Integrand size = 39, antiderivative size = 195 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {1}{8} a^4 (48 A+35 B+28 C) x+\frac {a^4 A \text {arctanh}(\sin (c+d x))}{d}+\frac {a^4 (40 A+35 B+28 C) \sin (c+d x)}{8 d}+\frac {a (5 B+4 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {(20 A+35 B+28 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{60 d}+\frac {(32 A+35 B+28 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d} \]
1/8*a^4*(48*A+35*B+28*C)*x+a^4*A*arctanh(sin(d*x+c))/d+1/8*a^4*(40*A+35*B+ 28*C)*sin(d*x+c)/d+1/20*a*(5*B+4*C)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+1/5*C* (a+a*cos(d*x+c))^4*sin(d*x+c)/d+1/60*(20*A+35*B+28*C)*(a^2+a^2*cos(d*x+c)) ^2*sin(d*x+c)/d+1/24*(32*A+35*B+28*C)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d
Time = 4.30 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.93 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {a^4 \left (2880 A d x+2100 B d x+1680 C d x-480 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 (54 A+56 B+49 C) \sin (c+d x)+120 (4 A+7 B+8 C) \sin (2 (c+d x))+40 A \sin (3 (c+d x))+160 B \sin (3 (c+d x))+290 C \sin (3 (c+d x))+15 B \sin (4 (c+d x))+60 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))\right )}{480 d} \]
(a^4*(2880*A*d*x + 2100*B*d*x + 1680*C*d*x - 480*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*(5 4*A + 56*B + 49*C)*Sin[c + d*x] + 120*(4*A + 7*B + 8*C)*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 160*B*Sin[3*(c + d*x)] + 290*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 60*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(4 80*d)
Time = 1.57 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.10, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3042, 3524, 3042, 3455, 3042, 3455, 27, 3042, 3455, 27, 3042, 3447, 3042, 3502, 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.
| \(\displaystyle \int \sec (c+d x) (a \cos (c+d x)+a)^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (5 a A+a (5 B+4 C) \cos (c+d x)) \sec (c+d x)dx}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (5 a A+a (5 B+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{4} \int (\cos (c+d x) a+a)^3 \left (20 A a^2+(20 A+35 B+28 C) \cos (c+d x) a^2\right ) \sec (c+d x)dx+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (20 A a^2+(20 A+35 B+28 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \int 5 (\cos (c+d x) a+a)^2 \left (12 A a^3+(32 A+35 B+28 C) \cos (c+d x) a^3\right ) \sec (c+d x)dx+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \int (\cos (c+d x) a+a)^2 \left (12 A a^3+(32 A+35 B+28 C) \cos (c+d x) a^3\right ) \sec (c+d x)dx+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (12 A a^3+(32 A+35 B+28 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \left (\frac {1}{2} \int 3 (\cos (c+d x) a+a) \left (8 A a^4+(40 A+35 B+28 C) \cos (c+d x) a^4\right ) \sec (c+d x)dx+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \left (\frac {3}{2} \int (\cos (c+d x) a+a) \left (8 A a^4+(40 A+35 B+28 C) \cos (c+d x) a^4\right ) \sec (c+d x)dx+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \left (\frac {3}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (8 A a^4+(40 A+35 B+28 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \left (\frac {3}{2} \int \left ((40 A+35 B+28 C) \cos ^2(c+d x) a^5+8 A a^5+\left (8 A a^5+(40 A+35 B+28 C) a^5\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \left (\frac {3}{2} \int \frac {(40 A+35 B+28 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+8 A a^5+\left (8 A a^5+(40 A+35 B+28 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \left (8 A a^5+(48 A+35 B+28 C) \cos (c+d x) a^5\right ) \sec (c+d x)dx+\frac {a^5 (40 A+35 B+28 C) \sin (c+d x)}{d}\right )+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \frac {8 A a^5+(48 A+35 B+28 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a^5 (40 A+35 B+28 C) \sin (c+d x)}{d}\right )+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (8 a^5 A \int \sec (c+d x)dx+\frac {a^5 (40 A+35 B+28 C) \sin (c+d x)}{d}+a^5 x (48 A+35 B+28 C)\right )+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (8 a^5 A \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {a^5 (40 A+35 B+28 C) \sin (c+d x)}{d}+a^5 x (48 A+35 B+28 C)\right )+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )+\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {a^2 (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac {1}{4} \left (\frac {5}{3} \left (\frac {3}{2} \left (\frac {8 a^5 A \text {arctanh}(\sin (c+d x))}{d}+\frac {a^5 (40 A+35 B+28 C) \sin (c+d x)}{d}+a^5 x (48 A+35 B+28 C)\right )+\frac {(32 A+35 B+28 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{2 d}\right )+\frac {a^3 (20 A+35 B+28 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}\right )}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}\) |
