Integrand size = 34, antiderivative size = 98 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {1}{8} a^2 (4 A+B) c x-\frac {a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}+\frac {a^2 (4 A+B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f} \]
1/8*a^2*(4*A+B)*c*x-1/12*a^2*(4*A+B)*c*cos(f*x+e)^3/f+1/8*a^2*(4*A+B)*c*co s(f*x+e)*sin(f*x+e)/f-1/4*B*c*cos(f*x+e)^3*(a^2+a^2*sin(f*x+e))/f
Time = 0.70 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.17 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {a^2 c \cos (e+f x) \left (12 (4 A+B) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} (8 A+8 B+8 (A+B) \cos (2 (e+f x))-3 (8 A+B) \sin (e+f x)+3 B \sin (3 (e+f x)))\right )}{48 f \sqrt {\cos ^2(e+f x)}} \]
-1/48*(a^2*c*Cos[e + f*x]*(12*(4*A + B)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt [2]] + Sqrt[Cos[e + f*x]^2]*(8*A + 8*B + 8*(A + B)*Cos[2*(e + f*x)] - 3*(8 *A + B)*Sin[e + f*x] + 3*B*Sin[3*(e + f*x)])))/(f*Sqrt[Cos[e + f*x]^2])
Time = 0.48 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {3042, 3446, 3042, 3339, 3042, 3148, 3042, 3115, 24}
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.
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c-c \sin (e+f x)) (A+B \sin (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \sin (e+f x)+a)^2 (c-c \sin (e+f x)) (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3446 |
| \(\displaystyle a c \int \cos ^2(e+f x) (\sin (e+f x) a+a) (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \int \cos (e+f x)^2 (\sin (e+f x) a+a) (A+B \sin (e+f x))dx\) |
| \(\Big \downarrow \) 3339 |
| \(\displaystyle a c \left (\frac {1}{4} (4 A+B) \int \cos ^2(e+f x) (\sin (e+f x) a+a)dx-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)}{4 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {1}{4} (4 A+B) \int \cos (e+f x)^2 (\sin (e+f x) a+a)dx-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)}{4 f}\right )\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle a c \left (\frac {1}{4} (4 A+B) \left (a \int \cos ^2(e+f x)dx-\frac {a \cos ^3(e+f x)}{3 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)}{4 f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a c \left (\frac {1}{4} (4 A+B) \left (a \int \sin \left (e+f x+\frac {\pi }{2}\right )^2dx-\frac {a \cos ^3(e+f x)}{3 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)}{4 f}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a c \left (\frac {1}{4} (4 A+B) \left (a \left (\frac {\int 1dx}{2}+\frac {\sin (e+f x) \cos (e+f x)}{2 f}\right )-\frac {a \cos ^3(e+f x)}{3 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)}{4 f}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a c \left (\frac {1}{4} (4 A+B) \left (a \left (\frac {\sin (e+f x) \cos (e+f x)}{2 f}+\frac {x}{2}\right )-\frac {a \cos ^3(e+f x)}{3 f}\right )-\frac {B \cos ^3(e+f x) (a \sin (e+f x)+a)}{4 f}\right )\) |
a*c*(-1/4*(B*Cos[e + f*x]^3*(a + a*Sin[e + f*x]))/f + ((4*A + B)*(-1/3*(a* Cos[e + f*x]^3)/f + a*(x/2 + (Cos[e + f*x]*Sin[e + f*x])/(2*f))))/4)
3.1.30.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[ a^2 - b^2, 0] && NeQ[m + p + 1, 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[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B*Sin [e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a* d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 1.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.84
| method | result | size |
| parallelrisch | \(\frac {c \left (\frac {\left (-A -B \right ) \cos \left (3 f x +3 e \right )}{3}+A \sin \left (2 f x +2 e \right )-\frac {B \sin \left (4 f x +4 e \right )}{8}+\left (-A -B \right ) \cos \left (f x +e \right )+2 f x A +\frac {f x B}{2}-\frac {4 A}{3}-\frac {4 B}{3}\right ) a^{2}}{4 f}\) | \(82\) |
| risch | \(\frac {a^{2} c x A}{2}+\frac {a^{2} c x B}{8}-\frac {a^{2} c \cos \left (f x +e \right ) A}{4 f}-\frac {a^{2} c \cos \left (f x +e \right ) B}{4 f}-\frac {B \,a^{2} c \sin \left (4 f x +4 e \right )}{32 f}-\frac {a^{2} c \cos \left (3 f x +3 e \right ) A}{12 f}-\frac {a^{2} c \cos \left (3 f x +3 e \right ) B}{12 f}+\frac {A \,a^{2} c \sin \left (2 f x +2 e \right )}{4 f}\) | \(126\) |
