin (3 (c+d x))+240 C \sin (3 (c+d x))+135 A \sin (4 (c+d x))+30 C \sin (4 (c+d x))+36 A \sin (5 (c+d x))+5 A \sin (6 (c+d x)))}{960 d} \]
(a^3*(900*A*c + 1380*A*d*x + 1800*C*d*x + 120*(21*A + 26*C)*Sin[c + d*x] + 15*(63*A + 64*C)*Sin[2*(c + d*x)] + 380*A*Sin[3*(c + d*x)] + 240*C*Sin[3* (c + d*x)] + 135*A*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 36*A*Sin[5*( c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)
Time = 1.37 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {3042, 4575, 3042, 4505, 3042, 4505, 27, 3042, 4484, 25, 3042, 4274, 3042, 3115, 24, 3117}
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.
| \(\displaystyle \int \cos ^6(c+d x) (a \sec (c+d x)+a)^3 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4575 |
| \(\displaystyle \frac {\int \cos ^5(c+d x) (\sec (c+d x) a+a)^3 (3 a A+2 a (A+3 C) \sec (c+d x))dx}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a A+2 a (A+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{5} \int \cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left ((31 A+30 C) a^2+2 (8 A+15 C) \sec (c+d x) a^2\right )dx+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((31 A+30 C) a^2+2 (8 A+15 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int 3 \cos ^3(c+d x) (\sec (c+d x) a+a) \left ((73 A+90 C) a^3+6 (7 A+10 C) \sec (c+d x) a^3\right )dx+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \cos ^3(c+d x) (\sec (c+d x) a+a) \left ((73 A+90 C) a^3+6 (7 A+10 C) \sec (c+d x) a^3\right )dx+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((73 A+90 C) a^3+6 (7 A+10 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {a^4 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (15 (23 A+30 C) a^4+8 (34 A+45 C) \sec (c+d x) a^4\right )dx\right )+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) \left (15 (23 A+30 C) a^4+8 (34 A+45 C) \sec (c+d x) a^4\right )dx+\frac {a^4 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \frac {15 (23 A+30 C) a^4+8 (34 A+45 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^4 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (15 a^4 (23 A+30 C) \int \cos ^2(c+d x)dx+8 a^4 (34 A+45 C) \int \cos (c+d x)dx\right )+\frac {a^4 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (8 a^4 (34 A+45 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^4 (23 A+30 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a^4 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (8 a^4 (34 A+45 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^4 (23 A+30 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a^4 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (8 a^4 (34 A+45 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^4 (23 A+30 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a^4 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {a^4 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{3} \left (\frac {8 a^4 (34 A+45 C) \sin (c+d x)}{d}+15 a^4 (23 A+30 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{4 d}\right )+\frac {3 A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}\) |
(A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*A*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (((31*A + 30*C) *Cos[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(4*d) + (3*((a^4*(7 3*A + 90*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + ((8*a^4*(34*A + 45*C)*Sin [c + d*x])/d + 15*a^4*(23*A + 30*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2* d)))/3))/4)/5)/(6*a)
3.2.10.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_. ))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Co t[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 *(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.49
| method | result | size |
| parallelrisch | \(\frac {9 a^{3} \left (\left (7 A +\frac {64 C}{9}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {76 A}{27}+\frac {16 C}{9}\right ) \sin \left (3 d x +3 c \right )+\left (A +\frac {2 C}{9}\right ) \sin \left (4 d x +4 c \right )+\frac {4 A \sin \left (5 d x +5 c \right )}{15}+\frac {A \sin \left (6 d x +6 c \right )}{27}+\left (\frac {56 A}{3}+\frac {208 C}{9}\right ) \sin \left (d x +c \right )+\frac {92 \left (A +\frac {30 C}{23}\right ) x d}{9}\right )}{64 d}\) | \(106\) |
| risch | \(\frac {23 a^{3} A x}{16}+\frac {15 a^{3} x C}{8}+\frac {21 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {13 a^{3} C \sin \left (d x +c \right )}{4 d}+\frac {a^{3} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 a^{3} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {9 a^{3} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {19 a^{3} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} C}{4 d}+\frac {63 \sin \left (2 d x +2 c \right ) a^{3} A}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{d}\) | \(189\) |
