3.8.0 ∫ (c+d sin(e+fx))5 _________________________

Integrand size = 25, antiderivative size = 481 \[ \int \frac {(c+d \sin (e+f x))^5}{(3+b \sin (e+f x))^3} \, dx=-\frac {d^3 \left (90 b c d-108 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}+\frac {(b c-3 d)^3 \left (162 b c d-36 b^3 c d+972 d^2+9 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right ) \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{b^5 \left (9-b^2\right )^{5/2} f}-\frac {d \left (2430 b c d^3-2916 d^4-27 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-9 b^3 c d \left (4 c^2+55 d^2\right )+3 b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (9-b^2\right )^2 f}+\frac {d^2 \left (189 b c d^2-486 d^3+b^4 d \left (8 c^2-d^2\right )+9 b^2 d \left (c^2+10 d^2\right )-3 b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (9-b^2\right )^2 f}+\frac {(b c-3 d)^2 \left (9 b c+36 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (9-b^2\right )^2 f (3+b \sin (e+f x))}+\frac {(b c-3 d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (9-b^2\right ) f (3+b \sin (e+f x))^2} \]

-1/2*d^3*(30*a*b*c*d-12*a^2*d^2-b^2*(20*c^2+d^2))*x/b^5+(-a*d+b*c)^3*(6*a^3*b*c*d-12*a*b^3*c*d+12*a^4*d^2+a^2*

b^2*(2*c^2-29*d^2)+b^4*(c^2+20*d^2))*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/b^5/(a^2-b^2)^(5/2)/f-1/

2*d*(30*a^4*b*c*d^3-12*a^5*d^4-a^3*b^2*d^2*(16*c^2-21*d^2)-b^5*c*d*(17*c^2-10*d^2)-a^2*b^3*c*d*(4*c^2+55*d^2)+

a*b^4*(6*c^4+43*c^2*d^2-6*d^4))*cos(f*x+e)/b^4/(a^2-b^2)^2/f+1/2*d^2*(7*a^3*b*c*d^2-6*a^4*d^3+b^4*d*(8*c^2-d^2

)+a^2*b^2*d*(c^2+10*d^2)-a*b^3*c*(3*c^2+16*d^2))*cos(f*x+e)*sin(f*x+e)/b^3/(a^2-b^2)^2/f+1/2*(-a*d+b*c)^2*(4*a

^2*d+3*a*b*c-7*b^2*d)*cos(f*x+e)*(c+d*sin(f*x+e))^2/b^2/(a^2-b^2)^2/f/(a+b*sin(f*x+e))+1/2*(-a*d+b*c)^2*cos(f*

x+e)*(c+d*sin(f*x+e))^3/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^2

Rubi [A] (veriﬁed)

Time = 1.38 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2871, 3126, 3112, 3102, 2814, 2739, 632, 210} \[ \int \frac {(c+d \sin (e+f x))^5}{(3+b \sin (e+f x))^3} \, dx=\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}-\frac {d^3 x \left (-12 a^2 d^2+30 a b c d-\left (b^2 \left (20 c^2+d^2\right )\right )\right )}{2 b^5}+\frac {(b c-a d)^3 \left (12 a^4 d^2+6 a^3 b c d+a^2 b^2 \left (2 c^2-29 d^2\right )-12 a b^3 c d+b^4 \left (c^2+20 d^2\right )\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 f \left (a^2-b^2\right )^{5/2}}+\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b^3 f \left (a^2-b^2\right )^2}-\frac {d \left (-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )-b^5 c d \left (17 c^2-10 d^2\right )\right ) \cos (e+f x)}{2 b^4 f \left (a^2-b^2\right )^2} \]

Int[(c + d*Sin[e + f*x])^5/(a + b*Sin[e + f*x])^3,x]

-1/2*(d^3*(30*a*b*c*d - 12*a^2*d^2 - b^2*(20*c^2 + d^2))*x)/b^5 + ((b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d +

12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])

/(b^5*(a^2 - b^2)^(5/2)*f) - (d*(30*a^4*b*c*d^3 - 12*a^5*d^4 - a^3*b^2*d^2*(16*c^2 - 21*d^2) - b^5*c*d*(17*c^2

- 10*d^2) - a^2*b^3*c*d*(4*c^2 + 55*d^2) + a*b^4*(6*c^4 + 43*c^2*d^2 - 6*d^4))*Cos[e + f*x])/(2*b^4*(a^2 - b^

2)^2*f) + (d^2*(7*a^3*b*c*d^2 - 6*a^4*d^3 + b^4*d*(8*c^2 - d^2) + a^2*b^2*d*(c^2 + 10*d^2) - a*b^3*c*(3*c^2 +

16*d^2))*Cos[e + f*x]*Sin[e + f*x])/(2*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(3*a*b*c + 4*a^2*d - 7*b^2*d)*Cos

[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(

c + d*Sin[e + f*x])^3)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])

], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]

, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis

t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)

, x] - Dist[(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

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si

mp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/

(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e

+ f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 +

c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 +

d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((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 + 1)/(b*f*(m + 2))), x] + Dist[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_)])*((A_.) + (B_.)*sin[(e

_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +

b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*

c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e

+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s

in[(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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {(c+d \sin (e+f x))^2 \left (7 b^2 c^2 d+3 a^2 d^3-2 a b c \left (c^2+4 d^2\right )-\left (a^2 c d^2+2 a b d \left (2 c^2+d^2\right )-b^2 \left (c^3+6 c d^2\right )\right ) \sin (e+f x)+2 d \left (2 a b c d-2 a^2 d^2-b^2 \left (c^2-d^2\right )\right ) \sin ^2(e+f x)\right )}{(a+b \sin (e+f x))^2} \, dx}{2 b \left (a^2-b^2\right )} \\ & = \frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {(c+d \sin (e+f x)) \left (11 a^3 b c d^3-8 a^4 d^4+b^4 c^2 \left (c^2+20 d^2\right )-a b^3 c d \left (15 c^2+32 d^2\right )+a^2 b^2 \left (2 c^4+7 c^2 d^2+14 d^4\right )+d \left (4 a^4 c d^2-b^4 c \left (c^2-8 d^2\right )+a^2 b^2 c \left (4 c^2-3 d^2\right )-a^3 b d \left (4 c^2-d^2\right )-a b^3 d \left (5 c^2+4 d^2\right )\right ) \sin (e+f x)-2 d \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2} \\ & = \frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {-2 \left (15 a^4 b c d^4-6 a^5 d^5-10 a^3 b^2 d^3 \left (c^2-d^2\right )-b^5 c^3 \left (c^2+20 d^2\right )+a b^4 d \left (15 c^4+40 c^2 d^2-d^4\right )-2 a^2 b^3 c \left (c^4+5 c^2 d^2+15 d^4\right )\right )+2 b d \left (b^4 d^2 \left (20 c^2+d^2\right )-a b^3 c d \left (17 c^2+20 d^2\right )-a^3 b \left (4 c^3 d-5 c d^3\right )+a^4 \left (4 c^2 d^2-2 d^4\right )+a^2 b^2 \left (6 c^4+3 c^2 d^2+4 d^4\right )\right ) \sin (e+f x)+2 d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2} \\ & = -\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {-2 b \left (15 a^4 b c d^4-6 a^5 d^5-10 a^3 b^2 d^3 \left (c^2-d^2\right )-b^5 c^3 \left (c^2+20 d^2\right )+a b^4 d \left (15 c^4+40 c^2 d^2-d^4\right )-2 a^2 b^3 c \left (c^4+5 c^2 d^2+15 d^4\right )\right )-2 \left (a^2-b^2\right )^2 d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2} \\ & = -\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left ((b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2} \\ & = -\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left ((b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{b^5 \left (a^2-b^2\right )^2 f} \\ & = -\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\left (2 (b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{b^5 \left (a^2-b^2\right )^2 f} \\ & = -\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}+\frac {(b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2} f}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2} \\ \end{align*}

Mathematica [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(1251\) vs. \(2(481)=962\).

Time = 10.36 (sec) , antiderivative size = 1251, normalized size of antiderivative = 2.60 \[ \int \frac {(c+d \sin (e+f x))^5}{(3+b \sin (e+f x))^3} \, dx=\frac {\frac {16 (b c-3 d)^3 \left (162 b c d-36 b^3 c d+972 d^2+9 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right ) \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{\sqrt {9-b^2}}+\frac {116640 b^2 c^2 d^3 e-19440 b^4 c^2 d^3 e+80 b^8 c^2 d^3 e-524880 b c d^4 e+87480 b^3 c d^4 e-360 b^7 c d^4 e+629856 d^5 e-99144 b^2 d^5 e-972 b^4 d^5 e+432 b^6 d^5 e+4 b^8 d^5 e+116640 b^2 c^2 d^3 f x-19440 b^4 c^2 d^3 f x+80 b^8 c^2 d^3 f x-524880 b c d^4 f x+87480 b^3 c d^4 f x-360 b^7 c d^4 f x+629856 d^5 f x-99144 b^2 d^5 f x-972 b^4 d^5 f x+432 b^6 d^5 f x+4 b^8 d^5 f x-4 b \left (43740 b c d^4-8505 b^3 c d^4-52488 d^5+9720 b^2 d^3 \left (-c^2+d^2\right )-18 b^5 \left (4 c^5+30 c^3 d^2-5 c d^4\right )+b^7 \left (2 c^5+5 c d^4\right )+108 b^4 \left (5 c^4 d+25 c^2 d^3-2 d^5\right )+6 b^6 \left (5 c^4 d-d^5\right )\right ) \cos (e+f x)-4 b^2 \left (-9+b^2\right )^2 d^3 \left (-90 b c d+108 d^2+b^2 \left (20 c^2+d^2\right )\right ) (e+f x) \cos (2 (e+f x))+1620 b^4 c d^4 \cos (3 (e+f x))-360 b^6 c d^4 \cos (3 (e+f x))+20 b^8 c d^4 \cos (3 (e+f x))-1944 b^3 d^5 \cos (3 (e+f x))+432 b^5 d^5 \cos (3 (e+f x))-24 b^7 d^5 \cos (3 (e+f x))+77760 b^3 c^2 d^3 e \sin (e+f x)-17280 b^5 c^2 d^3 e \sin (e+f x)+960 b^7 c^2 d^3 e \sin (e+f x)-349920 b^2 c d^4 e \sin (e+f x)+77760 b^4 c d^4 e \sin (e+f x)-4320 b^6 c d^4 e \sin (e+f x)+419904 b d^5 e \sin (e+f x)-89424 b^3 d^5 e \sin (e+f x)+4320 b^5 d^5 e \sin (e+f x)+48 b^7 d^5 e \sin (e+f x)+77760 b^3 c^2 d^3 f x \sin (e+f x)-17280 b^5 c^2 d^3 f x \sin (e+f x)+960 b^7 c^2 d^3 f x \sin (e+f x)-349920 b^2 c d^4 f x \sin (e+f x)+77760 b^4 c d^4 f x \sin (e+f x)-4320 b^6 c d^4 f x \sin (e+f x)+419904 b d^5 f x \sin (e+f x)-89424 b^3 d^5 f x \sin (e+f x)+4320 b^5 d^5 f x \sin (e+f x)+48 b^7 d^5 f x \sin (e+f x)+36 b^7 c^5 \sin (2 (e+f x))-180 b^6 c^4 d \sin (2 (e+f x))-40 b^8 c^4 d \sin (2 (e+f x))-1080 b^5 c^3 d^2 \sin (2 (e+f x))+480 b^7 c^3 d^2 \sin (2 (e+f x))+9720 b^4 c^2 d^3 \sin (2 (e+f x))-2160 b^6 c^2 d^3 \sin (2 (e+f x))-43740 b^3 c d^4 \sin (2 (e+f x))+8640 b^5 c d^4 \sin (2 (e+f x))-240 b^7 c d^4 \sin (2 (e+f x))+52488 b^2 d^5 \sin (2 (e+f x))-10530 b^4 d^5 \sin (2 (e+f x))+432 b^6 d^5 \sin (2 (e+f x))-2 b^8 d^5 \sin (2 (e+f x))+81 b^4 d^5 \sin (4 (e+f x))-18 b^6 d^5 \sin (4 (e+f x))+b^8 d^5 \sin (4 (e+f x))}{(3+b \sin (e+f x))^2}}{16 b^5 \left (-9+b^2\right )^2 f} \]

Integrate[(c + d*Sin[e + f*x])^5/(3 + b*Sin[e + f*x])^3,x]

