Integrand size = 41, antiderivative size = 244 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^3 (23 A+26 B+30 C) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^3 (34 A+38 B+45 C) \tan (c+d x)}{15 d}+\frac {a^3 (23 A+26 B+30 C) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^3 (73 A+86 B+90 C) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(31 A+42 B+30 C) \left (a^3+a^3 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 a d}+\frac {A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d} \]
1/16*a^3*(23*A+26*B+30*C)*arctanh(sin(d*x+c))/d+1/15*a^3*(34*A+38*B+45*C)* tan(d*x+c)/d+1/16*a^3*(23*A+26*B+30*C)*sec(d*x+c)*tan(d*x+c)/d+1/120*a^3*( 73*A+86*B+90*C)*sec(d*x+c)^2*tan(d*x+c)/d+1/120*(31*A+42*B+30*C)*(a^3+a^3* cos(d*x+c))*sec(d*x+c)^3*tan(d*x+c)/d+1/10*(A+2*B)*(a^2+a^2*cos(d*x+c))^2* sec(d*x+c)^4*tan(d*x+c)/a/d+1/6*A*(a+a*cos(d*x+c))^3*sec(d*x+c)^5*tan(d*x+ c)/d
Time = 6.50 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.54 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^3 \left (15 (23 A+26 B+30 C) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (15 (23 A+26 B+30 C) \sec (c+d x)+10 (23 A+6 (3 B+C)) \sec ^3(c+d x)+40 A \sec ^5(c+d x)+16 \left (60 (A+B+C)+5 (7 A+5 B+3 C) \tan ^2(c+d x)+3 (3 A+B) \tan ^4(c+d x)\right )\right )\right )}{240 d} \]
(a^3*(15*(23*A + 26*B + 30*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(23 *A + 26*B + 30*C)*Sec[c + d*x] + 10*(23*A + 6*(3*B + C))*Sec[c + d*x]^3 + 40*A*Sec[c + d*x]^5 + 16*(60*(A + B + C) + 5*(7*A + 5*B + 3*C)*Tan[c + d*x ]^2 + 3*(3*A + B)*Tan[c + d*x]^4))))/(240*d)
Time = 1.86 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.463, Rules used = {3042, 3522, 3042, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3500, 3042, 3227, 3042, 4254, 24, 4255, 3042, 4257}
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.
| \(\displaystyle \int \sec ^7(c+d x) (a \cos (c+d x)+a)^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^3 (3 a (A+2 B)+2 a (A+3 C) \cos (c+d x)) \sec ^6(c+d x)dx}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a (A+2 B)+2 a (A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{5} \int (\cos (c+d x) a+a)^2 \left ((31 A+42 B+30 C) a^2+2 (8 A+6 B+15 C) \cos (c+d x) a^2\right ) \sec ^5(c+d x)dx+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((31 A+42 B+30 C) a^2+2 (8 A+6 B+15 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{4} \int 3 (\cos (c+d x) a+a) \left ((73 A+86 B+90 C) a^3+2 (21 A+22 B+30 C) \cos (c+d x) a^3\right ) \sec ^4(c+d x)dx+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int (\cos (c+d x) a+a) \left ((73 A+86 B+90 C) a^3+2 (21 A+22 B+30 C) \cos (c+d x) a^3\right ) \sec ^4(c+d x)dx+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((73 A+86 B+90 C) a^3+2 (21 A+22 B+30 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \left (2 (21 A+22 B+30 C) \cos ^2(c+d x) a^4+(73 A+86 B+90 C) a^4+\left (2 (21 A+22 B+30 C) a^4+(73 A+86 B+90 C) a^4\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \int \frac {2 (21 A+22 B+30 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+(73 A+86 B+90 C) a^4+\left (2 (21 A+22 B+30 C) a^4+(73 A+86 B+90 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \left (15 (23 A+26 B+30 C) a^4+8 (34 A+38 B+45 C) \cos (c+d x) a^4\right ) \sec ^3(c+d x)dx+\frac {a^4 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \frac {15 (23 A+26 B+30 C) a^4+8 (34 A+38 B+45 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^4 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (15 a^4 (23 A+26 B+30 C) \int \sec ^3(c+d x)dx+8 a^4 (34 A+38 B+45 C) \int \sec ^2(c+d x)dx\right )+\frac {a^4 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (8 a^4 (34 A+38 B+45 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+15 a^4 (23 A+26 B+30 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^4 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (15 a^4 (23 A+26 B+30 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {8 a^4 (34 A+38 B+45 C) \int 