(C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((a^2*(5*B + 4*C)*(a + a*C os[c + d*x])^3*Sin[c + d*x])/(4*d) + ((a^3*(20*A + 35*B + 28*C)*(a + a*Cos [c + d*x])^2*Sin[c + d*x])/(3*d) + (5*(((32*A + 35*B + 28*C)*(a^5 + a^5*Co s[c + d*x])*Sin[c + d*x])/(2*d) + (3*(a^5*(48*A + 35*B + 28*C)*x + (8*a^5* A*ArcTanh[Sin[c + d*x]])/d + (a^5*(40*A + 35*B + 28*C)*Sin[c + d*x])/d))/2 ))/3)/4)/(5*a)
3.4.31.3.1 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_)]), 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)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, 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[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !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)*Cos[e + f*x]*(a + b*Sin[e + f*x] )^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 7.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.70
| method | result | size |
| parallelrisch | \(\frac {a^{4} \left (-12 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+12 A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (12 A +21 B +24 C \right ) \sin \left (2 d x +2 c \right )+\left (A +4 B +\frac {29 C}{4}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {3 B}{8}+\frac {3 C}{2}\right ) \sin \left (4 d x +4 c \right )+\frac {3 \sin \left (5 d x +5 c \right ) C}{20}+\left (81 A +84 B +\frac {147 C}{2}\right ) \sin \left (d x +c \right )+72 x \left (A +\frac {35 B}{48}+\frac {7 C}{12}\right ) d \right )}{12 d}\) | \(136\) |
| parts | \(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (d x +c \right )}{d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(241\) |
| derivativedivides | \(\frac {\frac {a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} A \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \sin \left (d x +c \right )+4 C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \left (d x +c \right )+C \,a^{4} \sin \left (d x +c \right )}{d}\) | \(340\) |
| default | \(\frac {\frac {a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} A \sin \left (d x +c \right )+6 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 C \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{4} A \left (d x +c \right )+4 B \,a^{4} \sin \left (d x +c \right )+4 C \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{4} \left (d x +c \right )+C \,a^{4} \sin \left (d x +c \right )}{d}\) | \(340\) |
| risch | \(6 a^{4} x A +\frac {35 a^{4} B x}{8}+\frac {7 a^{4} C x}{2}-\frac {27 i {\mathrm e}^{i \left (d x +c \right )} a^{4} A}{8 d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{4}}{2 d}-\frac {49 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{4}}{16 d}+\frac {49 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{4}}{16 d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{4}}{2 d}+\frac {27 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} A}{8 d}+\frac {a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (5 d x +5 c \right ) C \,a^{4}}{80 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{32 d}+\frac {\sin \left (4 d x +4 c \right ) C \,a^{4}}{8 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} A}{12 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{4}}{3 d}+\frac {29 \sin \left (3 d x +3 c \right ) C \,a^{4}}{48 d}+\frac {\sin \left (2 d x +2 c \right ) a^{4} A}{d}+\frac {7 \sin \left (2 d x +2 c \right ) B \,a^{4}}{4 d}+\frac {2 \sin \left (2 d x +2 c \right ) C \,a^{4}}{d}\) | \(341\) |
