Integrand size = 31, antiderivative size = 229 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {7 a^4 (7 A+8 B) \text {arctanh}(\sin (c+d x))}{16 d}+\frac {a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac {7 a^4 (7 A+8 B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^4 (159 A+176 B) \sec ^2(c+d x) \tan (c+d x)}{120 d}+\frac {(73 A+72 B) \left (a^4+a^4 \cos (c+d x)\right ) \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac {(3 A+2 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec ^4(c+d x) \tan (c+d x)}{10 d}+\frac {a A (a+a \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{6 d} \] Output:
7/16*a^4*(7*A+8*B)*arctanh(sin(d*x+c))/d+1/15*a^4*(72*A+83*B)*tan(d*x+c)/d +7/16*a^4*(7*A+8*B)*sec(d*x+c)*tan(d*x+c)/d+1/120*a^4*(159*A+176*B)*sec(d* x+c)^2*tan(d*x+c)/d+1/120*(73*A+72*B)*(a^4+a^4*cos(d*x+c))*sec(d*x+c)^3*ta n(d*x+c)/d+1/10*(3*A+2*B)*(a^2+a^2*cos(d*x+c))^2*sec(d*x+c)^4*tan(d*x+c)/d +1/6*a*A*(a+a*cos(d*x+c))^3*sec(d*x+c)^5*tan(d*x+c)/d
Time = 5.16 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.13 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {49 a^4 A \text {arctanh}(\sin (c+d x))}{16 d}+\frac {7 a^4 B \text {arctanh}(\sin (c+d x))}{2 d}+\frac {8 a^4 A \tan (c+d x)}{d}+\frac {8 a^4 B \tan (c+d x)}{d}+\frac {49 a^4 A \sec (c+d x) \tan (c+d x)}{16 d}+\frac {7 a^4 B \sec (c+d x) \tan (c+d x)}{2 d}+\frac {41 a^4 A \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 B \sec ^3(c+d x) \tan (c+d x)}{d}+\frac {a^4 A \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 A \tan ^3(c+d x)}{d}+\frac {8 a^4 B \tan ^3(c+d x)}{3 d}+\frac {4 a^4 A \tan ^5(c+d x)}{5 d}+\frac {a^4 B \tan ^5(c+d x)}{5 d} \] Input:
Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^7,x]
Output:
(49*a^4*A*ArcTanh[Sin[c + d*x]])/(16*d) + (7*a^4*B*ArcTanh[Sin[c + d*x]])/ (2*d) + (8*a^4*A*Tan[c + d*x])/d + (8*a^4*B*Tan[c + d*x])/d + (49*a^4*A*Se c[c + d*x]*Tan[c + d*x])/(16*d) + (7*a^4*B*Sec[c + d*x]*Tan[c + d*x])/(2*d ) + (41*a^4*A*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (a^4*B*Sec[c + d*x]^3* Tan[c + d*x])/d + (a^4*A*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (4*a^4*A*Tan [c + d*x]^3)/d + (8*a^4*B*Tan[c + d*x]^3)/(3*d) + (4*a^4*A*Tan[c + d*x]^5) /(5*d) + (a^4*B*Tan[c + d*x]^5)/(5*d)
Time = 1.75 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.05, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.613, Rules used = {3042, 3454, 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.