((16*(b*c - 3*d)^3*(162*b*c*d - 36*b^3*c*d + 972*d^2 + 9*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTan[(b

+ 3*Tan[(e + f*x)/2])/Sqrt[9 - b^2]])/Sqrt[9 - b^2] + (116640*b^2*c^2*d^3*e - 19440*b^4*c^2*d^3*e + 80*b^8*c^2

*d^3*e - 524880*b*c*d^4*e + 87480*b^3*c*d^4*e - 360*b^7*c*d^4*e + 629856*d^5*e - 99144*b^2*d^5*e - 972*b^4*d^5

*e + 432*b^6*d^5*e + 4*b^8*d^5*e + 116640*b^2*c^2*d^3*f*x - 19440*b^4*c^2*d^3*f*x + 80*b^8*c^2*d^3*f*x - 52488

0*b*c*d^4*f*x + 87480*b^3*c*d^4*f*x - 360*b^7*c*d^4*f*x + 629856*d^5*f*x - 99144*b^2*d^5*f*x - 972*b^4*d^5*f*x

+ 432*b^6*d^5*f*x + 4*b^8*d^5*f*x - 4*b*(43740*b*c*d^4 - 8505*b^3*c*d^4 - 52488*d^5 + 9720*b^2*d^3*(-c^2 + d^

2) - 18*b^5*(4*c^5 + 30*c^3*d^2 - 5*c*d^4) + b^7*(2*c^5 + 5*c*d^4) + 108*b^4*(5*c^4*d + 25*c^2*d^3 - 2*d^5) +

6*b^6*(5*c^4*d - d^5))*Cos[e + f*x] - 4*b^2*(-9 + b^2)^2*d^3*(-90*b*c*d + 108*d^2 + b^2*(20*c^2 + d^2))*(e + f

*x)*Cos[2*(e + f*x)] + 1620*b^4*c*d^4*Cos[3*(e + f*x)] - 360*b^6*c*d^4*Cos[3*(e + f*x)] + 20*b^8*c*d^4*Cos[3*(

e + f*x)] - 1944*b^3*d^5*Cos[3*(e + f*x)] + 432*b^5*d^5*Cos[3*(e + f*x)] - 24*b^7*d^5*Cos[3*(e + f*x)] + 77760

*b^3*c^2*d^3*e*Sin[e + f*x] - 17280*b^5*c^2*d^3*e*Sin[e + f*x] + 960*b^7*c^2*d^3*e*Sin[e + f*x] - 349920*b^2*c

*d^4*e*Sin[e + f*x] + 77760*b^4*c*d^4*e*Sin[e + f*x] - 4320*b^6*c*d^4*e*Sin[e + f*x] + 419904*b*d^5*e*Sin[e +

f*x] - 89424*b^3*d^5*e*Sin[e + f*x] + 4320*b^5*d^5*e*Sin[e + f*x] + 48*b^7*d^5*e*Sin[e + f*x] + 77760*b^3*c^2*

d^3*f*x*Sin[e + f*x] - 17280*b^5*c^2*d^3*f*x*Sin[e + f*x] + 960*b^7*c^2*d^3*f*x*Sin[e + f*x] - 349920*b^2*c*d^

4*f*x*Sin[e + f*x] + 77760*b^4*c*d^4*f*x*Sin[e + f*x] - 4320*b^6*c*d^4*f*x*Sin[e + f*x] + 419904*b*d^5*f*x*Sin

[e + f*x] - 89424*b^3*d^5*f*x*Sin[e + f*x] + 4320*b^5*d^5*f*x*Sin[e + f*x] + 48*b^7*d^5*f*x*Sin[e + f*x] + 36*

b^7*c^5*Sin[2*(e + f*x)] - 180*b^6*c^4*d*Sin[2*(e + f*x)] - 40*b^8*c^4*d*Sin[2*(e + f*x)] - 1080*b^5*c^3*d^2*S

in[2*(e + f*x)] + 480*b^7*c^3*d^2*Sin[2*(e + f*x)] + 9720*b^4*c^2*d^3*Sin[2*(e + f*x)] - 2160*b^6*c^2*d^3*Sin[

2*(e + f*x)] - 43740*b^3*c*d^4*Sin[2*(e + f*x)] + 8640*b^5*c*d^4*Sin[2*(e + f*x)] - 240*b^7*c*d^4*Sin[2*(e + f

*x)] + 52488*b^2*d^5*Sin[2*(e + f*x)] - 10530*b^4*d^5*Sin[2*(e + f*x)] + 432*b^6*d^5*Sin[2*(e + f*x)] - 2*b^8*

d^5*Sin[2*(e + f*x)] + 81*b^4*d^5*Sin[4*(e + f*x)] - 18*b^6*d^5*Sin[4*(e + f*x)] + b^8*d^5*Sin[4*(e + f*x)])/(

3 + b*Sin[e + f*x])^2)/(16*b^5*(-9 + b^2)^2*f)

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1122\) vs. \(2(519)=1038\).

Time = 4.08 (sec) , antiderivative size = 1123, normalized size of antiderivative = 2.33


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1123\)
default	\(\text {Expression too large to display}\)	\(1123\)
risch	\(\text {Expression too large to display}\)	\(3634\)

int((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

1/f*(-2/b^5*((-1/2*b^2*(5*a^7*d^5-15*a^6*b*c*d^4+10*a^5*b^2*c^2*d^3-8*a^5*b^2*d^5+10*a^4*b^3*c^3*d^2+30*a^4*b^

3*c*d^4-15*a^3*b^4*c^4*d-40*a^3*b^4*c^2*d^3+5*a^2*b^5*c^5+20*a^2*b^5*c^3*d^2-2*b^7*c^5)/a/(a^4-2*a^2*b^2+b^4)*

tan(1/2*f*x+1/2*e)^3-1/2*b*(6*a^9*d^5-20*a^8*b*c*d^4+20*a^7*b^2*c^2*d^3+3*a^7*b^2*d^5-5*a^6*b^3*c*d^4-10*a^5*b

^4*c^4*d-10*a^5*b^4*c^2*d^3-18*a^5*b^4*d^5+4*a^4*b^5*c^5+30*a^4*b^5*c^3*d^2+70*a^4*b^5*c*d^4-25*a^3*b^6*c^4*d-

100*a^3*b^6*c^2*d^3+7*a^2*b^7*c^5+60*a^2*b^7*c^3*d^2-10*a*b^8*c^4*d-2*b^9*c^5)/(a^4-2*a^2*b^2+b^4)/a^2*tan(1/2

*f*x+1/2*e)^2-1/2*b^2*(19*a^7*d^5-65*a^6*b*c*d^4+70*a^5*b^2*c^2*d^3-28*a^5*b^2*d^5-10*a^4*b^3*c^3*d^2+110*a^4*

b^3*c*d^4-25*a^3*b^4*c^4*d-160*a^3*b^4*c^2*d^3+11*a^2*b^5*c^5+100*a^2*b^5*c^3*d^2-20*a*b^6*c^4*d-2*b^7*c^5)/(a

^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)-1/2*b*(6*a^7*d^5-20*a^6*b*c*d^4+20*a^5*b^2*c^2*d^3-9*a^5*b^2*d^5+35*a^4

*b^3*c*d^4-10*a^3*b^4*c^4*d-50*a^3*b^4*c^2*d^3+4*a^2*b^5*c^5+30*a^2*b^5*c^3*d^2-5*a*b^6*c^4*d-b^7*c^5)/(a^4-2*

a^2*b^2+b^4))/(tan(1/2*f*x+1/2*e)^2*a+2*b*tan(1/2*f*x+1/2*e)+a)^2+1/2*(12*a^7*d^5-30*a^6*b*c*d^4+20*a^5*b^2*c^

2*d^3-29*a^5*b^2*d^5+75*a^4*b^3*c*d^4-50*a^3*b^4*c^2*d^3+20*a^3*b^4*d^5-2*a^2*b^5*c^5-10*a^2*b^5*c^3*d^2-60*a^

2*b^5*c*d^4+15*a*b^6*c^4*d+60*a*b^6*c^2*d^3-b^7*c^5-20*b^7*c^3*d^2)/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan

(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2)))+2*d^3/b^5*((1/2*tan(1/2*f*x+1/2*e)^3*b^2*d^2+(3*a*b*d^2-5*

b^2*c*d)*tan(1/2*f*x+1/2*e)^2-1/2*tan(1/2*f*x+1/2*e)*b^2*d^2+3*a*b*d^2-5*b^2*c*d)/(1+tan(1/2*f*x+1/2*e)^2)^2+1

/2*(12*a^2*d^2-30*a*b*c*d+20*b^2*c^2+b^2*d^2)*arctan(tan(1/2*f*x+1/2*e))))

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1544 vs. \(2 (519) = 1038\).

Time = 0.48 (sec) , antiderivative size = 3174, normalized size of antiderivative = 6.60 \[ \int \frac {(c+d \sin (e+f x))^5}{(3+b \sin (e+f x))^3} \, dx=\text {Too large to display} \]

integrate((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x, algorithm="fricas")

[1/4*(2*(20*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*c^2*d^3 - 30*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c*

d^4 + (12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*d^5)*f*x*cos(f*x + e)^2 - 4*(5*(a^6*b^4 - 3*a^

4*b^6 + 3*a^2*b^8 - b^10)*c*d^4 - 2*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*d^5)*cos(f*x + e)^3 - 2*(20*(a^8

*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10)*c^2*d^3 - 30*(a^9*b - 2*a^7*b^3 + 2*a^3*b^7 - a*b^9)*c*d^4 + (12*a^10 - 2

3*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8 - b^10)*d^5)*f*x - ((2*a^4*b^5 + 3*a^2*b^7 + b^9)*c^5 - 15*(a^

3*b^6 + a*b^8)*c^4*d + 10*(a^4*b^5 + 3*a^2*b^7 + 2*b^9)*c^3*d^2 - 10*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6 + 6*a*b^

8)*c^2*d^3 + 15*(2*a^8*b - 3*a^6*b^3 - a^4*b^5 + 4*a^2*b^7)*c*d^4 - (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*

b^6)*d^5 + (15*a*b^8*c^4*d - (2*a^2*b^7 + b^9)*c^5 - 10*(a^2*b^7 + 2*b^9)*c^3*d^2 + 10*(2*a^5*b^4 - 5*a^3*b^6

+ 6*a*b^8)*c^2*d^3 - 15*(2*a^6*b^3 - 5*a^4*b^5 + 4*a^2*b^7)*c*d^4 + (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*d^5

)*cos(f*x + e)^2 - 2*(15*a^2*b^7*c^4*d - (2*a^3*b^6 + a*b^8)*c^5 - 10*(a^3*b^6 + 2*a*b^8)*c^3*d^2 + 10*(2*a^6*

b^3 - 5*a^4*b^5 + 6*a^2*b^7)*c^2*d^3 - 15*(2*a^7*b^2 - 5*a^5*b^4 + 4*a^3*b^6)*c*d^4 + (12*a^8*b - 29*a^6*b^3 +

20*a^4*b^5)*d^5)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2

- b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*

x + e) - a^2 - b^2)) - 2*((4*a^4*b^6 - 5*a^2*b^8 + b^10)*c^5 - 5*(2*a^5*b^5 - a^3*b^7 - a*b^9)*c^4*d + 30*(a^4

*b^6 - a^2*b^8)*c^3*d^2 + 10*(2*a^7*b^3 - 7*a^5*b^5 + 5*a^3*b^7)*c^2*d^3 - 5*(6*a^8*b^2 - 15*a^6*b^4 + 7*a^4*b

^6 + 4*a^2*b^8 - 2*b^10)*c*d^4 + (12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*d^5)*cos(f*x + e)

- 2*((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*d^5*cos(f*x + e)^3 + 2*(20*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*

b^9)*c^2*d^3 - 30*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*c*d^4 + (12*a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*

a^3*b^7 - a*b^9)*d^5)*f*x + (3*(a^3*b^7 - a*b^9)*c^5 - 5*(a^4*b^6 + a^2*b^8 - 2*b^10)*c^4*d - 10*(a^5*b^5 - 5*

a^3*b^7 + 4*a*b^9)*c^3*d^2 + 30*(a^6*b^4 - 3*a^4*b^6 + 2*a^2*b^8)*c^2*d^3 - 5*(9*a^7*b^3 - 25*a^5*b^5 + 20*a^3

*b^7 - 4*a*b^9)*c*d^4 + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*d^5)*cos(f*x + e))*sin(f*x

+ e))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*f*cos(f*x + e)^2 - 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^