| derivativedivides | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \sin \left (d x +c \right )}{d}\) | \(245\) |
| default | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \sin \left (d x +c \right )}{d}\) | \(245\) |
9/64*a^3*((7*A+64/9*C)*sin(2*d*x+2*c)+(76/27*A+16/9*C)*sin(3*d*x+3*c)+(A+2 /9*C)*sin(4*d*x+4*c)+4/15*A*sin(5*d*x+5*c)+1/27*A*sin(6*d*x+6*c)+(56/3*A+2 08/9*C)*sin(d*x+c)+92/9*(A+30/23*C)*x*d)/d
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.58 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (23 \, A + 30 \, C\right )} a^{3} d x + {\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 144 \, A a^{3} \cos \left (d x + c\right )^{4} + 10 \, {\left (23 \, A + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (17 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (23 \, A + 30 \, C\right )} a^{3} \cos \left (d x + c\right ) + 16 \, {\left (34 \, A + 45 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{240 \, d} \]
1/240*(15*(23*A + 30*C)*a^3*d*x + (40*A*a^3*cos(d*x + c)^5 + 144*A*a^3*cos (d*x + c)^4 + 10*(23*A + 6*C)*a^3*cos(d*x + c)^3 + 16*(17*A + 15*C)*a^3*co s(d*x + c)^2 + 15*(23*A + 30*C)*a^3*cos(d*x + c) + 16*(34*A + 45*C)*a^3)*s in(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.11 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 960 \, C a^{3} \sin \left (d x + c\right )}{960 \, d} \]
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 - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 + 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*si n(2*d*x + 2*c))*C*a^3 + 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^3 + 960*C *a^3*sin(d*x + c))/d
Time = 0.33 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.13 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (23 \, A a^{3} + 30 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (345 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 450 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2550 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5940 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5814 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7500 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3165 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5130 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1470 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
1/240*(15*(23*A*a^3 + 30*C*a^3)*(d*x + c) + 2*(345*A*a^3*tan(1/2*d*x + 1/2 *c)^11 + 450*C*a^3*tan(1/2*d*x + 1/2*c)^11 + 1955*A*a^3*tan(1/2*d*x + 1/2* c)^9 + 2550*C*a^3*tan(1/2*d*x + 1/2*c)^9 + 4554*A*a^3*tan(1/2*d*x + 1/2*c) ^7 + 5940*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 5814*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 7500*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 3165*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 5130*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 1575*A*a^3*tan(1/2*d*x + 1/2*c) + 147 0*C*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d
Time = 18.33 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.32 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {23\,A\,a^3}{8}+\frac {15\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {391\,A\,a^3}{24}+\frac {85\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {759\,A\,a^3}{20}+\frac {99\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {969\,A\,a^3}{20}+\frac {125\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {211\,A\,a^3}{8}+\frac {171\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {105\,A\,a^3}{8}+\frac {49\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,A+30\,C\right )}{8\,\left (\frac {23\,A\,a^3}{8}+\frac {15\,C\,a^3}{4}\right )}\right )\,\left (23\,A+30\,C\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)*((105*A*a^3)/8 + (49*C*a^3)/4) + tan(c/2 + (d*x)/2)^11 *((23*A*a^3)/8 + (15*C*a^3)/4) + tan(c/2 + (d*x)/2)^3*((211*A*a^3)/8 + (17 1*C*a^3)/4) + tan(c/2 + (d*x)/2)^9*((391*A*a^3)/24 + (85*C*a^3)/4) + tan(c /2 + (d*x)/2)^7*((759*A*a^3)/20 + (99*C*a^3)/2) + tan(c/2 + (d*x)/2)^5*((9 69*A*a^3)/20 + (125*C*a^3)/2))/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + ( d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (a^3*atan((a^3*tan(c/2 + (d* x)/2)*(23*A + 30*C))/(8*((23*A*a^3)/8 + (15*C*a^3)/4)))*(23*A + 30*C))/(8* d)