1d(-\tan (c+d x))}{d}\right )+\frac {a^4 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (15 a^4 (23 A+26 B+30 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {8 a^4 (34 A+38 B+45 C) \tan (c+d x)}{d}\right )+\frac {a^4 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (15 a^4 (23 A+26 B+30 C) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 a^4 (34 A+38 B+45 C) \tan (c+d x)}{d}\right )+\frac {a^4 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (15 a^4 (23 A+26 B+30 C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 a^4 (34 A+38 B+45 C) \tan (c+d x)}{d}\right )+\frac {a^4 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (15 a^4 (23 A+26 B+30 C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 a^4 (34 A+38 B+45 C) \tan (c+d x)}{d}\right )+\frac {a^4 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}}{6 a}+\frac {A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
(A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((3*(A + 2* B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (((31*A + 42*B + 30*C)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d ) + (3*((a^4*(73*A + 86*B + 90*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((8 *a^4*(34*A + 38*B + 45*C)*Tan[c + d*x])/d + 15*a^4*(23*A + 26*B + 30*C)*(A rcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/3))/4)/5) /(6*a)
3.4.27.3.1 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[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m* (c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* (n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ [m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 12.59 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.11
| method | result | size |
| parallelrisch | \(-\frac {23 \left (\left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (A +\frac {26 B}{23}+\frac {30 C}{23}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right ) \left (A +\frac {26 B}{23}+\frac {30 C}{23}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-16 C -\frac {480 A}{23}-\frac {416 B}{23}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {252 B}{23}-\frac {34 A}{3}-\frac {212 C}{23}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {1088 A}{115}-\frac {1216 B}{115}-\frac {256 C}{23}\right ) \sin \left (4 d x +4 c \right )+\left (-2 A -\frac {60 C}{23}-\frac {52 B}{23}\right ) \sin \left (5 d x +5 c \right )+\left (-\frac {544 A}{345}-\frac {608 B}{345}-\frac {48 C}{23}\right ) \sin \left (6 d x +6 c \right )-\frac {300 \left (\frac {2 B}{3}+\frac {38 C}{75}+A \right ) \sin \left (d x +c \right )}{23}\right ) a^{3}}{16 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(270\) |
| parts | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (3 A \,a^{3}+B \,a^{3}\right ) \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,a^{3}+3 C \,a^{3}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (A \,a^{3}+3 B \,a^{3}+3 C \,a^{3}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 A \,a^{3}+3 B \,a^{3}+C \,a^{3}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C \,a^{3} \tan \left (d x +c \right )}{d}\) | \(277\) |
| derivativedivides | \(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \,a^{3} \tan \left (d x +c \right )+3 A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 A \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 C \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+A \,a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+C \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(438\) |
| default | \(\frac {-A \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+B \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+C \,a^{3} \tan \left (d x +c \right )+3 A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 B \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 C \,a^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 A \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+3 B \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-3 C \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+A \,a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-B \,a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+C \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(438\) |