12)*f*sin(f*x + e) - (a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13)*f), 1/2*((20*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 -

b^10)*c^2*d^3 - 30*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c*d^4 + (12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 -

9*a^2*b^8 - b^10)*d^5)*f*x*cos(f*x + e)^2 - 2*(5*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*c*d^4 - 2*(a^7*b^3 -

3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*d^5)*cos(f*x + e)^3 - (20*(a^8*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10)*c^2*d^3 - 3

0*(a^9*b - 2*a^7*b^3 + 2*a^3*b^7 - a*b^9)*c*d^4 + (12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8

- b^10)*d^5)*f*x + ((2*a^4*b^5 + 3*a^2*b^7 + b^9)*c^5 - 15*(a^3*b^6 + a*b^8)*c^4*d + 10*(a^4*b^5 + 3*a^2*b^7 +

2*b^9)*c^3*d^2 - 10*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6 + 6*a*b^8)*c^2*d^3 + 15*(2*a^8*b - 3*a^6*b^3 - a^4*b^5 +

4*a^2*b^7)*c*d^4 - (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*b^6)*d^5 + (15*a*b^8*c^4*d - (2*a^2*b^7 + b^9)*c

^5 - 10*(a^2*b^7 + 2*b^9)*c^3*d^2 + 10*(2*a^5*b^4 - 5*a^3*b^6 + 6*a*b^8)*c^2*d^3 - 15*(2*a^6*b^3 - 5*a^4*b^5 +

4*a^2*b^7)*c*d^4 + (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*d^5)*cos(f*x + e)^2 - 2*(15*a^2*b^7*c^4*d - (2*a^3*

b^6 + a*b^8)*c^5 - 10*(a^3*b^6 + 2*a*b^8)*c^3*d^2 + 10*(2*a^6*b^3 - 5*a^4*b^5 + 6*a^2*b^7)*c^2*d^3 - 15*(2*a^7

*b^2 - 5*a^5*b^4 + 4*a^3*b^6)*c*d^4 + (12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*d^5)*sin(f*x + e))*sqrt(a^2 - b^2)*

arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) - ((4*a^4*b^6 - 5*a^2*b^8 + b^10)*c^5 - 5*(2*a^5*

b^5 - a^3*b^7 - a*b^9)*c^4*d + 30*(a^4*b^6 - a^2*b^8)*c^3*d^2 + 10*(2*a^7*b^3 - 7*a^5*b^5 + 5*a^3*b^7)*c^2*d^3

- 5*(6*a^8*b^2 - 15*a^6*b^4 + 7*a^4*b^6 + 4*a^2*b^8 - 2*b^10)*c*d^4 + (12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6

*a^3*b^7 - 4*a*b^9)*d^5)*cos(f*x + e) - ((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*d^5*cos(f*x + e)^3 + 2*(20*(

a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c^2*d^3 - 30*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*c*d^4 + (12*

a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*d^5)*f*x + (3*(a^3*b^7 - a*b^9)*c^5 - 5*(a^4*b^6 + a^2*b^

8 - 2*b^10)*c^4*d - 10*(a^5*b^5 - 5*a^3*b^7 + 4*a*b^9)*c^3*d^2 + 30*(a^6*b^4 - 3*a^4*b^6 + 2*a^2*b^8)*c^2*d^3

- 5*(9*a^7*b^3 - 25*a^5*b^5 + 20*a^3*b^7 - 4*a*b^9)*c*d^4 + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8

+ b^10)*d^5)*cos(f*x + e))*sin(f*x + e))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*f*cos(f*x + e)^2 - 2*(a^7

*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*f*sin(f*x + e) - (a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13)*f)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^5}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]

integrate((c+d*sin(f*x+e))**5/(a+b*sin(f*x+e))**3,x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d \sin (e+f x))^5}{(3+b \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]

integrate((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x, algorithm="maxima")

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3043 vs. \(2 (519) = 1038\).

Time = 0.38 (sec) , antiderivative size = 3043, normalized size of antiderivative = 6.33 \[ \int \frac {(c+d \sin (e+f x))^5}{(3+b \sin (e+f x))^3} \, dx=\text {Too large to display} \]

integrate((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x, algorithm="giac")

1/2*(2*(2*a^2*b^5*c^5 + b^7*c^5 - 15*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 + 20*b^7*c^3*d^2 - 20*a^5*b^2*c^2*d^3 +

50*a^3*b^4*c^2*d^3 - 60*a*b^6*c^2*d^3 + 30*a^6*b*c*d^4 - 75*a^4*b^3*c*d^4 + 60*a^2*b^5*c*d^4 - 12*a^7*d^5 + 29

*a^5*b^2*d^5 - 20*a^3*b^4*d^5)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*f*x + 1/2*e) + b)/

sqrt(a^2 - b^2)))/((a^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(a^2 - b^2)) + 2*(5*a^3*b^6*c^5*tan(1/2*f*x + 1/2*e)^7 - 2*

a*b^8*c^5*tan(1/2*f*x + 1/2*e)^7 - 15*a^4*b^5*c^4*d*tan(1/2*f*x + 1/2*e)^7 + 10*a^5*b^4*c^3*d^2*tan(1/2*f*x +

1/2*e)^7 + 20*a^3*b^6*c^3*d^2*tan(1/2*f*x + 1/2*e)^7 + 10*a^6*b^3*c^2*d^3*tan(1/2*f*x + 1/2*e)^7 - 40*a^4*b^5*

c^2*d^3*tan(1/2*f*x + 1/2*e)^7 - 15*a^7*b^2*c*d^4*tan(1/2*f*x + 1/2*e)^7 + 30*a^5*b^4*c*d^4*tan(1/2*f*x + 1/2*

e)^7 + 6*a^8*b*d^5*tan(1/2*f*x + 1/2*e)^7 - 10*a^6*b^3*d^5*tan(1/2*f*x + 1/2*e)^7 + a^4*b^5*d^5*tan(1/2*f*x +

1/2*e)^7 + 4*a^4*b^5*c^5*tan(1/2*f*x + 1/2*e)^6 + 7*a^2*b^7*c^5*tan(1/2*f*x + 1/2*e)^6 - 2*b^9*c^5*tan(1/2*f*x

+ 1/2*e)^6 - 10*a^5*b^4*c^4*d*tan(1/2*f*x + 1/2*e)^6 - 25*a^3*b^6*c^4*d*tan(1/2*f*x + 1/2*e)^6 - 10*a*b^8*c^4

*d*tan(1/2*f*x + 1/2*e)^6 + 30*a^4*b^5*c^3*d^2*tan(1/2*f*x + 1/2*e)^6 + 60*a^2*b^7*c^3*d^2*tan(1/2*f*x + 1/2*e

)^6 + 20*a^7*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^6 - 10*a^5*b^4*c^2*d^3*tan(1/2*f*x + 1/2*e)^6 - 100*a^3*b^6*c^2*

d^3*tan(1/2*f*x + 1/2*e)^6 - 30*a^8*b*c*d^4*tan(1/2*f*x + 1/2*e)^6 + 15*a^6*b^3*c*d^4*tan(1/2*f*x + 1/2*e)^6 +

60*a^4*b^5*c*d^4*tan(1/2*f*x + 1/2*e)^6 + 12*a^9*d^5*tan(1/2*f*x + 1/2*e)^6 - 5*a^7*b^2*d^5*tan(1/2*f*x + 1/2

*e)^6 - 20*a^5*b^4*d^5*tan(1/2*f*x + 1/2*e)^6 + 4*a^3*b^6*d^5*tan(1/2*f*x + 1/2*e)^6 + 21*a^3*b^6*c^5*tan(1/2*

f*x + 1/2*e)^5 - 6*a*b^8*c^5*tan(1/2*f*x + 1/2*e)^5 - 55*a^4*b^5*c^4*d*tan(1/2*f*x + 1/2*e)^5 - 20*a^2*b^7*c^4

*d*tan(1/2*f*x + 1/2*e)^5 + 10*a^5*b^4*c^3*d^2*tan(1/2*f*x + 1/2*e)^5 + 140*a^3*b^6*c^3*d^2*tan(1/2*f*x + 1/2*

e)^5 + 90*a^6*b^3*c^2*d^3*tan(1/2*f*x + 1/2*e)^5 - 240*a^4*b^5*c^2*d^3*tan(1/2*f*x + 1/2*e)^5 - 135*a^7*b^2*c*

d^4*tan(1/2*f*x + 1/2*e)^5 + 250*a^5*b^4*c*d^4*tan(1/2*f*x + 1/2*e)^5 - 40*a^3*b^6*c*d^4*tan(1/2*f*x + 1/2*e)^

5 + 54*a^8*b*d^5*tan(1/2*f*x + 1/2*e)^5 - 90*a^6*b^3*d^5*tan(1/2*f*x + 1/2*e)^5 + 17*a^4*b^5*d^5*tan(1/2*f*x +

1/2*e)^5 + 4*a^2*b^7*d^5*tan(1/2*f*x + 1/2*e)^5 + 12*a^4*b^5*c^5*tan(1/2*f*x + 1/2*e)^4 + 13*a^2*b^7*c^5*tan(

1/2*f*x + 1/2*e)^4 - 4*b^9*c^5*tan(1/2*f*x + 1/2*e)^4 - 30*a^5*b^4*c^4*d*tan(1/2*f*x + 1/2*e)^4 - 55*a^3*b^6*c

^4*d*tan(1/2*f*x + 1/2*e)^4 - 20*a*b^8*c^4*d*tan(1/2*f*x + 1/2*e)^4 + 90*a^4*b^5*c^3*d^2*tan(1/2*f*x + 1/2*e)^

4 + 120*a^2*b^7*c^3*d^2*tan(1/2*f*x + 1/2*e)^4 + 60*a^7*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^4 - 70*a^5*b^4*c^2*d^

3*tan(1/2*f*x + 1/2*e)^4 - 200*a^3*b^6*c^2*d^3*tan(1/2*f*x + 1/2*e)^4 - 90*a^8*b*c*d^4*tan(1/2*f*x + 1/2*e)^4

+ 45*a^6*b^3*c*d^4*tan(1/2*f*x + 1/2*e)^4 + 190*a^4*b^5*c*d^4*tan(1/2*f*x + 1/2*e)^4 - 40*a^2*b^7*c*d^4*tan(1/

2*f*x + 1/2*e)^4 + 36*a^9*d^5*tan(1/2*f*x + 1/2*e)^4 - 15*a^7*b^2*d^5*tan(1/2*f*x + 1/2*e)^4 - 66*a^5*b^4*d^5*

tan(1/2*f*x + 1/2*e)^4 + 24*a^3*b^6*d^5*tan(1/2*f*x + 1/2*e)^4 + 27*a^3*b^6*c^5*tan(1/2*f*x + 1/2*e)^3 - 6*a*b

^8*c^5*tan(1/2*f*x + 1/2*e)^3 - 65*a^4*b^5*c^4*d*tan(1/2*f*x + 1/2*e)^3 - 40*a^2*b^7*c^4*d*tan(1/2*f*x + 1/2*e

)^3 - 10*a^5*b^4*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 + 220*a^3*b^6*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 + 150*a^6*b^3*c^2

*d^3*tan(1/2*f*x + 1/2*e)^3 - 360*a^4*b^5*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 - 225*a^7*b^2*c*d^4*tan(1/2*f*x + 1/2

*e)^3 + 410*a^5*b^4*c*d^4*tan(1/2*f*x + 1/2*e)^3 - 80*a^3*b^6*c*d^4*tan(1/2*f*x + 1/2*e)^3 + 90*a^8*b*d^5*tan(

1/2*f*x + 1/2*e)^3 - 162*a^6*b^3*d^5*tan(1/2*f*x + 1/2*e)^3 + 55*a^4*b^5*d^5*tan(1/2*f*x + 1/2*e)^3 - 4*a^2*b^

7*d^5*tan(1/2*f*x + 1/2*e)^3 + 12*a^4*b^5*c^5*tan(1/2*f*x + 1/2*e)^2 + 5*a^2*b^7*c^5*tan(1/2*f*x + 1/2*e)^2 -

2*b^9*c^5*tan(1/2*f*x + 1/2*e)^2 - 30*a^5*b^4*c^4*d*tan(1/2*f*x + 1/2*e)^2 - 35*a^3*b^6*c^4*d*tan(1/2*f*x + 1/

2*e)^2 - 10*a*b^8*c^4*d*tan(1/2*f*x + 1/2*e)^2 + 90*a^4*b^5*c^3*d^2*tan(1/2*f*x + 1/2*e)^2 + 60*a^2*b^7*c^3*d^

