Integrand size = 25, antiderivative size = 148 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{8} a^3 (20 A+13 C) x+\frac {a^3 (20 A+13 C) \sin (c+d x)}{5 d}+\frac {3 a^3 (20 A+13 C) \cos (c+d x) \sin (c+d x)}{40 d}-\frac {C (a+a \cos (c+d x))^3 \sin (c+d x)}{20 d}+\frac {C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 a d}-\frac {a^3 (20 A+13 C) \sin ^3(c+d x)}{60 d} \]
1/8*a^3*(20*A+13*C)*x+1/5*a^3*(20*A+13*C)*sin(d*x+c)/d+3/40*a^3*(20*A+13*C )*cos(d*x+c)*sin(d*x+c)/d-1/20*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)/a/d-1/60*a^3*(20*A+13*C)*sin(d*x+c)^3/d
Time = 0.43 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.66 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^3 (1200 A d x+780 C d x+60 (30 A+23 C) \sin (c+d x)+120 (3 A+4 C) \sin (2 (c+d x))+40 A \sin (3 (c+d x))+170 C \sin (3 (c+d x))+45 C \sin (4 (c+d x))+6 C \sin (5 (c+d x)))}{480 d} \]
(a^3*(1200*A*d*x + 780*C*d*x + 60*(30*A + 23*C)*Sin[c + d*x] + 120*(3*A + 4*C)*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 170*C*Sin[3*(c + d*x)] + 4 5*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d)
Time = 0.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3503, 3042, 3230, 3042, 3124, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a \cos (c+d x)+a)^3 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3503 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a)^3 (a (5 A+4 C)-a C \cos (c+d x))dx}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (5 A+4 C)-a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3230 |
| \(\displaystyle \frac {\frac {1}{4} a (20 A+13 C) \int (\cos (c+d x) a+a)^3dx-\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} a (20 A+13 C) \int \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3dx-\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 3124 |
| \(\displaystyle \frac {\frac {1}{4} a (20 A+13 C) \int \left (\cos ^3(c+d x) a^3+3 \cos ^2(c+d x) a^3+3 \cos (c+d x) a^3+a^3\right )dx-\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{4} a (20 A+13 C) \left (-\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {5 a^3 x}{2}\right )-\frac {a C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}}{5 a}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}\) |
(C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*a*d) + (-1/4*(a*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/d + (a*(20*A + 13*C)*((5*a^3*x)/2 + (4*a^3*Sin[c + d*x])/d + (3*a^3*Cos[c + d*x]*Sin[c + d*x])/(2*d) - (a^3*Sin[c + d*x]^3)/ (3*d)))/4)/(5*a)
3.1.20.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTri g[(a + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(b*(m + 1)) Int[(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[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) - a*C*Sin[e + f*x], x], x], x] /; FreeQ[{a , b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 6.54 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.61
| method | result | size |
| parallelrisch | \(\frac {5 \left (\frac {\left (\frac {3 A}{2}+2 C \right ) \sin \left (2 d x +2 c \right )}{5}+\frac {\left (A +\frac {17 C}{4}\right ) \sin \left (3 d x +3 c \right )}{30}+\frac {3 \sin \left (4 d x +4 c \right ) C}{80}+\frac {\sin \left (5 d x +5 c \right ) C}{200}+\frac {\left (3 A +\frac {23 C}{10}\right ) \sin \left (d x +c \right )}{2}+d x \left (A +\frac {13 C}{20}\right )\right ) a^{3}}{2 d}\) | \(91\) |
| risch | \(\frac {5 a^{3} A x}{2}+\frac {13 a^{3} C x}{8}+\frac {15 a^{3} A \sin \left (d x +c \right )}{4 d}+\frac {23 a^{3} C \sin \left (d x +c \right )}{8 d}+\frac {\sin \left (5 d x +5 c \right ) C \,a^{3}}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) C \,a^{3}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{3}}{12 d}+\frac {17 \sin \left (3 d x +3 c \right ) C \,a^{3}}{48 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{3}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C \,a^{3}}{d}\) | \(153\) |
| parts | \(a^{3} A x +\frac {\left (A \,a^{3}+3 C \,a^{3}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (3 A \,a^{3}+C \,a^{3}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {3 a^{3} A \sin \left (d x +c \right )}{d}+\frac {3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(171\) |
| derivativedivides | \(\frac {\frac {A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 A \,a^{3} \sin \left (d x +c \right )+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,a^{3} \left (d x +c \right )+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(197\) |
| default | \(\frac {\frac {A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {C \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 C \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 A \,a^{3} \sin \left (d x +c \right )+C \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,a^{3} \left (d x +c \right )+C \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(197\) |
| norman | \(\frac {\frac {a^{3} \left (20 A +13 C \right ) x}{8}+\frac {32 a^{3} \left (20 A +13 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {7 a^{3} \left (20 A +13 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {a^{3} \left (20 A +13 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {5 a^{3} \left (20 A +13 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {5 a^{3} \left (20 A +13 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a^{3} \left (20 A +13 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {5 a^{3} \left (20 A +13 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (20 A +13 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} \left (44 A +51 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (212 A +133 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(279\) |
5/2*(1/5*(3/2*A+2*C)*sin(2*d*x+2*c)+1/30*(A+17/4*C)*sin(3*d*x+3*c)+3/80*si n(4*d*x+4*c)*C+1/200*sin(5*d*x+5*c)*C+1/2*(3*A+23/10*C)*sin(d*x+c)+d*x*(A+ 13/20*C))*a^3/d
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.72 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (20 \, A + 13 \, C\right )} a^{3} d x + {\left (24 \, C a^{3} \cos \left (d x + c\right )^{4} + 90 \, C a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (12 \, A + 13 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (55 \, A + 38 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \]
1/120*(15*(20*A + 13*C)*a^3*d*x + (24*C*a^3*cos(d*x + c)^4 + 90*C*a^3*cos( d*x + c)^3 + 8*(5*A + 19*C)*a^3*cos(d*x + c)^2 + 15*(12*A + 13*C)*a^3*cos( d*x + c) + 8*(55*A + 38*C)*a^3)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (133) = 266\).