| norman | \(\frac {\left (6 a^{4} A +\frac {35}{8} B \,a^{4}+\frac {7}{2} C \,a^{4}\right ) x +\left (6 a^{4} A +\frac {35}{8} B \,a^{4}+\frac {7}{2} C \,a^{4}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (36 a^{4} A +\frac {105}{4} B \,a^{4}+21 C \,a^{4}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (36 a^{4} A +\frac {105}{4} B \,a^{4}+21 C \,a^{4}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (90 a^{4} A +\frac {525}{8} B \,a^{4}+\frac {105}{2} C \,a^{4}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (90 a^{4} A +\frac {525}{8} B \,a^{4}+\frac {105}{2} C \,a^{4}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (120 a^{4} A +\frac {175}{2} B \,a^{4}+70 C \,a^{4}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {a^{4} \left (40 A +35 B +28 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {a^{4} \left (72 A +93 B +100 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{4} \left (664 A +595 B +476 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{4} \left (952 A +1069 B +932 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {a^{4} \left (1240 A +1155 B +924 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 d}+\frac {a^{4} \left (1400 A +1405 B +1124 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{10 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {a^{4} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {a^{4} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\) | \(445\) |
1/12*a^4*(-12*A*ln(tan(1/2*d*x+1/2*c)-1)+12*A*ln(tan(1/2*d*x+1/2*c)+1)+(12 *A+21*B+24*C)*sin(2*d*x+2*c)+(A+4*B+29/4*C)*sin(3*d*x+3*c)+(3/8*B+3/2*C)*s in(4*d*x+4*c)+3/20*sin(5*d*x+5*c)*C+(81*A+84*B+147/2*C)*sin(d*x+c)+72*x*(A +35/48*B+7/12*C)*d)/d
Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.79 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {15 \, {\left (48 \, A + 35 \, B + 28 \, C\right )} a^{4} d x + 60 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (24 \, C a^{4} \cos \left (d x + c\right )^{4} + 30 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 20 \, B + 34 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (16 \, A + 27 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right ) + 8 \, {\left (100 \, A + 100 \, B + 83 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{120 \, d} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x, algorithm="fricas")
1/120*(15*(48*A + 35*B + 28*C)*a^4*d*x + 60*A*a^4*log(sin(d*x + c) + 1) - 60*A*a^4*log(-sin(d*x + c) + 1) + (24*C*a^4*cos(d*x + c)^4 + 30*(B + 4*C)* a^4*cos(d*x + c)^3 + 8*(5*A + 20*B + 34*C)*a^4*cos(d*x + c)^2 + 15*(16*A + 27*B + 28*C)*a^4*cos(d*x + c) + 8*(100*A + 100*B + 83*C)*a^4)*sin(d*x + c ))/d
\[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=a^{4} \left (\int A \sec {\left (c + d x \right )}\, dx + \int 4 A \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 A \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 A \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int A \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos {\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 B \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 B \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int B \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 C \cos ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 6 C \cos ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int 4 C \cos ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int C \cos ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx\right ) \]
a**4*(Integral(A*sec(c + d*x), x) + Integral(4*A*cos(c + d*x)*sec(c + d*x) , x) + Integral(6*A*cos(c + d*x)**2*sec(c + d*x), x) + Integral(4*A*cos(c + d*x)**3*sec(c + d*x), x) + Integral(A*cos(c + d*x)**4*sec(c + d*x), x) + Integral(B*cos(c + d*x)*sec(c + d*x), x) + Integral(4*B*cos(c + d*x)**2*s ec(c + d*x), x) + Integral(6*B*cos(c + d*x)**3*sec(c + d*x), x) + Integral (4*B*cos(c + d*x)**4*sec(c + d*x), x) + Integral(B*cos(c + d*x)**5*sec(c + d*x), x) + Integral(C*cos(c + d*x)**2*sec(c + d*x), x) + Integral(4*C*cos (c + d*x)**3*sec(c + d*x), x) + Integral(6*C*cos(c + d*x)**4*sec(c + d*x), x) + Integral(4*C*cos(c + d*x)**5*sec(c + d*x), x) + Integral(C*cos(c + d *x)**6*sec(c + d*x), x))
Time = 0.23 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.67 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=-\frac {160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1920 \, {\left (d x + c\right )} A a^{4} + 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 480 \, {\left (d x + c\right )} B a^{4} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 480 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 2880 \, A a^{4} \sin \left (d x + c\right ) - 1920 \, B a^{4} \sin \left (d x + c\right ) - 480 \, C a^{4} \sin \left (d x + c\right )}{480 \, d} \]
integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x, algorithm="maxima")
