Integrand size = 33, antiderivative size = 257 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) x+\frac {b \left (9 a^2 (4 A+5 C)+b^2 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {A b \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {b \left (A b^2+3 a^2 (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d} \]
1/16*a*(6*b^2*(3*A+4*C)+a^2*(5*A+6*C))*x+1/15*b*(9*a^2*(4*A+5*C)+b^2*(11*A +15*C))*sin(d*x+c)/d+1/16*a*(6*b^2*(3*A+4*C)+a^2*(5*A+6*C))*cos(d*x+c)*sin (d*x+c)/d+1/120*a*(6*A*b^2+5*a^2*(5*A+6*C))*cos(d*x+c)^3*sin(d*x+c)/d+1/10 *A*b*cos(d*x+c)^4*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/6*A*cos(d*x+c)^5*(a+b* sec(d*x+c))^3*sin(d*x+c)/d-1/15*b*(A*b^2+3*a^2*(4*A+5*C))*sin(d*x+c)^3/d
Time = 1.79 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.98 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {300 a^3 A c+1080 a A b^2 c+360 a^3 c C+1440 a b^2 c C+300 a^3 A d x+1080 a A b^2 d x+360 a^3 C d x+1440 a b^2 C d x+120 b \left (2 b^2 (3 A+4 C)+3 a^2 (5 A+6 C)\right ) \sin (c+d x)+15 a \left (48 b^2 (A+C)+a^2 (15 A+16 C)\right ) \sin (2 (c+d x))+300 a^2 A b \sin (3 (c+d x))+80 A b^3 \sin (3 (c+d x))+240 a^2 b C \sin (3 (c+d x))+45 a^3 A \sin (4 (c+d x))+90 a A b^2 \sin (4 (c+d x))+30 a^3 C \sin (4 (c+d x))+36 a^2 A b \sin (5 (c+d x))+5 a^3 A \sin (6 (c+d x))}{960 d} \]
(300*a^3*A*c + 1080*a*A*b^2*c + 360*a^3*c*C + 1440*a*b^2*c*C + 300*a^3*A*d *x + 1080*a*A*b^2*d*x + 360*a^3*C*d*x + 1440*a*b^2*C*d*x + 120*b*(2*b^2*(3 *A + 4*C) + 3*a^2*(5*A + 6*C))*Sin[c + d*x] + 15*a*(48*b^2*(A + C) + a^2*( 15*A + 16*C))*Sin[2*(c + d*x)] + 300*a^2*A*b*Sin[3*(c + d*x)] + 80*A*b^3*S in[3*(c + d*x)] + 240*a^2*b*C*Sin[3*(c + d*x)] + 45*a^3*A*Sin[4*(c + d*x)] + 90*a*A*b^2*Sin[4*(c + d*x)] + 30*a^3*C*Sin[4*(c + d*x)] + 36*a^2*A*b*Si n[5*(c + d*x)] + 5*a^3*A*Sin[6*(c + d*x)])/(960*d)
Time = 1.62 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.96, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.485, Rules used = {3042, 4583, 3042, 4582, 3042, 4562, 25, 3042, 4535, 3042, 3115, 24, 4532, 3042, 3492, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4583 |
| \(\displaystyle \frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (2 b (A+3 C) \sec ^2(c+d x)+a (5 A+6 C) \sec (c+d x)+3 A b\right )dx+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (2 b (A+3 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a (5 A+6 C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 A b\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (5 (5 A+6 C) a^2+b (47 A+60 C) \sec (c+d x) a+6 A b^2+2 b^2 (8 A+15 C) \sec ^2(c+d x)\right )dx+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (5 (5 A+6 C) a^2+b (47 A+60 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a+6 A b^2+2 b^2 (8 A+15 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) \left (8 (8 A+15 C) \sec ^2(c+d x) b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b+15 a \left ((5 A+6 C) a^2+6 b^2 (3 A+4 C)\right ) \sec (c+d x)\right )dx\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) \left (8 (8 A+15 C) \sec ^2(c+d x) b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b+15 a \left ((5 A+6 C) a^2+6 b^2 (3 A+4 C)\right ) \sec (c+d x)\right )dx+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \frac {8 (8 A+15 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b+15 a \left ((5 A+6 C) a^2+6 b^2 (3 A+4 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \int \cos ^2(c+d x)dx+\int \cos ^3(c+d x) \left (8 (8 A+15 C) \sec ^2(c+d x) b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b\right )dx\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\int \frac {8 (8 A+15 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\int \frac {8 (8 A+15 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {8 (8 A+15 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2 b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) b}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 4532 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \cos (c+d x) \left (8 (8 A+15 