| risch | \(-\frac {i a^{3} \left (-480 A \,{\mathrm e}^{8 i \left (d x +c \right )}-720 C -544 A -608 B -7680 A \,{\mathrm e}^{4 i \left (d x +c \right )}-7680 B \,{\mathrm e}^{4 i \left (d x +c \right )}-3648 B \,{\mathrm e}^{2 i \left (d x +c \right )}-8160 C \,{\mathrm e}^{4 i \left (d x +c \right )}-4080 C \,{\mathrm e}^{2 i \left (d x +c \right )}+1500 B \,{\mathrm e}^{7 i \left (d x +c \right )}-1955 A \,{\mathrm e}^{3 i \left (d x +c \right )}-2250 A \,{\mathrm e}^{5 i \left (d x +c \right )}-1500 B \,{\mathrm e}^{5 i \left (d x +c \right )}-3264 A \,{\mathrm e}^{2 i \left (d x +c \right )}-345 A \,{\mathrm e}^{i \left (d x +c \right )}-390 B \,{\mathrm e}^{i \left (d x +c \right )}-6080 B \,{\mathrm e}^{6 i \left (d x +c \right )}-1890 B \,{\mathrm e}^{3 i \left (d x +c \right )}-5440 A \,{\mathrm e}^{6 i \left (d x +c \right )}-7200 C \,{\mathrm e}^{6 i \left (d x +c \right )}+2250 A \,{\mathrm e}^{7 i \left (d x +c \right )}-450 C \,{\mathrm e}^{i \left (d x +c \right )}-1440 B \,{\mathrm e}^{8 i \left (d x +c \right )}+1140 C \,{\mathrm e}^{7 i \left (d x +c \right )}-1140 C \,{\mathrm e}^{5 i \left (d x +c \right )}-1590 C \,{\mathrm e}^{3 i \left (d x +c \right )}+390 B \,{\mathrm e}^{11 i \left (d x +c \right )}+345 A \,{\mathrm e}^{11 i \left (d x +c \right )}+450 C \,{\mathrm e}^{11 i \left (d x +c \right )}-240 C \,{\mathrm e}^{10 i \left (d x +c \right )}-2640 C \,{\mathrm e}^{8 i \left (d x +c \right )}+1590 C \,{\mathrm e}^{9 i \left (d x +c \right )}+1955 A \,{\mathrm e}^{9 i \left (d x +c \right )}+1890 B \,{\mathrm e}^{9 i \left (d x +c \right )}\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {23 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {13 B \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{8 d}-\frac {23 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {13 B \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {15 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{8 d}\) | \(538\) |
-23/16*((cos(6*d*x+6*c)+6*cos(4*d*x+4*c)+15*cos(2*d*x+2*c)+10)*(A+26/23*B+ 30/23*C)*ln(tan(1/2*d*x+1/2*c)-1)-(cos(6*d*x+6*c)+6*cos(4*d*x+4*c)+15*cos( 2*d*x+2*c)+10)*(A+26/23*B+30/23*C)*ln(tan(1/2*d*x+1/2*c)+1)+(-16*C-480/23* A-416/23*B)*sin(2*d*x+2*c)+(-252/23*B-34/3*A-212/23*C)*sin(3*d*x+3*c)+(-10 88/115*A-1216/115*B-256/23*C)*sin(4*d*x+4*c)+(-2*A-60/23*C-52/23*B)*sin(5* d*x+5*c)+(-544/345*A-608/345*B-48/23*C)*sin(6*d*x+6*c)-300/23*(2/3*B+38/75 *C+A)*sin(d*x+c))*a^3/d/(cos(6*d*x+6*c)+6*cos(4*d*x+4*c)+15*cos(2*d*x+2*c) +10)
Time = 0.28 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.83 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left (23 \, A + 26 \, B + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (23 \, A + 26 \, B + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (34 \, A + 38 \, B + 45 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 15 \, {\left (23 \, A + 26 \, B + 30 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 16 \, {\left (17 \, A + 19 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 10 \, {\left (23 \, A + 18 \, B + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 48 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 40 \, A a^{3}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7, x, algorithm="fricas")
1/480*(15*(23*A + 26*B + 30*C)*a^3*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 15*(23*A + 26*B + 30*C)*a^3*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(16* (34*A + 38*B + 45*C)*a^3*cos(d*x + c)^5 + 15*(23*A + 26*B + 30*C)*a^3*cos( d*x + c)^4 + 16*(17*A + 19*B + 15*C)*a^3*cos(d*x + c)^3 + 10*(23*A + 18*B + 6*C)*a^3*cos(d*x + c)^2 + 48*(3*A + B)*a^3*cos(d*x + c) + 40*A*a^3)*sin( d*x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (230) = 460\).