| parts | \(-\frac {\left (-A \,a^{2} c -B \,a^{2} c \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (-A \,a^{2} c +B \,a^{2} c \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (A \,a^{2} c +B \,a^{2} c \right ) \cos \left (f x +e \right )}{f}+a^{2} c x A -\frac {B \,a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(152\) |
| derivativedivides | \(\frac {\frac {A \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {B \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \cos \left (f x +e \right ) a^{2} c +B \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{2} c \left (f x +e \right )-B \cos \left (f x +e \right ) a^{2} c}{f}\) | \(186\) |
| default | \(\frac {\frac {A \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {B \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A \cos \left (f x +e \right ) a^{2} c +B \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{2} c \left (f x +e \right )-B \cos \left (f x +e \right ) a^{2} c}{f}\) | \(186\) |
| norman | \(\frac {\left (\frac {1}{2} A \,a^{2} c +\frac {1}{8} B \,a^{2} c \right ) x +\left (2 A \,a^{2} c +\frac {1}{2} B \,a^{2} c \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (2 A \,a^{2} c +\frac {1}{2} B \,a^{2} c \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3 A \,a^{2} c +\frac {3}{4} B \,a^{2} c \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {1}{2} A \,a^{2} c +\frac {1}{8} B \,a^{2} c \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {2 A \,a^{2} c +2 B \,a^{2} c}{3 f}-\frac {2 \left (A \,a^{2} c +B \,a^{2} c \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {2 \left (A \,a^{2} c +B \,a^{2} c \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (A \,a^{2} c +B \,a^{2} c \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {a^{2} c \left (4 A -B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {a^{2} c \left (4 A -B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a^{2} c \left (4 A +7 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{2} c \left (4 A +7 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\) | \(360\) |
1/4*c*(1/3*(-A-B)*cos(3*f*x+3*e)+A*sin(2*f*x+2*e)-1/8*B*sin(4*f*x+4*e)+(-A -B)*cos(f*x+e)+2*f*x*A+1/2*f*x*B-4/3*A-4/3*B)*a^2/f
Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {8 \, {\left (A + B\right )} a^{2} c \cos \left (f x + e\right )^{3} - 3 \, {\left (4 \, A + B\right )} a^{2} c f x + 3 \, {\left (2 \, B a^{2} c \cos \left (f x + e\right )^{3} - {\left (4 \, A + B\right )} a^{2} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
-1/24*(8*(A + B)*a^2*c*cos(f*x + e)^3 - 3*(4*A + B)*a^2*c*f*x + 3*(2*B*a^2 *c*cos(f*x + e)^3 - (4*A + B)*a^2*c*cos(f*x + e))*sin(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (90) = 180\).
Time = 0.20 (sec) , antiderivative size = 396, normalized size of antiderivative = 4.04 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\begin {cases} - \frac {A a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {A a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + A a^{2} c x + \frac {A a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {A a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 A a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {A a^{2} c \cos {\left (e + f x \right )}}{f} - \frac {3 B a^{2} c x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {3 B a^{2} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {B a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {3 B a^{2} c x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {B a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {5 B a^{2} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {B a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {3 B a^{2} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {B a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 B a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a^{2} c \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right ) & \text {otherwise} \end {cases} \]