C) b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) \cos ^2(c+d x) b\right )dx+15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 (8 A+15 C) b^3+24 \left (3 (4 A+5 C) a^2+A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b\right )dx+15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 3492 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\int \left (8 b \left (9 (4 A+5 C) a^2+b^2 (11 A+15 C)\right )-24 b \left (3 (4 A+5 C) a^2+A b^2\right ) \sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (15 a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 b \left (3 a^2 (4 A+5 C)+A b^2\right ) \sin ^3(c+d x)-8 b \left (9 a^2 (4 A+5 C)+b^2 (11 A+15 C)\right ) \sin (c+d x)}{d}\right )\right )+\frac {3 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\) |
(A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*A*b*Cos [c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((a*(6*A*b^2 + 5* a^2*(5*A + 6*C))*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (15*a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (-8*b* (9*a^2*(4*A + 5*C) + b^2*(11*A + 15*C))*Sin[c + d*x] + 8*b*(A*b^2 + 3*a^2* (4*A + 5*C))*Sin[c + d*x]^3)/d)/4)/5)/6
3.7.62.3.1 Defintions of rubi rules used
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[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2 ), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[ {e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
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 _)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si mp[1/(d*n) Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, d, e, f, A, B, C}, x] && 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/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[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/( d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*(C*n + A*(n + 1))*Csc[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^ 2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && Gt Q[m, 0] && LeQ[n, -1]
Time = 0.87 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.69
| method | result | size |
| parallelrisch | \(\frac {225 a \left (\left (A +\frac {16 C}{15}\right ) a^{2}+\frac {16 b^{2} \left (A +C \right )}{5}\right ) \sin \left (2 d x +2 c \right )+300 \left (\left (A +\frac {4 C}{5}\right ) a^{2}+\frac {4 A \,b^{2}}{15}\right ) b \sin \left (3 d x +3 c \right )+45 \left (a^{2} \left (A +\frac {2 C}{3}\right )+2 A \,b^{2}\right ) a \sin \left (4 d x +4 c \right )+36 A \,a^{2} b \sin \left (5 d x +5 c \right )+5 a^{3} A \sin \left (6 d x +6 c \right )+1800 b \left (a^{2} \left (A +\frac {6 C}{5}\right )+\frac {2 b^{2} \left (A +\frac {4 C}{3}\right )}{5}\right ) \sin \left (d x +c \right )+300 a x d \left (a^{2} \left (A +\frac {6 C}{5}\right )+\frac {18 b^{2} \left (A +\frac {4 C}{3}\right )}{5}\right )}{960 d}\) | \(178\) |
| derivativedivides | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a A \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{2} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 C a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(249\) |
| default | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a A \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{2} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 C a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(249\) |
| risch | \(\frac {5 a^{3} A x}{16}+\frac {9 A a \,b^{2} x}{8}+\frac {3 a^{3} x C}{8}+\frac {3 C a \,b^{2} x}{2}+\frac {15 \sin \left (d x +c \right ) A \,a^{2} b}{8 d}+\frac {3 \sin \left (d x +c \right ) A \,b^{3}}{4 d}+\frac {9 \sin \left (d x +c \right ) a^{2} b C}{4 d}+\frac {\sin \left (d x +c \right ) C \,b^{3}}{d}+\frac {a^{3} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 A \,a^{2} b \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 a^{3} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) a A \,b^{2}}{32 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {5 A \,a^{2} b \sin \left (3 d x +3 c \right )}{16 d}+\frac {\sin \left (3 d x +3 c \right ) A \,b^{3}}{12 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} b C}{4 d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{3} A}{64 d}+\frac {3 \sin \left (2 d x +2 c \right ) a A \,b^{2}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) C a \,b^{2}}{4 d}\) | \(315\) |