2*tan(1/2*f*x + 1/2*e)^2 + 60*a^7*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^2 - 110*a^5*b^4*c^2*d^3*tan(1/2*f*x + 1/2*e

)^2 - 100*a^3*b^6*c^2*d^3*tan(1/2*f*x + 1/2*e)^2 - 90*a^8*b*c*d^4*tan(1/2*f*x + 1/2*e)^2 + 85*a^6*b^3*c*d^4*ta

n(1/2*f*x + 1/2*e)^2 + 120*a^4*b^5*c*d^4*tan(1/2*f*x + 1/2*e)^2 - 40*a^2*b^7*c*d^4*tan(1/2*f*x + 1/2*e)^2 + 36

*a^9*d^5*tan(1/2*f*x + 1/2*e)^2 - 31*a^7*b^2*d^5*tan(1/2*f*x + 1/2*e)^2 - 40*a^5*b^4*d^5*tan(1/2*f*x + 1/2*e)^

2 + 20*a^3*b^6*d^5*tan(1/2*f*x + 1/2*e)^2 + 11*a^3*b^6*c^5*tan(1/2*f*x + 1/2*e) - 2*a*b^8*c^5*tan(1/2*f*x + 1/

2*e) - 25*a^4*b^5*c^4*d*tan(1/2*f*x + 1/2*e) - 20*a^2*b^7*c^4*d*tan(1/2*f*x + 1/2*e) - 10*a^5*b^4*c^3*d^2*tan(

1/2*f*x + 1/2*e) + 100*a^3*b^6*c^3*d^2*tan(1/2*f*x + 1/2*e) + 70*a^6*b^3*c^2*d^3*tan(1/2*f*x + 1/2*e) - 160*a^

4*b^5*c^2*d^3*tan(1/2*f*x + 1/2*e) - 105*a^7*b^2*c*d^4*tan(1/2*f*x + 1/2*e) + 190*a^5*b^4*c*d^4*tan(1/2*f*x +

1/2*e) - 40*a^3*b^6*c*d^4*tan(1/2*f*x + 1/2*e) + 42*a^8*b*d^5*tan(1/2*f*x + 1/2*e) - 74*a^6*b^3*d^5*tan(1/2*f*

x + 1/2*e) + 23*a^4*b^5*d^5*tan(1/2*f*x + 1/2*e) + 4*a^4*b^5*c^5 - a^2*b^7*c^5 - 10*a^5*b^4*c^4*d - 5*a^3*b^6*

c^4*d + 30*a^4*b^5*c^3*d^2 + 20*a^7*b^2*c^2*d^3 - 50*a^5*b^4*c^2*d^3 - 30*a^8*b*c*d^4 + 55*a^6*b^3*c*d^4 - 10*

a^4*b^5*c*d^4 + 12*a^9*d^5 - 21*a^7*b^2*d^5 + 6*a^5*b^4*d^5)/((a^6*b^4 - 2*a^4*b^6 + a^2*b^8)*(a*tan(1/2*f*x +

1/2*e)^4 + 2*b*tan(1/2*f*x + 1/2*e)^3 + 2*a*tan(1/2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)^2) + (20*b

^2*c^2*d^3 - 30*a*b*c*d^4 + 12*a^2*d^5 + b^2*d^5)*(f*x + e)/b^5)/f

Mupad [B] (veriﬁcation not implemented)

Time = 26.51 (sec) , antiderivative size = 23910, normalized size of antiderivative = 49.71 \[ \int \frac {(c+d \sin (e+f x))^5}{(3+b \sin (e+f x))^3} \, dx=\text {Too large to display} \]

int((c + d*sin(e + f*x))^5/(a + b*sin(e + f*x))^3,x)

- ((b^7*c^5 - 12*a^7*d^5 - 4*a^2*b^5*c^5 - 6*a^3*b^4*d^5 + 21*a^5*b^2*d^5 + 10*a^2*b^5*c*d^4 + 10*a^3*b^4*c^4*

d - 55*a^4*b^3*c*d^4 - 30*a^2*b^5*c^3*d^2 + 50*a^3*b^4*c^2*d^3 - 20*a^5*b^2*c^2*d^3 + 5*a*b^6*c^4*d + 30*a^6*b

*c*d^4)/(b^4*(a^4 + b^4 - 2*a^2*b^2)) - (tan(e/2 + (f*x)/2)^6*(12*a^9*d^5 - 2*b^9*c^5 + 7*a^2*b^7*c^5 + 4*a^4*

b^5*c^5 + 4*a^3*b^6*d^5 - 20*a^5*b^4*d^5 - 5*a^7*b^2*d^5 - 25*a^3*b^6*c^4*d + 60*a^4*b^5*c*d^4 - 10*a^5*b^4*c^

4*d + 15*a^6*b^3*c*d^4 + 60*a^2*b^7*c^3*d^2 - 100*a^3*b^6*c^2*d^3 + 30*a^4*b^5*c^3*d^2 - 10*a^5*b^4*c^2*d^3 +

20*a^7*b^2*c^2*d^3 - 10*a*b^8*c^4*d - 30*a^8*b*c*d^4))/(a^2*b^4*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)

^2*(2*b^9*c^5 - 36*a^9*d^5 - 5*a^2*b^7*c^5 - 12*a^4*b^5*c^5 - 20*a^3*b^6*d^5 + 40*a^5*b^4*d^5 + 31*a^7*b^2*d^5

+ 40*a^2*b^7*c*d^4 + 35*a^3*b^6*c^4*d - 120*a^4*b^5*c*d^4 + 30*a^5*b^4*c^4*d - 85*a^6*b^3*c*d^4 - 60*a^2*b^7*

c^3*d^2 + 100*a^3*b^6*c^2*d^3 - 90*a^4*b^5*c^3*d^2 + 110*a^5*b^4*c^2*d^3 - 60*a^7*b^2*c^2*d^3 + 10*a*b^8*c^4*d

+ 90*a^8*b*c*d^4))/(a^2*b^4*(a^4 + b^4 - 2*a^2*b^2)) - (tan(e/2 + (f*x)/2)^5*(54*a^7*d^5 - 6*b^7*c^5 + 4*a*b^

6*d^5 + 21*a^2*b^5*c^5 + 17*a^3*b^4*d^5 - 90*a^5*b^2*d^5 - 40*a^2*b^5*c*d^4 - 55*a^3*b^4*c^4*d + 250*a^4*b^3*c

*d^4 + 140*a^2*b^5*c^3*d^2 - 240*a^3*b^4*c^2*d^3 + 10*a^4*b^3*c^3*d^2 + 90*a^5*b^2*c^2*d^3 - 20*a*b^6*c^4*d -

135*a^6*b*c*d^4))/(a*b^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)^3*(6*b^7*c^5 - 90*a^7*d^5 + 4*a*b^6*d^

5 - 27*a^2*b^5*c^5 - 55*a^3*b^4*d^5 + 162*a^5*b^2*d^5 + 80*a^2*b^5*c*d^4 + 65*a^3*b^4*c^4*d - 410*a^4*b^3*c*d^

4 - 220*a^2*b^5*c^3*d^2 + 360*a^3*b^4*c^2*d^3 + 10*a^4*b^3*c^3*d^2 - 150*a^5*b^2*c^2*d^3 + 40*a*b^6*c^4*d + 22

5*a^6*b*c*d^4))/(a*b^3*(a^4 + b^4 - 2*a^2*b^2)) - (tan(e/2 + (f*x)/2)^7*(6*a^7*d^5 - 2*b^7*c^5 + 5*a^2*b^5*c^5

+ a^3*b^4*d^5 - 10*a^5*b^2*d^5 - 15*a^3*b^4*c^4*d + 30*a^4*b^3*c*d^4 + 20*a^2*b^5*c^3*d^2 - 40*a^3*b^4*c^2*d^

3 + 10*a^4*b^3*c^3*d^2 + 10*a^5*b^2*c^2*d^3 - 15*a^6*b*c*d^4))/(a*b^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f

*x)/2)*(2*b^7*c^5 - 42*a^7*d^5 - 11*a^2*b^5*c^5 - 23*a^3*b^4*d^5 + 74*a^5*b^2*d^5 + 40*a^2*b^5*c*d^4 + 25*a^3*

b^4*c^4*d - 190*a^4*b^3*c*d^4 - 100*a^2*b^5*c^3*d^2 + 160*a^3*b^4*c^2*d^3 + 10*a^4*b^3*c^3*d^2 - 70*a^5*b^2*c^

2*d^3 + 20*a*b^6*c^4*d + 105*a^6*b*c*d^4))/(a*b^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)^4*(3*a^2 + 4*

b^2)*(b^7*c^5 - 12*a^7*d^5 - 4*a^2*b^5*c^5 - 6*a^3*b^4*d^5 + 21*a^5*b^2*d^5 + 10*a^2*b^5*c*d^4 + 10*a^3*b^4*c^

4*d - 55*a^4*b^3*c*d^4 - 30*a^2*b^5*c^3*d^2 + 50*a^3*b^4*c^2*d^3 - 20*a^5*b^2*c^2*d^3 + 5*a*b^6*c^4*d + 30*a^6

*b*c*d^4))/(a^2*b^4*(a^4 + b^4 - 2*a^2*b^2)))/(f*(tan(e/2 + (f*x)/2)^2*(4*a^2 + 4*b^2) + tan(e/2 + (f*x)/2)^6*

(4*a^2 + 4*b^2) + tan(e/2 + (f*x)/2)^4*(6*a^2 + 8*b^2) + a^2*tan(e/2 + (f*x)/2)^8 + a^2 + 12*a*b*tan(e/2 + (f*

x)/2)^3 + 12*a*b*tan(e/2 + (f*x)/2)^5 + 4*a*b*tan(e/2 + (f*x)/2)^7 + 4*a*b*tan(e/2 + (f*x)/2))) - (atan(((((4*

(2*a^2*b^16*d^10 + 40*a^4*b^14*d^10 + 108*a^6*b^12*d^10 - 872*a^8*b^10*d^10 + 1538*a^10*b^8*d^10 - 1104*a^12*b

^6*d^10 + 288*a^14*b^4*d^10 - 120*a^3*b^15*c*d^9 - 960*a^5*b^13*c*d^9 + 5040*a^7*b^11*c*d^9 - 8160*a^9*b^9*c*d

^9 + 5640*a^11*b^7*c*d^9 - 1440*a^13*b^5*c*d^9 + 80*a^2*b^16*c^2*d^8 + 800*a^2*b^16*c^4*d^6 - 2400*a^3*b^15*c^

3*d^7 + 2440*a^4*b^14*c^2*d^8 - 3200*a^4*b^14*c^4*d^6 + 9600*a^5*b^13*c^3*d^7 - 10560*a^6*b^12*c^2*d^8 + 4800*

a^6*b^12*c^4*d^6 - 14400*a^7*b^11*c^3*d^7 + 16240*a^8*b^10*c^2*d^8 - 3200*a^8*b^10*c^4*d^6 + 9600*a^9*b^9*c^3*

d^7 - 10960*a^10*b^8*c^2*d^8 + 800*a^10*b^8*c^4*d^6 - 2400*a^11*b^7*c^3*d^7 + 2760*a^12*b^6*c^2*d^8))/(b^19 -

4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/2 + (f*x)/2)*(a*b^18*c^10 - 2*a*b^18*d^10 + 4*a^3*

b^16*c^10 + 4*a^5*b^14*c^10 - 39*a^3*b^16*d^10 - 88*a^5*b^14*d^10 + 1326*a^7*b^12*d^10 - 3134*a^9*b^10*d^10 +

3194*a^11*b^8*d^10 - 1536*a^13*b^6*d^10 + 288*a^15*b^4*d^10 - 80*a*b^18*c^2*d^8 - 800*a*b^18*c^4*d^6 + 400*a*b

^18*c^6*d^4 + 40*a*b^18*c^8*d^2 + 120*a^2*b^17*c*d^9 - 30*a^2*b^17*c^9*d + 900*a^4*b^15*c*d^9 - 60*a^4*b^15*c^

9*d - 7920*a^6*b^13*c*d^9 + 17160*a^8*b^11*c*d^9 - 16710*a^10*b^9*c*d^9 + 7800*a^12*b^7*c*d^9 - 1440*a^14*b^5*

c*d^9 + 2400*a^2*b^17*c^3*d^7 - 2400*a^2*b^17*c^5*d^5 - 720*a^2*b^17*c^7*d^3 - 2400*a^3*b^16*c^2*d^8 + 9600*a^