Piecewise((-A*a**2*c*x*sin(e + f*x)**2/2 - A*a**2*c*x*cos(e + f*x)**2/2 + A*a**2*c*x + A*a**2*c*sin(e + f*x)**2*cos(e + f*x)/f + A*a**2*c*sin(e + f* x)*cos(e + f*x)/(2*f) + 2*A*a**2*c*cos(e + f*x)**3/(3*f) - A*a**2*c*cos(e + f*x)/f - 3*B*a**2*c*x*sin(e + f*x)**4/8 - 3*B*a**2*c*x*sin(e + f*x)**2*c os(e + f*x)**2/4 + B*a**2*c*x*sin(e + f*x)**2/2 - 3*B*a**2*c*x*cos(e + f*x )**4/8 + B*a**2*c*x*cos(e + f*x)**2/2 + 5*B*a**2*c*sin(e + f*x)**3*cos(e + f*x)/(8*f) + B*a**2*c*sin(e + f*x)**2*cos(e + f*x)/f + 3*B*a**2*c*sin(e + f*x)*cos(e + f*x)**3/(8*f) - B*a**2*c*sin(e + f*x)*cos(e + f*x)/(2*f) + 2 *B*a**2*c*cos(e + f*x)**3/(3*f) - B*a**2*c*cos(e + f*x)/f, Ne(f, 0)), (x*( A + B*sin(e))*(a*sin(e) + a)**2*(-c*sin(e) + c), True))
Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.83 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c + 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c - 96 \, {\left (f x + e\right )} A a^{2} c + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c - 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c + 96 \, A a^{2} c \cos \left (f x + e\right ) + 96 \, B a^{2} c \cos \left (f x + e\right )}{96 \, f} \]
-1/96*(32*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^2*c + 24*(2*f*x + 2*e - si n(2*f*x + 2*e))*A*a^2*c - 96*(f*x + e)*A*a^2*c + 32*(cos(f*x + e)^3 - 3*co s(f*x + e))*B*a^2*c + 3*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^2*c - 24*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^2*c + 96*A*a^2*c*c os(f*x + e) + 96*B*a^2*c*cos(f*x + e))/f
Time = 0.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {B a^{2} c \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {A a^{2} c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (4 \, A a^{2} c + B a^{2} c\right )} x - \frac {{\left (A a^{2} c + B a^{2} c\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {{\left (A a^{2} c + B a^{2} c\right )} \cos \left (f x + e\right )}{4 \, f} \]
-1/32*B*a^2*c*sin(4*f*x + 4*e)/f + 1/4*A*a^2*c*sin(2*f*x + 2*e)/f + 1/8*(4 *A*a^2*c + B*a^2*c)*x - 1/12*(A*a^2*c + B*a^2*c)*cos(3*f*x + 3*e)/f - 1/4* (A*a^2*c + B*a^2*c)*cos(f*x + e)/f
Time = 13.88 (sec) , antiderivative size = 339, normalized size of antiderivative = 3.46 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {a^2\,c\,\mathrm {atan}\left (\frac {a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,A+B\right )}{4\,\left (A\,a^2\,c+\frac {B\,a^2\,c}{4}\right )}\right )\,\left (4\,A+B\right )}{4\,f}-\frac {a^2\,c\,\left (4\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,A\,a^2\,c+2\,B\,a^2\,c\right )-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,a^2\,c-\frac {B\,a^2\,c}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,A\,a^2\,c+2\,B\,a^2\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {2\,A\,a^2\,c}{3}+\frac {2\,B\,a^2\,c}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (A\,a^2\,c-\frac {B\,a^2\,c}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (A\,a^2\,c+\frac {7\,B\,a^2\,c}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (A\,a^2\,c+\frac {7\,B\,a^2\,c}{4}\right )+\frac {2\,A\,a^2\,c}{3}+\frac {2\,B\,a^2\,c}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
(a^2*c*atan((a^2*c*tan(e/2 + (f*x)/2)*(4*A + B))/(4*(A*a^2*c + (B*a^2*c)/4 )))*(4*A + B))/(4*f) - (a^2*c*(4*A + B)*(atan(tan(e/2 + (f*x)/2)) - (f*x)/ 2))/(4*f) - (tan(e/2 + (f*x)/2)^4*(2*A*a^2*c + 2*B*a^2*c) - tan(e/2 + (f*x )/2)*(A*a^2*c - (B*a^2*c)/4) + tan(e/2 + (f*x)/2)^6*(2*A*a^2*c + 2*B*a^2*c ) + tan(e/2 + (f*x)/2)^2*((2*A*a^2*c)/3 + (2*B*a^2*c)/3) + tan(e/2 + (f*x) /2)^7*(A*a^2*c - (B*a^2*c)/4) - tan(e/2 + (f*x)/2)^3*(A*a^2*c + (7*B*a^2*c )/4) + tan(e/2 + (f*x)/2)^5*(A*a^2*c + (7*B*a^2*c)/4) + (2*A*a^2*c)/3 + (2 *B*a^2*c)/3)/(f*(4*tan(e/2 + (f*x)/2)^2 + 6*tan(e/2 + (f*x)/2)^4 + 4*tan(e /2 + (f*x)/2)^6 + tan(e/2 + (f*x)/2)^8 + 1))