1/960*(225*a*((A+16/15*C)*a^2+16/5*b^2*(A+C))*sin(2*d*x+2*c)+300*((A+4/5*C )*a^2+4/15*A*b^2)*b*sin(3*d*x+3*c)+45*(a^2*(A+2/3*C)+2*A*b^2)*a*sin(4*d*x+ 4*c)+36*A*a^2*b*sin(5*d*x+5*c)+5*a^3*A*sin(6*d*x+6*c)+1800*b*(a^2*(A+6/5*C )+2/5*b^2*(A+4/3*C))*sin(d*x+c)+300*a*x*d*(a^2*(A+6/5*C)+18/5*b^2*(A+4/3*C )))/d
Time = 0.29 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.74 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} d x + {\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 144 \, A a^{2} b \cos \left (d x + c\right )^{4} + 96 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 80 \, {\left (2 \, A + 3 \, C\right )} b^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, A a b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (3 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
1/240*(15*((5*A + 6*C)*a^3 + 6*(3*A + 4*C)*a*b^2)*d*x + (40*A*a^3*cos(d*x + c)^5 + 144*A*a^2*b*cos(d*x + c)^4 + 96*(4*A + 5*C)*a^2*b + 80*(2*A + 3*C )*b^3 + 10*((5*A + 6*C)*a^3 + 18*A*a*b^2)*cos(d*x + c)^3 + 16*(3*(4*A + 5* C)*a^2*b + 5*A*b^3)*cos(d*x + c)^2 + 15*((5*A + 6*C)*a^3 + 6*(3*A + 4*C)*a *b^2)*cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.95 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} b + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{2} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{3} - 960 \, C b^{3} \sin \left (d x + c\right )}{960 \, d} \]
-1/960*(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 - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d *x + 2*c))*C*a^3 - 192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^2*b + 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^2*b - 90*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a*b^2 - 720*(2*d*x + 2* c + sin(2*d*x + 2*c))*C*a*b^2 + 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*b^ 3 - 960*C*b^3*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (243) = 486\).
Time = 0.35 (sec) , antiderivative size = 882, normalized size of antiderivative = 3.43 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
1/240*(15*(5*A*a^3 + 6*C*a^3 + 18*A*a*b^2 + 24*C*a*b^2)*(d*x + c) - 2*(165 *A*a^3*tan(1/2*d*x + 1/2*c)^11 + 150*C*a^3*tan(1/2*d*x + 1/2*c)^11 - 720*A *a^2*b*tan(1/2*d*x + 1/2*c)^11 - 720*C*a^2*b*tan(1/2*d*x + 1/2*c)^11 + 450 *A*a*b^2*tan(1/2*d*x + 1/2*c)^11 + 360*C*a*b^2*tan(1/2*d*x + 1/2*c)^11 - 2 40*A*b^3*tan(1/2*d*x + 1/2*c)^11 - 240*C*b^3*tan(1/2*d*x + 1/2*c)^11 - 25* A*a^3*tan(1/2*d*x + 1/2*c)^9 + 210*C*a^3*tan(1/2*d*x + 1/2*c)^9 - 1680*A*a ^2*b*tan(1/2*d*x + 1/2*c)^9 - 2640*C*a^2*b*tan(1/2*d*x + 1/2*c)^9 + 630*A* a*b^2*tan(1/2*d*x + 1/2*c)^9 + 1080*C*a*b^2*tan(1/2*d*x + 1/2*c)^9 - 880*A *b^3*tan(1/2*d*x + 1/2*c)^9 - 1200*C*b^3*tan(1/2*d*x + 1/2*c)^9 + 450*A*a^ 3*tan(1/2*d*x + 1/2*c)^7 + 60*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 3744*A*a^2*b* tan(1/2*d*x + 1/2*c)^7 - 4320*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 180*A*a*b^2 *tan(1/2*d*x + 1/2*c)^7 + 720*C*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 1440*A*b^3* tan(1/2*d*x + 1/2*c)^7 - 2400*C*b^3*tan(1/2*d*x + 1/2*c)^7 - 450*A*a^3*tan (1/2*d*x + 1/2*c)^5 - 60*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 3744*A*a^2*b*tan(1 /2*d*x + 1/2*c)^5 - 4320*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 180*A*a*b^2*tan( 1/2*d*x + 1/2*c)^5 - 720*C*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 1440*A*b^3*tan(1 /2*d*x + 1/2*c)^5 - 2400*C*b^3*tan(1/2*d*x + 1/2*c)^5 + 25*A*a^3*tan(1/2*d *x + 1/2*c)^3 - 210*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 1680*A*a^2*b*tan(1/2*d* x + 1/2*c)^3 - 2640*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 630*A*a*b^2*tan(1/2*d *x + 1/2*c)^3 - 1080*C*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 880*A*b^3*tan(1/2...