3*b^16*c^4*d^6 + 2320*a^3*b^16*c^6*d^4 + 325*a^3*b^16*c^8*d^2 - 18800*a^4*b^15*c^3*d^7 - 1040*a^4*b^15*c^5*d^5

- 440*a^4*b^15*c^7*d^3 + 17780*a^5*b^14*c^2*d^8 - 13600*a^5*b^14*c^4*d^6 - 1310*a^5*b^14*c^6*d^4 + 40*a^5*b^1

4*c^8*d^2 + 34960*a^6*b^13*c^3*d^7 + 2428*a^6*b^13*c^5*d^5 + 160*a^6*b^13*c^7*d^3 - 36000*a^7*b^12*c^2*d^8 + 9

330*a^7*b^12*c^4*d^6 + 360*a^7*b^12*c^6*d^4 - 30200*a^8*b^11*c^3*d^7 - 1208*a^8*b^11*c^5*d^5 - 80*a^8*b^11*c^7

*d^3 + 33445*a^9*b^10*c^2*d^8 - 3440*a^9*b^10*c^4*d^6 + 120*a^9*b^10*c^6*d^4 + 12960*a^10*b^9*c^3*d^7 - 48*a^1

0*b^9*c^5*d^5 - 15100*a^11*b^8*c^2*d^8 + 800*a^11*b^8*c^4*d^6 - 2400*a^12*b^7*c^3*d^7 + 2760*a^13*b^6*c^2*d^8)

)/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + ((a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*

b*c*d^4*15i)*((8*tan(e/2 + (f*x)/2)*(4*a*b^21*c^5 - 12*a^5*b^17*c^5 + 8*a^7*b^15*c^5 - 80*a^4*b^18*d^5 + 276*a

^6*b^16*d^5 - 360*a^8*b^14*d^5 + 212*a^10*b^12*d^5 - 48*a^12*b^10*d^5 + 80*a*b^21*c^3*d^2 - 60*a^2*b^20*c^4*d

+ 240*a^3*b^19*c*d^4 + 120*a^4*b^18*c^4*d - 780*a^5*b^17*c*d^4 - 60*a^6*b^16*c^4*d + 960*a^7*b^15*c*d^4 - 540*

a^9*b^13*c*d^4 + 120*a^11*b^11*c*d^4 - 240*a^2*b^20*c^2*d^3 - 120*a^3*b^19*c^3*d^2 + 680*a^4*b^18*c^2*d^3 - 72

0*a^6*b^16*c^2*d^3 + 40*a^7*b^15*c^3*d^2 + 360*a^8*b^14*c^2*d^3 - 80*a^10*b^12*c^2*d^3))/(b^20 - 4*a^2*b^18 +

6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) - (4*(4*a*b^20*d^5 - 4*a^2*b^19*c^5 + 12*a^6*b^15*c^5 - 8*a^8*b^13*c^5 + 2

8*a^3*b^18*d^5 - 120*a^5*b^16*d^5 + 164*a^7*b^14*d^5 - 100*a^9*b^12*d^5 + 24*a^11*b^10*d^5 + 80*a*b^20*c^2*d^3

- 120*a^2*b^19*c*d^4 + 60*a^3*b^18*c^4*d + 360*a^4*b^17*c*d^4 - 120*a^5*b^16*c^4*d - 420*a^6*b^15*c*d^4 + 60*

a^7*b^14*c^4*d + 240*a^8*b^13*c*d^4 - 60*a^10*b^11*c*d^4 - 80*a^2*b^19*c^3*d^2 - 160*a^3*b^18*c^2*d^3 + 120*a^

4*b^17*c^3*d^2 + 120*a^5*b^16*c^2*d^3 - 80*a^7*b^14*c^2*d^3 - 40*a^8*b^13*c^3*d^2 + 40*a^9*b^12*c^2*d^3))/(b^1

9 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b

^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(e/2 + (f*x)/2)*(12*a*b^2

4 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4

*a^6*b^14 + a^8*b^12))*(a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i))/b^5))/b^5)*(a^2*d^5*6i +

(b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i)*1i)/b^5 + (((4*(2*a^2*b^16*d^10 + 40*a^4*b^14*d^10 + 108*a^6*b^

12*d^10 - 872*a^8*b^10*d^10 + 1538*a^10*b^8*d^10 - 1104*a^12*b^6*d^10 + 288*a^14*b^4*d^10 - 120*a^3*b^15*c*d^9

- 960*a^5*b^13*c*d^9 + 5040*a^7*b^11*c*d^9 - 8160*a^9*b^9*c*d^9 + 5640*a^11*b^7*c*d^9 - 1440*a^13*b^5*c*d^9 +

80*a^2*b^16*c^2*d^8 + 800*a^2*b^16*c^4*d^6 - 2400*a^3*b^15*c^3*d^7 + 2440*a^4*b^14*c^2*d^8 - 3200*a^4*b^14*c^

4*d^6 + 9600*a^5*b^13*c^3*d^7 - 10560*a^6*b^12*c^2*d^8 + 4800*a^6*b^12*c^4*d^6 - 14400*a^7*b^11*c^3*d^7 + 1624

0*a^8*b^10*c^2*d^8 - 3200*a^8*b^10*c^4*d^6 + 9600*a^9*b^9*c^3*d^7 - 10960*a^10*b^8*c^2*d^8 + 800*a^10*b^8*c^4*

d^6 - 2400*a^11*b^7*c^3*d^7 + 2760*a^12*b^6*c^2*d^8))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11)

- (8*tan(e/2 + (f*x)/2)*(a*b^18*c^10 - 2*a*b^18*d^10 + 4*a^3*b^16*c^10 + 4*a^5*b^14*c^10 - 39*a^3*b^16*d^10 -

88*a^5*b^14*d^10 + 1326*a^7*b^12*d^10 - 3134*a^9*b^10*d^10 + 3194*a^11*b^8*d^10 - 1536*a^13*b^6*d^10 + 288*a^

15*b^4*d^10 - 80*a*b^18*c^2*d^8 - 800*a*b^18*c^4*d^6 + 400*a*b^18*c^6*d^4 + 40*a*b^18*c^8*d^2 + 120*a^2*b^17*c

*d^9 - 30*a^2*b^17*c^9*d + 900*a^4*b^15*c*d^9 - 60*a^4*b^15*c^9*d - 7920*a^6*b^13*c*d^9 + 17160*a^8*b^11*c*d^9

- 16710*a^10*b^9*c*d^9 + 7800*a^12*b^7*c*d^9 - 1440*a^14*b^5*c*d^9 + 2400*a^2*b^17*c^3*d^7 - 2400*a^2*b^17*c^

5*d^5 - 720*a^2*b^17*c^7*d^3 - 2400*a^3*b^16*c^2*d^8 + 9600*a^3*b^16*c^4*d^6 + 2320*a^3*b^16*c^6*d^4 + 325*a^3

*b^16*c^8*d^2 - 18800*a^4*b^15*c^3*d^7 - 1040*a^4*b^15*c^5*d^5 - 440*a^4*b^15*c^7*d^3 + 17780*a^5*b^14*c^2*d^8

- 13600*a^5*b^14*c^4*d^6 - 1310*a^5*b^14*c^6*d^4 + 40*a^5*b^14*c^8*d^2 + 34960*a^6*b^13*c^3*d^7 + 2428*a^6*b^

13*c^5*d^5 + 160*a^6*b^13*c^7*d^3 - 36000*a^7*b^12*c^2*d^8 + 9330*a^7*b^12*c^4*d^6 + 360*a^7*b^12*c^6*d^4 - 30

200*a^8*b^11*c^3*d^7 - 1208*a^8*b^11*c^5*d^5 - 80*a^8*b^11*c^7*d^3 + 33445*a^9*b^10*c^2*d^8 - 3440*a^9*b^10*c^

4*d^6 + 120*a^9*b^10*c^6*d^4 + 12960*a^10*b^9*c^3*d^7 - 48*a^10*b^9*c^5*d^5 - 15100*a^11*b^8*c^2*d^8 + 800*a^1

1*b^8*c^4*d^6 - 2400*a^12*b^7*c^3*d^7 + 2760*a^13*b^6*c^2*d^8))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 +

a^8*b^12) + ((a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i)*((4*(4*a*b^20*d^5 - 4*a^2*b^19*c^5

+ 12*a^6*b^15*c^5 - 8*a^8*b^13*c^5 + 28*a^3*b^18*d^5 - 120*a^5*b^16*d^5 + 164*a^7*b^14*d^5 - 100*a^9*b^12*d^5

+ 24*a^11*b^10*d^5 + 80*a*b^20*c^2*d^3 - 120*a^2*b^19*c*d^4 + 60*a^3*b^18*c^4*d + 360*a^4*b^17*c*d^4 - 120*a^5

*b^16*c^4*d - 420*a^6*b^15*c*d^4 + 60*a^7*b^14*c^4*d + 240*a^8*b^13*c*d^4 - 60*a^10*b^11*c*d^4 - 80*a^2*b^19*c

^3*d^2 - 160*a^3*b^18*c^2*d^3 + 120*a^4*b^17*c^3*d^2 + 120*a^5*b^16*c^2*d^3 - 80*a^7*b^14*c^2*d^3 - 40*a^8*b^1

3*c^3*d^2 + 40*a^9*b^12*c^2*d^3))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/2 + (f*x

)/2)*(4*a*b^21*c^5 - 12*a^5*b^17*c^5 + 8*a^7*b^15*c^5 - 80*a^4*b^18*d^5 + 276*a^6*b^16*d^5 - 360*a^8*b^14*d^5

+ 212*a^10*b^12*d^5 - 48*a^12*b^10*d^5 + 80*a*b^21*c^3*d^2 - 60*a^2*b^20*c^4*d + 240*a^3*b^19*c*d^4 + 120*a^4*

b^18*c^4*d - 780*a^5*b^17*c*d^4 - 60*a^6*b^16*c^4*d + 960*a^7*b^15*c*d^4 - 540*a^9*b^13*c*d^4 + 120*a^11*b^11*

c*d^4 - 240*a^2*b^20*c^2*d^3 - 120*a^3*b^19*c^3*d^2 + 680*a^4*b^18*c^2*d^3 - 720*a^6*b^16*c^2*d^3 + 40*a^7*b^1

5*c^3*d^2 + 360*a^8*b^14*c^2*d^3 - 80*a^10*b^12*c^2*d^3))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b

^12) + (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b

^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(e/2 + (f*x)/2)*(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 4

4*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12))*(a^2*d^5*6i + (b^2*d^3*(2

0*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i))/b^5))/b^5)*(a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i)*1

i)/b^5)/((8*(2326*a^9*b^6*d^15 - 20*a^5*b^10*d^15 - 11*a^7*b^8*d^15 - 864*a^15*d^15 - 4770*a^11*b^4*d^15 + 345

6*a^13*b^2*d^15 + 400*a*b^14*c^6*d^9 + 8040*a*b^14*c^8*d^7 + 801*a*b^14*c^10*d^5 + 20*a*b^14*c^12*d^3 + 60*a^4

*b^11*c*d^14 + 45*a^6*b^9*c*d^14 - 19860*a^8*b^7*c*d^14 + 38835*a^10*b^5*c*d^14 - 27000*a^12*b^3*c*d^14 + 20*a

^2*b^13*c^3*d^12 - 1599*a^2*b^13*c^5*d^10 - 52680*a^2*b^13*c^7*d^8 - 15230*a^2*b^13*c^9*d^6 - 630*a^2*b^13*c^1

1*d^4 - 60*a^3*b^12*c^2*d^13 + 2385*a^3*b^12*c^4*d^11 + 150460*a^3*b^12*c^6*d^9 + 61605*a^3*b^12*c^8*d^7 + 741

6*a^3*b^12*c^10*d^5 + 80*a^3*b^12*c^12*d^3 - 1550*a^4*b^11*c^3*d^12 - 246516*a^4*b^11*c^5*d^10 - 92100*a^4*b^1

1*c^7*d^8 - 18970*a^4*b^11*c^9*d^6 - 1320*a^4*b^11*c^11*d^4 + 330*a^5*b^10*c^2*d^13 + 255870*a^5*b^10*c^4*d^11

- 2490*a^5*b^10*c^6*d^9 + 2940*a^5*b^10*c^8*d^7 + 2652*a^5*b^10*c^10*d^5 + 80*a^5*b^10*c^12*d^3 - 174080*a^6*

b^9*c^3*d^12 + 206889*a^6*b^9*c^5*d^10 + 46620*a^6*b^9*c^7*d^8 + 80*a^6*b^9*c^9*d^6 - 120*a^6*b^9*c^11*d^4 + 7