| \(\displaystyle \int \sec ^7(c+d x) (a \cos (c+d x)+a)^4 (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{6} \int (\cos (c+d x) a+a)^3 (3 a (3 A+2 B)+2 a (A+3 B) \cos (c+d x)) \sec ^6(c+d x)dx+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a (3 A+2 B)+2 a (A+3 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int (\cos (c+d x) a+a)^2 \left ((73 A+72 B) a^2+14 (2 A+3 B) \cos (c+d x) a^2\right ) \sec ^5(c+d x)dx+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((73 A+72 B) a^2+14 (2 A+3 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int 3 (\cos (c+d x) a+a) \left ((159 A+176 B) a^3+2 (43 A+52 B) \cos (c+d x) a^3\right ) \sec ^4(c+d x)dx+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int (\cos (c+d x) a+a) \left ((159 A+176 B) a^3+2 (43 A+52 B) \cos (c+d x) a^3\right ) \sec ^4(c+d x)dx+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((159 A+176 B) a^3+2 (43 A+52 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \left (2 (43 A+52 B) \cos ^2(c+d x) a^4+(159 A+176 B) a^4+\left (2 (43 A+52 B) a^4+(159 A+176 B) a^4\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \int \frac {2 (43 A+52 B) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+(159 A+176 B) a^4+\left (2 (43 A+52 B) a^4+(159 A+176 B) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \left (105 (7 A+8 B) a^4+8 (72 A+83 B) \cos (c+d x) a^4\right ) \sec ^3(c+d x)dx+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \int \frac {105 (7 A+8 B) a^4+8 (72 A+83 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^4 (7 A+8 B) \int \sec ^3(c+d x)dx+8 a^4 (72 A+83 B) \int \sec ^2(c+d x)dx\right )+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (8 a^4 (72 A+83 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+105 a^4 (7 A+8 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^4 (7 A+8 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {8 a^4 (72 A+83 B) \int 1d(-\tan (c+d x))}{d}\right )+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^4 (7 A+8 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {8 a^4 (72 A+83 B) \tan (c+d x)}{d}\right )+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^4 (7 A+8 B) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 a^4 (72 A+83 B) \tan (c+d x)}{d}\right )+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^4 (7 A+8 B) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 a^4 (72 A+83 B) \tan (c+d x)}{d}\right )+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {3}{4} \left (\frac {1}{3} \left (105 a^4 (7 A+8 B) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {8 a^4 (72 A+83 B) \tan (c+d x)}{d}\right )+\frac {a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{3 d}\right )+\frac {(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{4 d}\right )+\frac {3 (3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{5 d}\right )+\frac {a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}\) |
Input:
Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^7,x]
Output:
(a*A*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + ((3*(3*A + 2*B)*(a^2 + a^2*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((( 73*A + 72*B)*(a^4 + a^4*Cos[c + d*x])*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*((a^4*(159*A + 176*B)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((8*a^4*(72 *A + 83*B)*Tan[c + d*x])/d + 105*a^4*(7*A + 8*B)*(ArcTanh[Sin[c + d*x]]/(2 *d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/3))/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[((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[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 = 19.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.10
| method | result | size |