Time = 18.98 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.23 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (2\,A\,b^3-\frac {11\,A\,a^3}{8}-\frac {5\,C\,a^3}{4}+2\,C\,b^3-\frac {15\,A\,a\,b^2}{4}+6\,A\,a^2\,b-3\,C\,a\,b^2+6\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,A\,a^3}{24}+\frac {22\,A\,b^3}{3}-\frac {7\,C\,a^3}{4}+10\,C\,b^3-\frac {21\,A\,a\,b^2}{4}+14\,A\,a^2\,b-9\,C\,a\,b^2+22\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (12\,A\,b^3-\frac {15\,A\,a^3}{4}-\frac {C\,a^3}{2}+20\,C\,b^3-\frac {3\,A\,a\,b^2}{2}+\frac {156\,A\,a^2\,b}{5}-6\,C\,a\,b^2+36\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {15\,A\,a^3}{4}+12\,A\,b^3+\frac {C\,a^3}{2}+20\,C\,b^3+\frac {3\,A\,a\,b^2}{2}+\frac {156\,A\,a^2\,b}{5}+6\,C\,a\,b^2+36\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {22\,A\,b^3}{3}-\frac {5\,A\,a^3}{24}+\frac {7\,C\,a^3}{4}+10\,C\,b^3+\frac {21\,A\,a\,b^2}{4}+14\,A\,a^2\,b+9\,C\,a\,b^2+22\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,A\,a^3}{8}+2\,A\,b^3+\frac {5\,C\,a^3}{4}+2\,C\,b^3+\frac {15\,A\,a\,b^2}{4}+6\,A\,a^2\,b+3\,C\,a\,b^2+6\,C\,a^2\,b\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\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A\,a^2+18\,A\,b^2+6\,C\,a^2+24\,C\,b^2\right )}{8\,\left (\frac {5\,A\,a^3}{8}+\frac {3\,C\,a^3}{4}+\frac {9\,A\,a\,b^2}{4}+3\,C\,a\,b^2\right )}\right )\,\left (5\,A\,a^2+18\,A\,b^2+6\,C\,a^2+24\,C\,b^2\right )}{8\,d} \]
(tan(c/2 + (d*x)/2)*((11*A*a^3)/8 + 2*A*b^3 + (5*C*a^3)/4 + 2*C*b^3 + (15* A*a*b^2)/4 + 6*A*a^2*b + 3*C*a*b^2 + 6*C*a^2*b) - tan(c/2 + (d*x)/2)^11*(( 11*A*a^3)/8 - 2*A*b^3 + (5*C*a^3)/4 - 2*C*b^3 + (15*A*a*b^2)/4 - 6*A*a^2*b + 3*C*a*b^2 - 6*C*a^2*b) + tan(c/2 + (d*x)/2)^3*((22*A*b^3)/3 - (5*A*a^3) /24 + (7*C*a^3)/4 + 10*C*b^3 + (21*A*a*b^2)/4 + 14*A*a^2*b + 9*C*a*b^2 + 2 2*C*a^2*b) + tan(c/2 + (d*x)/2)^9*((5*A*a^3)/24 + (22*A*b^3)/3 - (7*C*a^3) /4 + 10*C*b^3 - (21*A*a*b^2)/4 + 14*A*a^2*b - 9*C*a*b^2 + 22*C*a^2*b) + ta n(c/2 + (d*x)/2)^5*((15*A*a^3)/4 + 12*A*b^3 + (C*a^3)/2 + 20*C*b^3 + (3*A* a*b^2)/2 + (156*A*a^2*b)/5 + 6*C*a*b^2 + 36*C*a^2*b) - tan(c/2 + (d*x)/2)^ 7*((15*A*a^3)/4 - 12*A*b^3 + (C*a^3)/2 - 20*C*b^3 + (3*A*a*b^2)/2 - (156*A *a^2*b)/5 + 6*C*a*b^2 - 36*C*a^2*b))/(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*atan((a*tan(c/2 + ( d*x)/2)*(5*A*a^2 + 18*A*b^2 + 6*C*a^2 + 24*C*b^2))/(8*((5*A*a^3)/8 + (3*C* a^3)/4 + (9*A*a*b^2)/4 + 3*C*a*b^2)))*(5*A*a^2 + 18*A*b^2 + 6*C*a^2 + 24*C *b^2))/(8*d)