Integrand size = 31, antiderivative size = 216 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=b^4 B x+\frac {\left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
b^4*B*x+1/8*(3*A*a^4+24*A*a^2*b^2+8*A*b^4+16*B*a^3*b+32*B*a*b^3)*arctanh(s in(d*x+c))/d+1/6*a*(16*A*a^2*b+19*A*b^3+4*B*a^3+34*B*a*b^2)*tan(d*x+c)/d+1 /24*a^2*(9*A*a^2+26*A*b^2+32*B*a*b)*sec(d*x+c)*tan(d*x+c)/d+1/12*a*(7*A*b+ 4*B*a)*(a+b*cos(d*x+c))^2*sec(d*x+c)^2*tan(d*x+c)/d+1/4*a*A*(a+b*cos(d*x+c ))^3*sec(d*x+c)^3*tan(d*x+c)/d
Time = 0.72 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.74 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {24 b^4 B d x+3 \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \text {arctanh}(\sin (c+d x))+3 a \left (8 \left (4 a^2 A b+4 A b^3+a^3 B+6 a b^2 B\right )+a \left (3 a^2 A+24 A b^2+16 a b B\right ) \sec (c+d x)+2 a^3 A \sec ^3(c+d x)\right ) \tan (c+d x)+8 a^3 (4 A b+a B) \tan ^3(c+d x)}{24 d} \]
(24*b^4*B*d*x + 3*(3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^ 3*B)*ArcTanh[Sin[c + d*x]] + 3*a*(8*(4*a^2*A*b + 4*A*b^3 + a^3*B + 6*a*b^2 *B) + a*(3*a^2*A + 24*A*b^2 + 16*a*b*B)*Sec[c + d*x] + 2*a^3*A*Sec[c + d*x ]^3)*Tan[c + d*x] + 8*a^3*(4*A*b + a*B)*Tan[c + d*x]^3)/(24*d)
Time = 1.49 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {3042, 3468, 3042, 3526, 3042, 3510, 25, 3042, 3500, 27, 3042, 3214, 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.
| \(\displaystyle \int \sec ^5(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {1}{4} \int (a+b \cos (c+d x))^2 \left (4 b^2 B \cos ^2(c+d x)+\left (3 A a^2+8 b B a+4 A b^2\right ) \cos (c+d x)+a (7 A b+4 a B)\right ) \sec ^4(c+d x)dx+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (4 b^2 B \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 A a^2+8 b B a+4 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (7 A b+4 a B)\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (12 B \cos ^2(c+d x) b^3+a \left (9 A a^2+32 b B a+26 A b^2\right )+\left (8 B a^3+23 A b a^2+36 b^2 B a+12 A b^3\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (12 B \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^3+a \left (9 A a^2+32 b B a+26 A b^2\right )+\left (8 B a^3+23 A b a^2+36 b^2 B a+12 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}-\frac {1}{2} \int -\left (\left (24 B \cos ^2(c+d x) b^4+4 a \left (4 B a^3+16 A b a^2+34 b^2 B a+19 A b^3\right )+3 \left (3 A a^4+16 b B a^3+24 A b^2 a^2+32 b^3 B a+8 A b^4\right ) \cos (c+d x)\right ) \sec ^2(c+d x)\right )dx\right )+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (24 B \cos ^2(c+d x) b^4+4 a \left (4 B a^3+16 A b a^2+34 b^2 B a+19 A b^3\right )+3 \left (3 A a^4+16 b B a^3+24 A b^2 a^2+32 b^3 B a+8 A b^4\right ) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {24 B \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^4+4 a \left (4 B a^3+16 A b a^2+34 b^2 B a+19 A b^3\right )+3 \left (3 A a^4+16 b B a^3+24 A b^2 a^2+32 b^3 B a+8 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left (3 A a^4+16 b B a^3+24 A b^2 a^2+32 b^3 B a+8 A b^4+8 b^4 B \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 a \left (4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right ) \tan (c+d x)}{d}\right )+\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (3 A a^4+16 b B a^3+24 A b^2 a^2+32 b^3 B a+8 A b^4+8 b^4 B \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 a \left (4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right ) \tan (c+d x)}{d}\right )+\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {3 A a^4+16 b B a^3+24 A b^2 a^2+32 b^3 B a+8 A b^4+8 b^4 B \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {4 a \left (4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right ) \tan (c+d x)}{d}\right )+\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \int \sec (c+d x)dx+8 b^4 B x\right )+\frac {4 a \left (4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right ) \tan (c+d x)}{d}\right )+\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\left (3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+8 b^4 B x\right )+\frac {4 a \left (4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right ) \tan (c+d x)}{d}\right )+\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{2 d}+\frac {1}{2} \left (\frac {4 a \left (4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right ) \tan (c+d x)}{d}+3 \left (\frac {\left (3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \text {arctanh}(\sin (c+d x))}{d}+8 b^4 B x\right )\right )\right )+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\) |
(a*A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((a*(7*A* b + 4*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((a ^2*(9*a^2*A + 26*A*b^2 + 32*a*b*B)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (3*( 8*b^4*B*x + ((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B)* ArcTanh[Sin[c + d*x]])/d) + (4*a*(16*a^2*A*b + 19*A*b^3 + 4*a^3*B + 34*a*b ^2*B)*Tan[c + d*x])/d)/2)/3)/4
3.3.47.3.1 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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, 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] :> Si mp[(-(b*c - a*d))*(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)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2 , 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