Time = 0.30 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.85 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 A a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + A a^{3} x + \frac {2 A a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 A a^{3} \sin {\left (c + d x \right )}}{d} + \frac {9 C a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 C a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {C a^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {9 C a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {C a^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {8 C a^{3} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {9 C a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 C a^{3} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {15 C a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {3 C a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{3} & \text {otherwise} \end {cases} \]
Piecewise((3*A*a**3*x*sin(c + d*x)**2/2 + 3*A*a**3*x*cos(c + d*x)**2/2 + A *a**3*x + 2*A*a**3*sin(c + d*x)**3/(3*d) + A*a**3*sin(c + d*x)*cos(c + d*x )**2/d + 3*A*a**3*sin(c + d*x)*cos(c + d*x)/(2*d) + 3*A*a**3*sin(c + d*x)/ d + 9*C*a**3*x*sin(c + d*x)**4/8 + 9*C*a**3*x*sin(c + d*x)**2*cos(c + d*x) **2/4 + C*a**3*x*sin(c + d*x)**2/2 + 9*C*a**3*x*cos(c + d*x)**4/8 + C*a**3 *x*cos(c + d*x)**2/2 + 8*C*a**3*sin(c + d*x)**5/(15*d) + 4*C*a**3*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*C*a**3*sin(c + d*x)**3*cos(c + d*x)/(8*d ) + 2*C*a**3*sin(c + d*x)**3/d + C*a**3*sin(c + d*x)*cos(c + d*x)**4/d + 1 5*C*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 3*C*a**3*sin(c + d*x)*cos(c + d*x)**2/d + C*a**3*sin(c + d*x)*cos(c + d*x)/(2*d), Ne(d, 0)), (x*(A + C *cos(c)**2)*(a*cos(c) + a)**3, True))
Time = 0.22 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.28 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} - 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 480 \, {\left (d x + c\right )} A a^{3} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} + 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 1440 \, A a^{3} \sin \left (d x + c\right )}{480 \, d} \]
-1/480*(160*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 - 360*(2*d*x + 2*c + s in(2*d*x + 2*c))*A*a^3 - 480*(d*x + c)*A*a^3 - 32*(3*sin(d*x + c)^5 - 10*s in(d*x + c)^3 + 15*sin(d*x + c))*C*a^3 + 480*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 - 45*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C* a^3 - 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^3 - 1440*A*a^3*sin(d*x + c) )/d
Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.89 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{3} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {3 \, C a^{3} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {1}{8} \, {\left (20 \, A a^{3} + 13 \, C a^{3}\right )} x + \frac {{\left (4 \, A a^{3} + 17 \, C a^{3}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (3 \, A a^{3} + 4 \, C a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (30 \, A a^{3} + 23 \, C a^{3}\right )} \sin \left (d x + c\right )}{8 \, d} \]
1/80*C*a^3*sin(5*d*x + 5*c)/d + 3/32*C*a^3*sin(4*d*x + 4*c)/d + 1/8*(20*A* a^3 + 13*C*a^3)*x + 1/48*(4*A*a^3 + 17*C*a^3)*sin(3*d*x + 3*c)/d + 1/4*(3* A*a^3 + 4*C*a^3)*sin(2*d*x + 2*c)/d + 1/8*(30*A*a^3 + 23*C*a^3)*sin(d*x + c)/d
Time = 2.23 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.87 \[ \int (a+a \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (5\,A\,a^3+\frac {13\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {70\,A\,a^3}{3}+\frac {91\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {128\,A\,a^3}{3}+\frac {416\,C\,a^3}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {106\,A\,a^3}{3}+\frac {133\,C\,a^3}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (11\,A\,a^3+\frac {51\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^3\,\left (20\,A+13\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (20\,A+13\,C\right )}{4\,\left (5\,A\,a^3+\frac {13\,C\,a^3}{4}\right )}\right )\,\left (20\,A+13\,C\right )}{4\,d} \]
(tan(c/2 + (d*x)/2)*(11*A*a^3 + (51*C*a^3)/4) + tan(c/2 + (d*x)/2)^9*(5*A* a^3 + (13*C*a^3)/4) + tan(c/2 + (d*x)/2)^7*((70*A*a^3)/3 + (91*C*a^3)/6) + tan(c/2 + (d*x)/2)^3*((106*A*a^3)/3 + (133*C*a^3)/6) + tan(c/2 + (d*x)/2) ^5*((128*A*a^3)/3 + (416*C*a^3)/15))/(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^3*(20*A + 13*C)*(atan(tan(c/2 + (d*x)/2)) - (d *x)/2))/(4*d) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(20*A + 13*C))/(4*(5*A*a ^3 + (13*C*a^3)/4)))*(20*A + 13*C))/(4*d)