| parallelrisch | \(\frac {125 a^{4} \left (-\frac {147 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {8 B}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{100}+\frac {147 \left (\frac {\cos \left (6 d x +6 c \right )}{15}+\frac {2 \cos \left (4 d x +4 c \right )}{5}+\cos \left (2 d x +2 c \right )+\frac {2}{3}\right ) \left (A +\frac {8 B}{7}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{100}+\frac {28 \left (8 A +7 B \right ) \sin \left (2 d x +2 c \right )}{125}+\frac {\left (116 B +\frac {769 A}{6}\right ) \sin \left (3 d x +3 c \right )}{125}+\frac {48 \left (12 A +13 B \right ) \sin \left (4 d x +4 c \right )}{625}+\frac {7 \left (4 B +\frac {7 A}{2}\right ) \sin \left (5 d x +5 c \right )}{125}+\frac {4 \left (24 A +\frac {83 B}{3}\right ) \sin \left (6 d x +6 c \right )}{625}+\sin \left (d x +c \right ) \left (\frac {88 B}{125}+A \right )\right )}{4 d \left (\cos \left (6 d x +6 c \right )+6 \cos \left (4 d x +4 c \right )+15 \cos \left (2 d x +2 c \right )+10\right )}\) | \(251\) |
| parts | \(\frac {a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}-\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {B \,a^{4} \tan \left (d x +c \right )}{d}\) | \(267\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-B \,a^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(365\) |
| default | \(\frac {a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+B \,a^{4} \tan \left (d x +c \right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 B \,a^{4} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-4 a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 B \,a^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-B \,a^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(365\) |
| risch | \(-\frac {i a^{4} \left (735 A \,{\mathrm e}^{11 i \left (d x +c \right )}+840 B \,{\mathrm e}^{11 i \left (d x +c \right )}-240 B \,{\mathrm e}^{10 i \left (d x +c \right )}+3845 A \,{\mathrm e}^{9 i \left (d x +c \right )}+3480 B \,{\mathrm e}^{9 i \left (d x +c \right )}-1920 A \,{\mathrm e}^{8 i \left (d x +c \right )}-4080 B \,{\mathrm e}^{8 i \left (d x +c \right )}+3750 A \,{\mathrm e}^{7 i \left (d x +c \right )}+2640 B \,{\mathrm e}^{7 i \left (d x +c \right )}-11520 A \,{\mathrm e}^{6 i \left (d x +c \right )}-13280 B \,{\mathrm e}^{6 i \left (d x +c \right )}-3750 A \,{\mathrm e}^{5 i \left (d x +c \right )}-2640 B \,{\mathrm e}^{5 i \left (d x +c \right )}-15360 A \,{\mathrm e}^{4 i \left (d x +c \right )}-15840 B \,{\mathrm e}^{4 i \left (d x +c \right )}-3845 A \,{\mathrm e}^{3 i \left (d x +c \right )}-3480 B \,{\mathrm e}^{3 i \left (d x +c \right )}-6912 A \,{\mathrm e}^{2 i \left (d x +c \right )}-7728 B \,{\mathrm e}^{2 i \left (d x +c \right )}-735 A \,{\mathrm e}^{i \left (d x +c \right )}-840 B \,{\mathrm e}^{i \left (d x +c \right )}-1152 A -1328 B \right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}+\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{2 d}-\frac {49 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{2 d}\) | \(371\) |
Input:
int((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^7,x,method=_RETURNVERBO SE)
Output:
125/4*a^4*(-147/100*(1/15*cos(6*d*x+6*c)+2/5*cos(4*d*x+4*c)+cos(2*d*x+2*c) +2/3)*(A+8/7*B)*ln(tan(1/2*d*x+1/2*c)-1)+147/100*(1/15*cos(6*d*x+6*c)+2/5* cos(4*d*x+4*c)+cos(2*d*x+2*c)+2/3)*(A+8/7*B)*ln(tan(1/2*d*x+1/2*c)+1)+28/1 25*(8*A+7*B)*sin(2*d*x+2*c)+1/125*(116*B+769/6*A)*sin(3*d*x+3*c)+48/625*(1 2*A+13*B)*sin(4*d*x+4*c)+7/125*(4*B+7/2*A)*sin(5*d*x+5*c)+4/625*(24*A+83/3 *B)*sin(6*d*x+6*c)+sin(d*x+c)*(88/125*B+A))/d/(cos(6*d*x+6*c)+6*cos(4*d*x+ 4*c)+15*cos(2*d*x+2*c)+10)
Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.81 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (72 \, A + 83 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} + 105 \, {\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (18 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 10 \, {\left (41 \, A + 24 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 48 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 40 \, A a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^7,x, algorithm="f ricas")
Output:
1/480*(105*(7*A + 8*B)*a^4*cos(d*x + c)^6*log(sin(d*x + c) + 1) - 105*(7*A + 8*B)*a^4*cos(d*x + c)^6*log(-sin(d*x + c) + 1) + 2*(16*(72*A + 83*B)*a^ 4*cos(d*x + c)^5 + 105*(7*A + 8*B)*a^4*cos(d*x + c)^4 + 32*(18*A + 17*B)*a ^4*cos(d*x + c)^3 + 10*(41*A + 24*B)*a^4*cos(d*x + c)^2 + 48*(4*A + B)*a^4 *cos(d*x + c) + 40*A*a^4)*sin(d*x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**4*(A+B*cos(d*x+c))*sec(d*x+c)**7,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (215) = 430\).