6440*a^7*b^8*c^2*d^13 - 335925*a^7*b^8*c^4*d^11 - 44620*a^7*b^8*c^6*d^9 + 480*a^7*b^8*c^8*d^7 + 48*a^7*b^8*c^1

0*d^5 + 281510*a^8*b^7*c^3*d^12 - 60342*a^8*b^7*c^5*d^10 - 15180*a^8*b^7*c^7*d^8 - 800*a^8*b^7*c^9*d^6 - 13912

5*a^9*b^6*c^2*d^13 + 167580*a^9*b^6*c^4*d^11 + 25220*a^9*b^6*c^6*d^9 + 2400*a^9*b^6*c^8*d^7 - 167550*a^10*b^5*

c^3*d^12 - 5928*a^10*b^5*c^5*d^10 - 2760*a^10*b^5*c^7*d^8 + 91080*a^11*b^4*c^2*d^13 - 24840*a^11*b^4*c^4*d^11

+ 1440*a^11*b^4*c^6*d^9 + 33660*a^12*b^3*c^3*d^12 - 288*a^12*b^3*c^5*d^10 - 20520*a^13*b^2*c^2*d^13 + 6480*a^1

4*b*c*d^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (16*tan(e/2 + (f*x)/2)*(7829*a^10*b^6*

d^15 - 20*a^4*b^12*d^15 - 411*a^6*b^10*d^15 - 1314*a^8*b^8*d^15 - 1728*a^16*d^15 - 11700*a^12*b^4*d^15 + 7344*

a^14*b^2*d^15 + 20*a*b^15*c^3*d^12 + 801*a*b^15*c^5*d^10 + 8040*a*b^15*c^7*d^8 + 400*a*b^15*c^9*d^6 + 60*a^3*b

^13*c*d^14 + 2445*a^5*b^11*c*d^14 + 14460*a^7*b^9*c*d^14 - 66735*a^9*b^7*c*d^14 + 92970*a^11*b^5*c*d^14 - 5616

0*a^13*b^3*c*d^14 - 60*a^2*b^14*c^2*d^13 - 3615*a^2*b^14*c^4*d^11 - 48660*a^2*b^14*c^6*d^9 - 7200*a^2*b^14*c^8

*d^7 + 6450*a^3*b^13*c^3*d^12 + 123324*a^3*b^13*c^5*d^10 + 7380*a^3*b^13*c^7*d^8 - 5670*a^4*b^12*c^2*d^13 - 16

8930*a^4*b^12*c^4*d^11 + 83780*a^4*b^12*c^6*d^9 + 12000*a^4*b^12*c^8*d^7 + 134160*a^5*b^11*c^3*d^12 - 314259*a

^5*b^11*c^5*d^10 - 36120*a^5*b^11*c^7*d^8 - 1200*a^5*b^11*c^9*d^6 - 61080*a^6*b^10*c^2*d^13 + 509145*a^6*b^10*

c^4*d^11 - 31020*a^6*b^10*c^6*d^9 - 2400*a^6*b^10*c^8*d^7 - 458210*a^7*b^9*c^3*d^12 + 291630*a^7*b^9*c^5*d^10

+ 17940*a^7*b^9*c^7*d^8 + 800*a^7*b^9*c^9*d^6 + 237870*a^8*b^8*c^2*d^13 - 565440*a^8*b^8*c^4*d^11 + 5340*a^8*b

^8*c^6*d^9 - 2400*a^8*b^8*c^8*d^7 + 558240*a^9*b^7*c^3*d^12 - 137784*a^9*b^7*c^5*d^10 + 2760*a^9*b^7*c^7*d^8 -

310560*a^10*b^6*c^2*d^13 + 297240*a^10*b^6*c^4*d^11 - 9440*a^10*b^6*c^6*d^9 - 310860*a^11*b^5*c^3*d^12 + 3628

8*a^11*b^5*c^5*d^10 + 180540*a^12*b^4*c^2*d^13 - 68400*a^12*b^4*c^4*d^11 + 70200*a^13*b^3*c^3*d^12 - 41040*a^1

4*b^2*c^2*d^13 + 12960*a^15*b*c*d^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + (((4*(2*a^2*

b^16*d^10 + 40*a^4*b^14*d^10 + 108*a^6*b^12*d^10 - 872*a^8*b^10*d^10 + 1538*a^10*b^8*d^10 - 1104*a^12*b^6*d^10

+ 288*a^14*b^4*d^10 - 120*a^3*b^15*c*d^9 - 960*a^5*b^13*c*d^9 + 5040*a^7*b^11*c*d^9 - 8160*a^9*b^9*c*d^9 + 56

40*a^11*b^7*c*d^9 - 1440*a^13*b^5*c*d^9 + 80*a^2*b^16*c^2*d^8 + 800*a^2*b^16*c^4*d^6 - 2400*a^3*b^15*c^3*d^7 +

2440*a^4*b^14*c^2*d^8 - 3200*a^4*b^14*c^4*d^6 + 9600*a^5*b^13*c^3*d^7 - 10560*a^6*b^12*c^2*d^8 + 4800*a^6*b^1

2*c^4*d^6 - 14400*a^7*b^11*c^3*d^7 + 16240*a^8*b^10*c^2*d^8 - 3200*a^8*b^10*c^4*d^6 + 9600*a^9*b^9*c^3*d^7 - 1

0960*a^10*b^8*c^2*d^8 + 800*a^10*b^8*c^4*d^6 - 2400*a^11*b^7*c^3*d^7 + 2760*a^12*b^6*c^2*d^8))/(b^19 - 4*a^2*b

^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/2 + (f*x)/2)*(a*b^18*c^10 - 2*a*b^18*d^10 + 4*a^3*b^16*c^

10 + 4*a^5*b^14*c^10 - 39*a^3*b^16*d^10 - 88*a^5*b^14*d^10 + 1326*a^7*b^12*d^10 - 3134*a^9*b^10*d^10 + 3194*a^

11*b^8*d^10 - 1536*a^13*b^6*d^10 + 288*a^15*b^4*d^10 - 80*a*b^18*c^2*d^8 - 800*a*b^18*c^4*d^6 + 400*a*b^18*c^6

*d^4 + 40*a*b^18*c^8*d^2 + 120*a^2*b^17*c*d^9 - 30*a^2*b^17*c^9*d + 900*a^4*b^15*c*d^9 - 60*a^4*b^15*c^9*d - 7

920*a^6*b^13*c*d^9 + 17160*a^8*b^11*c*d^9 - 16710*a^10*b^9*c*d^9 + 7800*a^12*b^7*c*d^9 - 1440*a^14*b^5*c*d^9 +

2400*a^2*b^17*c^3*d^7 - 2400*a^2*b^17*c^5*d^5 - 720*a^2*b^17*c^7*d^3 - 2400*a^3*b^16*c^2*d^8 + 9600*a^3*b^16*

c^4*d^6 + 2320*a^3*b^16*c^6*d^4 + 325*a^3*b^16*c^8*d^2 - 18800*a^4*b^15*c^3*d^7 - 1040*a^4*b^15*c^5*d^5 - 440*

a^4*b^15*c^7*d^3 + 17780*a^5*b^14*c^2*d^8 - 13600*a^5*b^14*c^4*d^6 - 1310*a^5*b^14*c^6*d^4 + 40*a^5*b^14*c^8*d

^2 + 34960*a^6*b^13*c^3*d^7 + 2428*a^6*b^13*c^5*d^5 + 160*a^6*b^13*c^7*d^3 - 36000*a^7*b^12*c^2*d^8 + 9330*a^7

*b^12*c^4*d^6 + 360*a^7*b^12*c^6*d^4 - 30200*a^8*b^11*c^3*d^7 - 1208*a^8*b^11*c^5*d^5 - 80*a^8*b^11*c^7*d^3 +

33445*a^9*b^10*c^2*d^8 - 3440*a^9*b^10*c^4*d^6 + 120*a^9*b^10*c^6*d^4 + 12960*a^10*b^9*c^3*d^7 - 48*a^10*b^9*c

^5*d^5 - 15100*a^11*b^8*c^2*d^8 + 800*a^11*b^8*c^4*d^6 - 2400*a^12*b^7*c^3*d^7 + 2760*a^13*b^6*c^2*d^8))/(b^20

- 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + ((a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4

*15i)*((8*tan(e/2 + (f*x)/2)*(4*a*b^21*c^5 - 12*a^5*b^17*c^5 + 8*a^7*b^15*c^5 - 80*a^4*b^18*d^5 + 276*a^6*b^16

*d^5 - 360*a^8*b^14*d^5 + 212*a^10*b^12*d^5 - 48*a^12*b^10*d^5 + 80*a*b^21*c^3*d^2 - 60*a^2*b^20*c^4*d + 240*a

^3*b^19*c*d^4 + 120*a^4*b^18*c^4*d - 780*a^5*b^17*c*d^4 - 60*a^6*b^16*c^4*d + 960*a^7*b^15*c*d^4 - 540*a^9*b^1

3*c*d^4 + 120*a^11*b^11*c*d^4 - 240*a^2*b^20*c^2*d^3 - 120*a^3*b^19*c^3*d^2 + 680*a^4*b^18*c^2*d^3 - 720*a^6*b

^16*c^2*d^3 + 40*a^7*b^15*c^3*d^2 + 360*a^8*b^14*c^2*d^3 - 80*a^10*b^12*c^2*d^3))/(b^20 - 4*a^2*b^18 + 6*a^4*b

^16 - 4*a^6*b^14 + a^8*b^12) - (4*(4*a*b^20*d^5 - 4*a^2*b^19*c^5 + 12*a^6*b^15*c^5 - 8*a^8*b^13*c^5 + 28*a^3*b

^18*d^5 - 120*a^5*b^16*d^5 + 164*a^7*b^14*d^5 - 100*a^9*b^12*d^5 + 24*a^11*b^10*d^5 + 80*a*b^20*c^2*d^3 - 120*

a^2*b^19*c*d^4 + 60*a^3*b^18*c^4*d + 360*a^4*b^17*c*d^4 - 120*a^5*b^16*c^4*d - 420*a^6*b^15*c*d^4 + 60*a^7*b^1

4*c^4*d + 240*a^8*b^13*c*d^4 - 60*a^10*b^11*c*d^4 - 80*a^2*b^19*c^3*d^2 - 160*a^3*b^18*c^2*d^3 + 120*a^4*b^17*

c^3*d^2 + 120*a^5*b^16*c^2*d^3 - 80*a^7*b^14*c^2*d^3 - 40*a^8*b^13*c^3*d^2 + 40*a^9*b^12*c^2*d^3))/(b^19 - 4*a

^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^16 + 8

*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(e/2 + (f*x)/2)*(12*a*b^24 - 56*

a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^

14 + a^8*b^12))*(a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i))/b^5))/b^5)*(a^2*d^5*6i + (b^2*d^

3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i))/b^5 - (((4*(2*a^2*b^16*d^10 + 40*a^4*b^14*d^10 + 108*a^6*b^12*d^10 -

872*a^8*b^10*d^10 + 1538*a^10*b^8*d^10 - 1104*a^12*b^6*d^10 + 288*a^14*b^4*d^10 - 120*a^3*b^15*c*d^9 - 960*a^5

*b^13*c*d^9 + 5040*a^7*b^11*c*d^9 - 8160*a^9*b^9*c*d^9 + 5640*a^11*b^7*c*d^9 - 1440*a^13*b^5*c*d^9 + 80*a^2*b^

16*c^2*d^8 + 800*a^2*b^16*c^4*d^6 - 2400*a^3*b^15*c^3*d^7 + 2440*a^4*b^14*c^2*d^8 - 3200*a^4*b^14*c^4*d^6 + 96

00*a^5*b^13*c^3*d^7 - 10560*a^6*b^12*c^2*d^8 + 4800*a^6*b^12*c^4*d^6 - 14400*a^7*b^11*c^3*d^7 + 16240*a^8*b^10

*c^2*d^8 - 3200*a^8*b^10*c^4*d^6 + 9600*a^9*b^9*c^3*d^7 - 10960*a^10*b^8*c^2*d^8 + 800*a^10*b^8*c^4*d^6 - 2400

*a^11*b^7*c^3*d^7 + 2760*a^12*b^6*c^2*d^8))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(

e/2 + (f*x)/2)*(a*b^18*c^10 - 2*a*b^18*d^10 + 4*a^3*b^16*c^10 + 4*a^5*b^14*c^10 - 39*a^3*b^16*d^10 - 88*a^5*b^