Time = 0.21 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.29 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {96 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{3} + 160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} + 32 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a^{3} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 5 \, A a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 90 \, B a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, C a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, B a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, C a^{3} \tan \left (d x + c\right )}{480 \, d} \]
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7, x, algorithm="maxima")
1/480*(96*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^3 + 160*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^3 + 32*(3*tan(d*x + c)^5 + 10*t an(d*x + c)^3 + 15*tan(d*x + c))*B*a^3 + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^3 + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^3 - 5*A*a^3*(2*(15* sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3* sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log (sin(d*x + c) - 1)) - 90*A*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin (d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin( d*x + c) - 1)) - 90*B*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 30*C*a^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^ 4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 120*B*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 360*C*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 480*C*a^3*tan(d*x + c))/d
Time = 0.40 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.61 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {15 \, {\left (23 \, A a^{3} + 26 \, B a^{3} + 30 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (23 \, A a^{3} + 26 \, B a^{3} + 30 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\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} + 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} - 2210 \, B 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} + 5148 \, B 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} - 5988 \, B 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} + 4190 \, B 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 ) - 1530 \, B 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} \]
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^7, x, algorithm="giac")
1/240*(15*(23*A*a^3 + 26*B*a^3 + 30*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(23*A*a^3 + 26*B*a^3 + 30*C*a^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1 )) - 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 + 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 - 2210*B*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 + 5148*B*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 - 59 88*B*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 + 4190*B*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) - 1530*B*a^3 *tan(1/2*d*x + 1/2*c) - 1470*C*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/ 2*c)^2 - 1)^6)/d
Time = 5.06 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.38 \[ \int (a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx=\frac {a^3\,\mathrm {atanh}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,A+26\,B+30\,C\right )}{4\,\left (\frac {23\,A\,a^3}{4}+\frac {13\,B\,a^3}{2}+\frac {15\,C\,a^3}{2}\right )}\right )\,\left (23\,A+26\,B+30\,C\right )}{8\,d}-\frac {\left (\frac {23\,A\,a^3}{8}+\frac {13\,B\,a^3}{4}+\frac {15\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (-\frac {391\,A\,a^3}{24}-\frac {221\,B\,a^3}{12}-\frac {85\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {759\,A\,a^3}{20}+\frac {429\,B\,a^3}{10}+\frac {99\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (-\frac {969\,A\,a^3}{20}-\frac {499\,B\,a^3}{10}-\frac {125\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {211\,A\,a^3}{8}+\frac {419\,B\,a^3}{12}+\frac {171\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (-\frac {105\,A\,a^3}{8}-\frac {51\,B\,a^3}{4}-\frac {49\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
(a^3*atanh((a^3*tan(c/2 + (d*x)/2)*(23*A + 26*B + 30*C))/(4*((23*A*a^3)/4 + (13*B*a^3)/2 + (15*C*a^3)/2)))*(23*A + 26*B + 30*C))/(8*d) - (tan(c/2 + (d*x)/2)^11*((23*A*a^3)/8 + (13*B*a^3)/4 + (15*C*a^3)/4) - tan(c/2 + (d*x) /2)^9*((391*A*a^3)/24 + (221*B*a^3)/12 + (85*C*a^3)/4) + tan(c/2 + (d*x)/2 )^3*((211*A*a^3)/8 + (419*B*a^3)/12 + (171*C*a^3)/4) + tan(c/2 + (d*x)/2)^ 7*((759*A*a^3)/20 + (429*B*a^3)/10 + (99*C*a^3)/2) - tan(c/2 + (d*x)/2)^5* ((969*A*a^3)/20 + (499*B*a^3)/10 + (125*C*a^3)/2) - tan(c/2 + (d*x)/2)*((1 05*A*a^3)/8 + (51*B*a^3)/4 + (49*C*a^3)/4))/(d*(15*tan(c/2 + (d*x)/2)^4 - 6*tan(c/2 + (d*x)/2)^2 - 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))