Time = 0.05 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.03 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {128 \, {\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^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 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^{4} + 960 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} - 5 \, A a^{4} {\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 )} - 180 \, A a^{4} {\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^{4} {\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 \, A a^{4} {\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 \, B a^{4} {\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 \, B a^{4} \tan \left (d x + c\right )}{480 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^7,x, algorithm="m axima")
Output:
1/480*(128*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^4 + 640*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 32*(3*tan(d*x + c)^5 + 10* tan(d*x + c)^3 + 15*tan(d*x + c))*B*a^4 + 960*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^4 - 5*A*a^4*(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*l og(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 180*A*a^4*(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^4*(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*A*a^4*(2*sin(d*x + c)/(sin( d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 480*B*a ^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin( d*x + c) - 1)) + 480*B*a^4*tan(d*x + c))/d
Time = 0.28 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.22 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {105 \, {\left (7 \, A a^{4} + 8 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (7 \, A a^{4} + 8 \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (735 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4165 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 4760 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 11802 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 13488 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3000 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \] Input:
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^7,x, algorithm="g iac")
Output:
1/240*(105*(7*A*a^4 + 8*B*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(7 *A*a^4 + 8*B*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(735*A*a^4*tan(1/ 2*d*x + 1/2*c)^11 + 840*B*a^4*tan(1/2*d*x + 1/2*c)^11 - 4165*A*a^4*tan(1/2 *d*x + 1/2*c)^9 - 4760*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 9702*A*a^4*tan(1/2*d *x + 1/2*c)^7 + 11088*B*a^4*tan(1/2*d*x + 1/2*c)^7 - 11802*A*a^4*tan(1/2*d *x + 1/2*c)^5 - 13488*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 7355*A*a^4*tan(1/2*d* x + 1/2*c)^3 + 9320*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 3105*A*a^4*tan(1/2*d*x + 1/2*c) - 3000*B*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^6 )/d
Time = 44.26 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.14 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx=\frac {\left (-\frac {49\,A\,a^4}{8}-7\,B\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {833\,A\,a^4}{24}+\frac {119\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {1617\,A\,a^4}{20}-\frac {462\,B\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1967\,A\,a^4}{20}+\frac {562\,B\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {1471\,A\,a^4}{24}-\frac {233\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+25\,B\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (7\,A+8\,B\right )}{8\,d} \] Input:
int(((A + B*cos(c + d*x))*(a + a*cos(c + d*x))^4)/cos(c + d*x)^7,x)
Output:
(tan(c/2 + (d*x)/2)*((207*A*a^4)/8 + 25*B*a^4) - tan(c/2 + (d*x)/2)^11*((4 9*A*a^4)/8 + 7*B*a^4) + tan(c/2 + (d*x)/2)^9*((833*A*a^4)/24 + (119*B*a^4) /3) - tan(c/2 + (d*x)/2)^3*((1471*A*a^4)/24 + (233*B*a^4)/3) - tan(c/2 + ( d*x)/2)^7*((1617*A*a^4)/20 + (462*B*a^4)/5) + tan(c/2 + (d*x)/2)^5*((1967* A*a^4)/20 + (562*B*a^4)/5))/(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)) + (7*a^4*atanh(tan(c/2 + (d*x)/2) )*(7*A + 8*B))/(8*d)
Time = 0.20 (sec) , antiderivative size = 663, normalized size of antiderivative = 2.90 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx =\text {Too large to display} \] Input:
int((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c))*sec(d*x+c)^7,x)
Output:
(a**4*( - 735*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a - 8 40*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*b + 2205*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a + 2520*cos(c + d*x)*log( tan((c + d*x)/2) - 1)*sin(c + d*x)**4*b - 2205*cos(c + d*x)*log(tan((c + d *x)/2) - 1)*sin(c + d*x)**2*a - 2520*cos(c + d*x)*log(tan((c + d*x)/2) - 1 )*sin(c + d*x)**2*b + 735*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a + 840*c os(c + d*x)*log(tan((c + d*x)/2) - 1)*b + 735*cos(c + d*x)*log(tan((c + d* x)/2) + 1)*sin(c + d*x)**6*a + 840*cos(c + d*x)*log(tan((c + d*x)/2) + 1)* sin(c + d*x)**6*b - 2205*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d* x)**4*a - 2520*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*b + 2205*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a + 2520*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*b - 735*cos(c + d*x)*log (tan((c + d*x)/2) + 1)*a - 840*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*b - 735*cos(c + d*x)*sin(c + d*x)**5*a - 840*cos(c + d*x)*sin(c + d*x)**5*b + 1880*cos(c + d*x)*sin(c + d*x)**3*a + 1920*cos(c + d*x)*sin(c + d*x)**3*b - 1185*cos(c + d*x)*sin(c + d*x)*a - 1080*cos(c + d*x)*sin(c + d*x)*b + 11 52*sin(c + d*x)**7*a + 1328*sin(c + d*x)**7*b - 4032*sin(c + d*x)**5*a - 4 528*sin(c + d*x)**5*b + 4800*sin(c + d*x)**3*a + 5120*sin(c + d*x)**3*b - 1920*sin(c + d*x)*a - 1920*sin(c + d*x)*b))/(240*cos(c + d*x)*d*(sin(c + d *x)**6 - 3*sin(c + d*x)**4 + 3*sin(c + d*x)**2 - 1))