14*d^10 + 1326*a^7*b^12*d^10 - 3134*a^9*b^10*d^10 + 3194*a^11*b^8*d^10 - 1536*a^13*b^6*d^10 + 288*a^15*b^4*d^1

0 - 80*a*b^18*c^2*d^8 - 800*a*b^18*c^4*d^6 + 400*a*b^18*c^6*d^4 + 40*a*b^18*c^8*d^2 + 120*a^2*b^17*c*d^9 - 30*

a^2*b^17*c^9*d + 900*a^4*b^15*c*d^9 - 60*a^4*b^15*c^9*d - 7920*a^6*b^13*c*d^9 + 17160*a^8*b^11*c*d^9 - 16710*a

^10*b^9*c*d^9 + 7800*a^12*b^7*c*d^9 - 1440*a^14*b^5*c*d^9 + 2400*a^2*b^17*c^3*d^7 - 2400*a^2*b^17*c^5*d^5 - 72

0*a^2*b^17*c^7*d^3 - 2400*a^3*b^16*c^2*d^8 + 9600*a^3*b^16*c^4*d^6 + 2320*a^3*b^16*c^6*d^4 + 325*a^3*b^16*c^8*

d^2 - 18800*a^4*b^15*c^3*d^7 - 1040*a^4*b^15*c^5*d^5 - 440*a^4*b^15*c^7*d^3 + 17780*a^5*b^14*c^2*d^8 - 13600*a

^5*b^14*c^4*d^6 - 1310*a^5*b^14*c^6*d^4 + 40*a^5*b^14*c^8*d^2 + 34960*a^6*b^13*c^3*d^7 + 2428*a^6*b^13*c^5*d^5

+ 160*a^6*b^13*c^7*d^3 - 36000*a^7*b^12*c^2*d^8 + 9330*a^7*b^12*c^4*d^6 + 360*a^7*b^12*c^6*d^4 - 30200*a^8*b^

11*c^3*d^7 - 1208*a^8*b^11*c^5*d^5 - 80*a^8*b^11*c^7*d^3 + 33445*a^9*b^10*c^2*d^8 - 3440*a^9*b^10*c^4*d^6 + 12

0*a^9*b^10*c^6*d^4 + 12960*a^10*b^9*c^3*d^7 - 48*a^10*b^9*c^5*d^5 - 15100*a^11*b^8*c^2*d^8 + 800*a^11*b^8*c^4*

d^6 - 2400*a^12*b^7*c^3*d^7 + 2760*a^13*b^6*c^2*d^8))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12)

+ ((a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i)*((4*(4*a*b^20*d^5 - 4*a^2*b^19*c^5 + 12*a^6*b

^15*c^5 - 8*a^8*b^13*c^5 + 28*a^3*b^18*d^5 - 120*a^5*b^16*d^5 + 164*a^7*b^14*d^5 - 100*a^9*b^12*d^5 + 24*a^11*

b^10*d^5 + 80*a*b^20*c^2*d^3 - 120*a^2*b^19*c*d^4 + 60*a^3*b^18*c^4*d + 360*a^4*b^17*c*d^4 - 120*a^5*b^16*c^4*

d - 420*a^6*b^15*c*d^4 + 60*a^7*b^14*c^4*d + 240*a^8*b^13*c*d^4 - 60*a^10*b^11*c*d^4 - 80*a^2*b^19*c^3*d^2 - 1

60*a^3*b^18*c^2*d^3 + 120*a^4*b^17*c^3*d^2 + 120*a^5*b^16*c^2*d^3 - 80*a^7*b^14*c^2*d^3 - 40*a^8*b^13*c^3*d^2

+ 40*a^9*b^12*c^2*d^3))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/2 + (f*x)/2)*(4*a*

b^21*c^5 - 12*a^5*b^17*c^5 + 8*a^7*b^15*c^5 - 80*a^4*b^18*d^5 + 276*a^6*b^16*d^5 - 360*a^8*b^14*d^5 + 212*a^10

*b^12*d^5 - 48*a^12*b^10*d^5 + 80*a*b^21*c^3*d^2 - 60*a^2*b^20*c^4*d + 240*a^3*b^19*c*d^4 + 120*a^4*b^18*c^4*d

- 780*a^5*b^17*c*d^4 - 60*a^6*b^16*c^4*d + 960*a^7*b^15*c*d^4 - 540*a^9*b^13*c*d^4 + 120*a^11*b^11*c*d^4 - 24

0*a^2*b^20*c^2*d^3 - 120*a^3*b^19*c^3*d^2 + 680*a^4*b^18*c^2*d^3 - 720*a^6*b^16*c^2*d^3 + 40*a^7*b^15*c^3*d^2

+ 360*a^8*b^14*c^2*d^3 - 80*a^10*b^12*c^2*d^3))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + (((

4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^

6*b^13 + a^8*b^11) + (8*tan(e/2 + (f*x)/2)*(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16

- 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12))*(a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^

2)*1i)/2 - a*b*c*d^4*15i))/b^5))/b^5)*(a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i))/b^5))*(a^2

*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*b*c*d^4*15i)*2i)/(b^5*f) - (atan((((a*d - b*c)^3*(-(a + b)^5*(a -

b)^5)^(1/2)*((4*(2*a^2*b^16*d^10 + 40*a^4*b^14*d^10 + 108*a^6*b^12*d^10 - 872*a^8*b^10*d^10 + 1538*a^10*b^8*d^

10 - 1104*a^12*b^6*d^10 + 288*a^14*b^4*d^10 - 120*a^3*b^15*c*d^9 - 960*a^5*b^13*c*d^9 + 5040*a^7*b^11*c*d^9 -

8160*a^9*b^9*c*d^9 + 5640*a^11*b^7*c*d^9 - 1440*a^13*b^5*c*d^9 + 80*a^2*b^16*c^2*d^8 + 800*a^2*b^16*c^4*d^6 -

2400*a^3*b^15*c^3*d^7 + 2440*a^4*b^14*c^2*d^8 - 3200*a^4*b^14*c^4*d^6 + 9600*a^5*b^13*c^3*d^7 - 10560*a^6*b^12

*c^2*d^8 + 4800*a^6*b^12*c^4*d^6 - 14400*a^7*b^11*c^3*d^7 + 16240*a^8*b^10*c^2*d^8 - 3200*a^8*b^10*c^4*d^6 + 9

600*a^9*b^9*c^3*d^7 - 10960*a^10*b^8*c^2*d^8 + 800*a^10*b^8*c^4*d^6 - 2400*a^11*b^7*c^3*d^7 + 2760*a^12*b^6*c^

2*d^8))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/2 + (f*x)/2)*(a*b^18*c^10 - 2*a*b^

18*d^10 + 4*a^3*b^16*c^10 + 4*a^5*b^14*c^10 - 39*a^3*b^16*d^10 - 88*a^5*b^14*d^10 + 1326*a^7*b^12*d^10 - 3134*

a^9*b^10*d^10 + 3194*a^11*b^8*d^10 - 1536*a^13*b^6*d^10 + 288*a^15*b^4*d^10 - 80*a*b^18*c^2*d^8 - 800*a*b^18*c

^4*d^6 + 400*a*b^18*c^6*d^4 + 40*a*b^18*c^8*d^2 + 120*a^2*b^17*c*d^9 - 30*a^2*b^17*c^9*d + 900*a^4*b^15*c*d^9

- 60*a^4*b^15*c^9*d - 7920*a^6*b^13*c*d^9 + 17160*a^8*b^11*c*d^9 - 16710*a^10*b^9*c*d^9 + 7800*a^12*b^7*c*d^9

- 1440*a^14*b^5*c*d^9 + 2400*a^2*b^17*c^3*d^7 - 2400*a^2*b^17*c^5*d^5 - 720*a^2*b^17*c^7*d^3 - 2400*a^3*b^16*c

^2*d^8 + 9600*a^3*b^16*c^4*d^6 + 2320*a^3*b^16*c^6*d^4 + 325*a^3*b^16*c^8*d^2 - 18800*a^4*b^15*c^3*d^7 - 1040*

a^4*b^15*c^5*d^5 - 440*a^4*b^15*c^7*d^3 + 17780*a^5*b^14*c^2*d^8 - 13600*a^5*b^14*c^4*d^6 - 1310*a^5*b^14*c^6*

d^4 + 40*a^5*b^14*c^8*d^2 + 34960*a^6*b^13*c^3*d^7 + 2428*a^6*b^13*c^5*d^5 + 160*a^6*b^13*c^7*d^3 - 36000*a^7*

b^12*c^2*d^8 + 9330*a^7*b^12*c^4*d^6 + 360*a^7*b^12*c^6*d^4 - 30200*a^8*b^11*c^3*d^7 - 1208*a^8*b^11*c^5*d^5 -

80*a^8*b^11*c^7*d^3 + 33445*a^9*b^10*c^2*d^8 - 3440*a^9*b^10*c^4*d^6 + 120*a^9*b^10*c^6*d^4 + 12960*a^10*b^9*

c^3*d^7 - 48*a^10*b^9*c^5*d^5 - 15100*a^11*b^8*c^2*d^8 + 800*a^11*b^8*c^4*d^6 - 2400*a^12*b^7*c^3*d^7 + 2760*a

^13*b^6*c^2*d^8))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + ((a*d - b*c)^3*(-(a + b)^5*(a - b

)^5)^(1/2)*((8*tan(e/2 + (f*x)/2)*(4*a*b^21*c^5 - 12*a^5*b^17*c^5 + 8*a^7*b^15*c^5 - 80*a^4*b^18*d^5 + 276*a^6

*b^16*d^5 - 360*a^8*b^14*d^5 + 212*a^10*b^12*d^5 - 48*a^12*b^10*d^5 + 80*a*b^21*c^3*d^2 - 60*a^2*b^20*c^4*d +

240*a^3*b^19*c*d^4 + 120*a^4*b^18*c^4*d - 780*a^5*b^17*c*d^4 - 60*a^6*b^16*c^4*d + 960*a^7*b^15*c*d^4 - 540*a^

9*b^13*c*d^4 + 120*a^11*b^11*c*d^4 - 240*a^2*b^20*c^2*d^3 - 120*a^3*b^19*c^3*d^2 + 680*a^4*b^18*c^2*d^3 - 720*

a^6*b^16*c^2*d^3 + 40*a^7*b^15*c^3*d^2 + 360*a^8*b^14*c^2*d^3 - 80*a^10*b^12*c^2*d^3))/(b^20 - 4*a^2*b^18 + 6*

a^4*b^16 - 4*a^6*b^14 + a^8*b^12) - (4*(4*a*b^20*d^5 - 4*a^2*b^19*c^5 + 12*a^6*b^15*c^5 - 8*a^8*b^13*c^5 + 28*

a^3*b^18*d^5 - 120*a^5*b^16*d^5 + 164*a^7*b^14*d^5 - 100*a^9*b^12*d^5 + 24*a^11*b^10*d^5 + 80*a*b^20*c^2*d^3 -

120*a^2*b^19*c*d^4 + 60*a^3*b^18*c^4*d + 360*a^4*b^17*c*d^4 - 120*a^5*b^16*c^4*d - 420*a^6*b^15*c*d^4 + 60*a^

7*b^14*c^4*d + 240*a^8*b^13*c*d^4 - 60*a^10*b^11*c*d^4 - 80*a^2*b^19*c^3*d^2 - 160*a^3*b^18*c^2*d^3 + 120*a^4*

b^17*c^3*d^2 + 120*a^5*b^16*c^2*d^3 - 80*a^7*b^14*c^2*d^3 - 40*a^8*b^13*c^3*d^2 + 40*a^9*b^12*c^2*d^3))/(b^19

- 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^1

6 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(e/2 + (f*x)/2)*(12*a*b^24

- 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a

^6*b^14 + a^8*b^12))*(a*d - b*c)^3*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4*d^2 + b^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2

*c^2 - 29*a^2*b^2*d^2 - 12*a*b^3*c*d + 6*a^3*b*c*d))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*

b^7 - a^10*b^5)))*(12*a^4*d^2 + b^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^2 - 12*a*b^3*c*d + 6*a^3*b

*c*d))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*a^4*d^2 + b^4*c^2 + 20*b

^4*d^2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^2 - 12*a*b^3*c*d + 6*a^3*b*c*d)*1i)/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11