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/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 5.47 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.99
| method | result | size |
| parts | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {B \,b^{4} \left (d x +c \right )}{d}\) | \(213\) |
| derivativedivides | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 B \tan \left (d x +c \right ) a^{2} b^{2}+4 A \tan \left (d x +c \right ) a \,b^{3}+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \left (d x +c \right )}{d}\) | \(260\) |
| default | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 B \tan \left (d x +c \right ) a^{2} b^{2}+4 A \tan \left (d x +c \right ) a \,b^{3}+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \left (d x +c \right )}{d}\) | \(260\) |
| parallelrisch | \(\frac {-36 \left (a^{4} A +8 A \,a^{2} b^{2}+\frac {8}{3} A \,b^{4}+\frac {16}{3} B \,a^{3} b +\frac {32}{3} B a \,b^{3}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+36 \left (a^{4} A +8 A \,a^{2} b^{2}+\frac {8}{3} A \,b^{4}+\frac {16}{3} B \,a^{3} b +\frac {32}{3} B a \,b^{3}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 B \,b^{4} d x \cos \left (2 d x +2 c \right )+24 B \,b^{4} d x \cos \left (4 d x +4 c \right )+\left (256 A \,a^{3} b +192 A a \,b^{3}+64 B \,a^{4}+288 B \,a^{2} b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (64 A \,a^{3} b +96 A a \,b^{3}+16 B \,a^{4}+144 B \,a^{2} b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (18 a^{4} A +144 A \,a^{2} b^{2}+96 B \,a^{3} b \right ) \sin \left (3 d x +3 c \right )+\left (66 a^{4} A +144 A \,a^{2} b^{2}+96 B \,a^{3} b \right ) \sin \left (d x +c \right )+72 B \,b^{4} d x}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(358\) |
| risch | \(b^{4} B x -\frac {i a \left (72 A a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+48 B \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-144 B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}-48 B \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+48 B \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-192 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-16 B \,a^{3}-64 A \,a^{2} b -144 B a \,b^{2}-96 A \,b^{3}-64 B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+33 A \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-288 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-33 A \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-9 A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+9 A \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-96 A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-288 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-48 B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-432 B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-256 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-432 B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}-48 B \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {3 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{4}}{d}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{3}}{d}-\frac {3 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{4}}{d}-\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{3}}{d}\) | \(627\) |
a^4*A/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+ tan(d*x+c)))+(A*b^4+4*B*a*b^3)/d*ln(sec(d*x+c)+tan(d*x+c))+(4*A*a*b^3+6*B* a^2*b^2)/d*tan(d*x+c)+(6*A*a^2*b^2+4*B*a^3*b)/d*(1/2*sec(d*x+c)*tan(d*x+c) +1/2*ln(sec(d*x+c)+tan(d*x+c)))-(4*A*a^3*b+B*a^4)/d*(-2/3-1/3*sec(d*x+c)^2 )*tan(d*x+c)+B*b^4/d*(d*x+c)
Time = 0.31 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.16 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {48 \, B b^{4} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{4} + 16 \, {\left (B a^{4} + 4 \, A a^{3} b + 9 \, B a^{2} b^{2} + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
1/48*(48*B*b^4*d*x*cos(d*x + c)^4 + 3*(3*A*a^4 + 16*B*a^3*b + 24*A*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(3*A*a^4 + 16*B*a^3*b + 24*A*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*cos(d*x + c)^4*log(-s in(d*x + c) + 1) + 2*(6*A*a^4 + 16*(B*a^4 + 4*A*a^3*b + 9*B*a^2*b^2 + 6*A* a*b^3)*cos(d*x + c)^3 + 3*(3*A*a^4 + 16*B*a^3*b + 24*A*a^2*b^2)*cos(d*x + c)^2 + 8*(B*a^4 + 4*A*a^3*b)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 )
Timed out. \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.47 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} b + 48 \, {\left (d x + c\right )} B b^{4} - 3 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 48 \, B a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 96 \, B a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 288 \, B a^{2} b^{2} \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \]
1/48*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^4 + 64*(tan(d*x + c)^3 + 3* tan(d*x + c))*A*a^3*b + 48*(d*x + c)*B*b^4 - 3*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)) - 48*B*a^3*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 72*A*a^2*b ^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin( d*x + c) - 1)) + 96*B*a*b^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1) ) + 24*A*b^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 288*B*a^2*b ^2*tan(d*x + c) + 192*A*a*b^3*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (206) = 412\).