-1/480*(160*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 480*(2*d*x + 2*c + s in(2*d*x + 2*c))*A*a^4 - 1920*(d*x + c)*A*a^4 + 640*(sin(d*x + c)^3 - 3*si n(d*x + c))*B*a^4 - 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2 *c))*B*a^4 - 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 480*(d*x + c)*B* a^4 - 32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 + 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 - 60*(12*d*x + 12*c + sin(4*d* x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 - 480*(2*d*x + 2*c + sin(2*d*x + 2*c) )*C*a^4 - 480*A*a^4*log(sec(d*x + c) + tan(d*x + c)) - 2880*A*a^4*sin(d*x + c) - 1920*B*a^4*sin(d*x + c) - 480*C*a^4*sin(d*x + c))/d
Time = 0.36 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.73 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {120 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 120 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 15 \, {\left (48 \, A a^{4} + 35 \, B a^{4} + 28 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (600 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 525 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 420 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2720 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2450 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1960 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4720 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4480 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3584 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3680 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3950 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3160 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1080 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1395 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1500 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
1/120*(120*A*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 120*A*a^4*log(abs(ta n(1/2*d*x + 1/2*c) - 1)) + 15*(48*A*a^4 + 35*B*a^4 + 28*C*a^4)*(d*x + c) + 2*(600*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 525*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 420*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 2720*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 245 0*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 1960*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 4720* A*a^4*tan(1/2*d*x + 1/2*c)^5 + 4480*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 3584*C* a^4*tan(1/2*d*x + 1/2*c)^5 + 3680*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 3950*B*a^ 4*tan(1/2*d*x + 1/2*c)^3 + 3160*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 1080*A*a^4* tan(1/2*d*x + 1/2*c) + 1395*B*a^4*tan(1/2*d*x + 1/2*c) + 1500*C*a^4*tan(1/ 2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d
Time = 2.98 (sec) , antiderivative size = 1151, normalized size of antiderivative = 5.90 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Too large to display} \]
(tan(c/2 + (d*x)/2)^9*(10*A*a^4 + (35*B*a^4)/4 + 7*C*a^4) + tan(c/2 + (d*x )/2)^7*((136*A*a^4)/3 + (245*B*a^4)/6 + (98*C*a^4)/3) + tan(c/2 + (d*x)/2) ^3*((184*A*a^4)/3 + (395*B*a^4)/6 + (158*C*a^4)/3) + tan(c/2 + (d*x)/2)^5* ((236*A*a^4)/3 + (224*B*a^4)/3 + (896*C*a^4)/15) + tan(c/2 + (d*x)/2)*(18* A*a^4 + (93*B*a^4)/4 + 25*C*a^4))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) - (A*a^4*atan((A*a^4*(tan(c/2 + (d*x)/2)*(1184*A^2*a^8 + (1225*B^2*a^8)/2 + 392*C^2*a^8 + 1680*A*B*a^8 + 1344*A*C*a^8 + 980*B*C* a^8) + A*a^4*(224*A*a^4 + 140*B*a^4 + 112*C*a^4))*1i + A*a^4*(tan(c/2 + (d *x)/2)*(1184*A^2*a^8 + (1225*B^2*a^8)/2 + 392*C^2*a^8 + 1680*A*B*a^8 + 134 4*A*C*a^8 + 980*B*C*a^8) - A*a^4*(224*A*a^4 + 140*B*a^4 + 112*C*a^4))*1i)/ (1920*A^3*a^12 + 1225*A*B^2*a^12 + 3080*A^2*B*a^12 + 784*A*C^2*a^12 + 2464 *A^2*C*a^12 + A*a^4*(tan(c/2 + (d*x)/2)*(1184*A^2*a^8 + (1225*B^2*a^8)/2 + 392*C^2*a^8 + 1680*A*B*a^8 + 1344*A*C*a^8 + 980*B*C*a^8) + A*a^4*(224*A*a ^4 + 140*B*a^4 + 112*C*a^4)) - A*a^4*(tan(c/2 + (d*x)/2)*(1184*A^2*a^8 + ( 1225*B^2*a^8)/2 + 392*C^2*a^8 + 1680*A*B*a^8 + 1344*A*C*a^8 + 980*B*C*a^8) - A*a^4*(224*A*a^4 + 140*B*a^4 + 112*C*a^4)) + 1960*A*B*C*a^12))*2i)/d - (a^4*atan(((a^4*(tan(c/2 + (d*x)/2)*(1184*A^2*a^8 + (1225*B^2*a^8)/2 + 392 *C^2*a^8 + 1680*A*B*a^8 + 1344*A*C*a^8 + 980*B*C*a^8) - (a^4*(48*A + 35*B + 28*C)*(224*A*a^4 + 140*B*a^4 + 112*C*a^4)*1i)/8)*(48*A + 35*B + 28*C)...