Time = 0.35 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.94 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {24 \, {\left (d x + c\right )} B b^{4} + 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 432 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 432 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a b^{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 )}^{4}}}{24 \, d} \]
1/24*(24*(d*x + c)*B*b^4 + 3*(3*A*a^4 + 16*B*a^3*b + 24*A*a^2*b^2 + 32*B*a *b^3 + 8*A*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(3*A*a^4 + 16*B*a^3 *b + 24*A*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1 )) + 2*(15*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 24*B*a^4*tan(1/2*d*x + 1/2*c)^7 - 96*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 48*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 144*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 96*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 9*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 40 *B*a^4*tan(1/2*d*x + 1/2*c)^5 + 160*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 48*B* a^3*b*tan(1/2*d*x + 1/2*c)^5 - 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 432*B *a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*A *a^4*tan(1/2*d*x + 1/2*c)^3 - 40*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 160*A*a^3* b*tan(1/2*d*x + 1/2*c)^3 - 48*B*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 72*A*a^2*b^ 2*tan(1/2*d*x + 1/2*c)^3 - 432*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 288*A*a* b^3*tan(1/2*d*x + 1/2*c)^3 + 15*A*a^4*tan(1/2*d*x + 1/2*c) + 24*B*a^4*tan( 1/2*d*x + 1/2*c) + 96*A*a^3*b*tan(1/2*d*x + 1/2*c) + 48*B*a^3*b*tan(1/2*d* x + 1/2*c) + 72*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 144*B*a^2*b^2*tan(1/2*d*x + 1/2*c) + 96*A*a*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^ 4)/d
Time = 3.17 (sec) , antiderivative size = 1969, normalized size of antiderivative = 9.12 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\text {Too large to display} \]
((27*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/8 + 9*A*b^4*atanh (sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + (9*A*a^4*sin(3*c + 3*d*x))/8 + 4 *B*a^4*sin(2*c + 2*d*x) + B*a^4*sin(4*c + 4*d*x) + 9*B*b^4*atan((9*A^2*a^8 *sin(c/2 + (d*x)/2) + 64*A^2*b^8*sin(c/2 + (d*x)/2) + 64*B^2*b^8*sin(c/2 + (d*x)/2) + 384*A^2*a^2*b^6*sin(c/2 + (d*x)/2) + 624*A^2*a^4*b^4*sin(c/2 + (d*x)/2) + 144*A^2*a^6*b^2*sin(c/2 + (d*x)/2) + 1024*B^2*a^2*b^6*sin(c/2 + (d*x)/2) + 1024*B^2*a^4*b^4*sin(c/2 + (d*x)/2) + 256*B^2*a^6*b^2*sin(c/2 + (d*x)/2) + 1792*A*B*a^3*b^5*sin(c/2 + (d*x)/2) + 960*A*B*a^5*b^3*sin(c/ 2 + (d*x)/2) + 512*A*B*a*b^7*sin(c/2 + (d*x)/2) + 96*A*B*a^7*b*sin(c/2 + ( d*x)/2))/(cos(c/2 + (d*x)/2)*(9*A^2*a^8 + 64*A^2*b^8 + 64*B^2*b^8 + 384*A^ 2*a^2*b^6 + 624*A^2*a^4*b^4 + 144*A^2*a^6*b^2 + 1024*B^2*a^2*b^6 + 1024*B^ 2*a^4*b^4 + 256*B^2*a^6*b^2 + 512*A*B*a*b^7 + 96*A*B*a^7*b + 1792*A*B*a^3* b^5 + 960*A*B*a^5*b^3))) + (33*A*a^4*sin(c + d*x))/8 + 12*B*b^4*cos(2*c + 2*d*x)*atan((9*A^2*a^8*sin(c/2 + (d*x)/2) + 64*A^2*b^8*sin(c/2 + (d*x)/2) + 64*B^2*b^8*sin(c/2 + (d*x)/2) + 384*A^2*a^2*b^6*sin(c/2 + (d*x)/2) + 624 *A^2*a^4*b^4*sin(c/2 + (d*x)/2) + 144*A^2*a^6*b^2*sin(c/2 + (d*x)/2) + 102 4*B^2*a^2*b^6*sin(c/2 + (d*x)/2) + 1024*B^2*a^4*b^4*sin(c/2 + (d*x)/2) + 2 56*B^2*a^6*b^2*sin(c/2 + (d*x)/2) + 1792*A*B*a^3*b^5*sin(c/2 + (d*x)/2) + 960*A*B*a^5*b^3*sin(c/2 + (d*x)/2) + 512*A*B*a*b^7*sin(c/2 + (d*x)/2) + 96 *A*B*a^7*b*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(9*A^2*a^8 + 64*A^2*...