- 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)) + ((a*d - b*c)^3*(-(a + b)^5*(a - b)^5)^(1/2)*((4*(2*a^2*b^16*d^10 + 40*

a^4*b^14*d^10 + 108*a^6*b^12*d^10 - 872*a^8*b^10*d^10 + 1538*a^10*b^8*d^10 - 1104*a^12*b^6*d^10 + 288*a^14*b^4

*d^10 - 120*a^3*b^15*c*d^9 - 960*a^5*b^13*c*d^9 + 5040*a^7*b^11*c*d^9 - 8160*a^9*b^9*c*d^9 + 5640*a^11*b^7*c*d

^9 - 1440*a^13*b^5*c*d^9 + 80*a^2*b^16*c^2*d^8 + 800*a^2*b^16*c^4*d^6 - 2400*a^3*b^15*c^3*d^7 + 2440*a^4*b^14*

c^2*d^8 - 3200*a^4*b^14*c^4*d^6 + 9600*a^5*b^13*c^3*d^7 - 10560*a^6*b^12*c^2*d^8 + 4800*a^6*b^12*c^4*d^6 - 144

00*a^7*b^11*c^3*d^7 + 16240*a^8*b^10*c^2*d^8 - 3200*a^8*b^10*c^4*d^6 + 9600*a^9*b^9*c^3*d^7 - 10960*a^10*b^8*c

^2*d^8 + 800*a^10*b^8*c^4*d^6 - 2400*a^11*b^7*c^3*d^7 + 2760*a^12*b^6*c^2*d^8))/(b^19 - 4*a^2*b^17 + 6*a^4*b^1

5 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/2 + (f*x)/2)*(a*b^18*c^10 - 2*a*b^18*d^10 + 4*a^3*b^16*c^10 + 4*a^5*b^14

*c^10 - 39*a^3*b^16*d^10 - 88*a^5*b^14*d^10 + 1326*a^7*b^12*d^10 - 3134*a^9*b^10*d^10 + 3194*a^11*b^8*d^10 - 1

536*a^13*b^6*d^10 + 288*a^15*b^4*d^10 - 80*a*b^18*c^2*d^8 - 800*a*b^18*c^4*d^6 + 400*a*b^18*c^6*d^4 + 40*a*b^1

8*c^8*d^2 + 120*a^2*b^17*c*d^9 - 30*a^2*b^17*c^9*d + 900*a^4*b^15*c*d^9 - 60*a^4*b^15*c^9*d - 7920*a^6*b^13*c*

d^9 + 17160*a^8*b^11*c*d^9 - 16710*a^10*b^9*c*d^9 + 7800*a^12*b^7*c*d^9 - 1440*a^14*b^5*c*d^9 + 2400*a^2*b^17*

c^3*d^7 - 2400*a^2*b^17*c^5*d^5 - 720*a^2*b^17*c^7*d^3 - 2400*a^3*b^16*c^2*d^8 + 9600*a^3*b^16*c^4*d^6 + 2320*

a^3*b^16*c^6*d^4 + 325*a^3*b^16*c^8*d^2 - 18800*a^4*b^15*c^3*d^7 - 1040*a^4*b^15*c^5*d^5 - 440*a^4*b^15*c^7*d^

3 + 17780*a^5*b^14*c^2*d^8 - 13600*a^5*b^14*c^4*d^6 - 1310*a^5*b^14*c^6*d^4 + 40*a^5*b^14*c^8*d^2 + 34960*a^6*

b^13*c^3*d^7 + 2428*a^6*b^13*c^5*d^5 + 160*a^6*b^13*c^7*d^3 - 36000*a^7*b^12*c^2*d^8 + 9330*a^7*b^12*c^4*d^6 +

360*a^7*b^12*c^6*d^4 - 30200*a^8*b^11*c^3*d^7 - 1208*a^8*b^11*c^5*d^5 - 80*a^8*b^11*c^7*d^3 + 33445*a^9*b^10*

c^2*d^8 - 3440*a^9*b^10*c^4*d^6 + 120*a^9*b^10*c^6*d^4 + 12960*a^10*b^9*c^3*d^7 - 48*a^10*b^9*c^5*d^5 - 15100*

a^11*b^8*c^2*d^8 + 800*a^11*b^8*c^4*d^6 - 2400*a^12*b^7*c^3*d^7 + 2760*a^13*b^6*c^2*d^8))/(b^20 - 4*a^2*b^18 +

6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + ((a*d - b*c)^3*(-(a + b)^5*(a - b)^5)^(1/2)*((4*(4*a*b^20*d^5 - 4*a^2*b

^19*c^5 + 12*a^6*b^15*c^5 - 8*a^8*b^13*c^5 + 28*a^3*b^18*d^5 - 120*a^5*b^16*d^5 + 164*a^7*b^14*d^5 - 100*a^9*b

^12*d^5 + 24*a^11*b^10*d^5 + 80*a*b^20*c^2*d^3 - 120*a^2*b^19*c*d^4 + 60*a^3*b^18*c^4*d + 360*a^4*b^17*c*d^4 -

120*a^5*b^16*c^4*d - 420*a^6*b^15*c*d^4 + 60*a^7*b^14*c^4*d + 240*a^8*b^13*c*d^4 - 60*a^10*b^11*c*d^4 - 80*a^

2*b^19*c^3*d^2 - 160*a^3*b^18*c^2*d^3 + 120*a^4*b^17*c^3*d^2 + 120*a^5*b^16*c^2*d^3 - 80*a^7*b^14*c^2*d^3 - 40

*a^8*b^13*c^3*d^2 + 40*a^9*b^12*c^2*d^3))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/

2 + (f*x)/2)*(4*a*b^21*c^5 - 12*a^5*b^17*c^5 + 8*a^7*b^15*c^5 - 80*a^4*b^18*d^5 + 276*a^6*b^16*d^5 - 360*a^8*b

^14*d^5 + 212*a^10*b^12*d^5 - 48*a^12*b^10*d^5 + 80*a*b^21*c^3*d^2 - 60*a^2*b^20*c^4*d + 240*a^3*b^19*c*d^4 +

120*a^4*b^18*c^4*d - 780*a^5*b^17*c*d^4 - 60*a^6*b^16*c^4*d + 960*a^7*b^15*c*d^4 - 540*a^9*b^13*c*d^4 + 120*a^

11*b^11*c*d^4 - 240*a^2*b^20*c^2*d^3 - 120*a^3*b^19*c^3*d^2 + 680*a^4*b^18*c^2*d^3 - 720*a^6*b^16*c^2*d^3 + 40

*a^7*b^15*c^3*d^2 + 360*a^8*b^14*c^2*d^3 - 80*a^10*b^12*c^2*d^3))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14

+ a^8*b^12) + (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 +

6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(e/2 + (f*x)/2)*(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*

b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12))*(a*d - b*c)^3*(-(

a + b)^5*(a - b)^5)^(1/2)*(12*a^4*d^2 + b^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^2 - 12*a*b^3*c*d +

6*a^3*b*c*d))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*a^4*d^2 + b^4*c^

2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^2 - 12*a*b^3*c*d + 6*a^3*b*c*d))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*

b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*a^4*d^2 + b^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^

2 - 12*a*b^3*c*d + 6*a^3*b*c*d)*1i)/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))

/((8*(2326*a^9*b^6*d^15 - 20*a^5*b^10*d^15 - 11*a^7*b^8*d^15 - 864*a^15*d^15 - 4770*a^11*b^4*d^15 + 3456*a^13*

b^2*d^15 + 400*a*b^14*c^6*d^9 + 8040*a*b^14*c^8*d^7 + 801*a*b^14*c^10*d^5 + 20*a*b^14*c^12*d^3 + 60*a^4*b^11*c

*d^14 + 45*a^6*b^9*c*d^14 - 19860*a^8*b^7*c*d^14 + 38835*a^10*b^5*c*d^14 - 27000*a^12*b^3*c*d^14 + 20*a^2*b^13

*c^3*d^12 - 1599*a^2*b^13*c^5*d^10 - 52680*a^2*b^13*c^7*d^8 - 15230*a^2*b^13*c^9*d^6 - 630*a^2*b^13*c^11*d^4 -

60*a^3*b^12*c^2*d^13 + 2385*a^3*b^12*c^4*d^11 + 150460*a^3*b^12*c^6*d^9 + 61605*a^3*b^12*c^8*d^7 + 7416*a^3*b

^12*c^10*d^5 + 80*a^3*b^12*c^12*d^3 - 1550*a^4*b^11*c^3*d^12 - 246516*a^4*b^11*c^5*d^10 - 92100*a^4*b^11*c^7*d

^8 - 18970*a^4*b^11*c^9*d^6 - 1320*a^4*b^11*c^11*d^4 + 330*a^5*b^10*c^2*d^13 + 255870*a^5*b^10*c^4*d^11 - 2490

*a^5*b^10*c^6*d^9 + 2940*a^5*b^10*c^8*d^7 + 2652*a^5*b^10*c^10*d^5 + 80*a^5*b^10*c^12*d^3 - 174080*a^6*b^9*c^3

*d^12 + 206889*a^6*b^9*c^5*d^10 + 46620*a^6*b^9*c^7*d^8 + 80*a^6*b^9*c^9*d^6 - 120*a^6*b^9*c^11*d^4 + 76440*a^

7*b^8*c^2*d^13 - 335925*a^7*b^8*c^4*d^11 - 44620*a^7*b^8*c^6*d^9 + 480*a^7*b^8*c^8*d^7 + 48*a^7*b^8*c^10*d^5 +

281510*a^8*b^7*c^3*d^12 - 60342*a^8*b^7*c^5*d^10 - 15180*a^8*b^7*c^7*d^8 - 800*a^8*b^7*c^9*d^6 - 139125*a^9*b

^6*c^2*d^13 + 167580*a^9*b^6*c^4*d^11 + 25220*a^9*b^6*c^6*d^9 + 2400*a^9*b^6*c^8*d^7 - 167550*a^10*b^5*c^3*d^1

2 - 5928*a^10*b^5*c^5*d^10 - 2760*a^10*b^5*c^7*d^8 + 91080*a^11*b^4*c^2*d^13 - 24840*a^11*b^4*c^4*d^11 + 1440*

a^11*b^4*c^6*d^9 + 33660*a^12*b^3*c^3*d^12 - 288*a^12*b^3*c^5*d^10 - 20520*a^13*b^2*c^2*d^13 + 6480*a^14*b*c*d

^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (16*tan(e/2 + (f*x)/2)*(7829*a^10*b^6*d^15 -

20*a^4*b^12*d^15 - 411*a^6*b^10*d^15 - 1314*a^8*b^8*d^15 - 1728*a^16*d^15 - 11700*a^12*b^4*d^15 + 7344*a^14*b^

2*d^15 + 20*a*b^15*c^3*d^12 + 801*a*b^15*c^5*d^10 + 8040*a*b^15*c^7*d^8 + 400*a*b^15*c^9*d^6 + 60*a^3*b^13*c*d

^14 + 2445*a^5*b^11*c*d^14 + 14460*a^7*b^9*c*d^14 - 66735*a^9*b^7*c*d^14 + 92970*a^11*b^5*c*d^14 - 56160*a^13*

b^3*c*d^14 - 60*a^2*b^14*c^2*d^13 - 3615*a^2*b^14*c^4*d^11 - 48660*a^2*b^14*c^6*d^9 - 7200*a^2*b^14*c^8*d^7 +

6450*a^3*b^13*c^3*d^12 + 123324*a^3*b^13*c^5*d^10 + 7380*a^3*b^13*c^7*d^8 - 5670*a^4*b^12*c^2*d^13 - 168930*a^

4*b^12*c^4*d^11 + 83780*a^4*b^12*c^6*d^9 + 12000*a^4*b^12*c^8*d^7 + 134160*a^5*b^11*c^3*d^12 - 314259*a^5*b^11

*c^5*d^10 - 36120*a^5*b^11*c^7*d^8 - 1200*a^5*b^11*c^9*d^6 - 61080*a^6*b^10*c^2*d^13 + 509145*a^6*b^10*c^4*d^1

1 - 31020*a^6*b^10*c^6*d^9 - 2400*a^6*b^10*c^8*d^7 - 458210*a^7*b^9*c^3*d^12 + 291630*a^7*b^9*c^5*d^10 + 17940

*a^7*b^9*c^7*d^8 + 800*a^7*b^9*c^9*d^6 + 237870*a^8*b^8*c^2*d^13 - 565440*a^8*b^8*c^4*d^11 + 5340*a^8*b^8*c^6*

d^9 - 2400*a^8*b^8*c^8*d^7 + 558240*a^9*b^7*c^3*d^12 - 137784*a^9*b^7*c^5*d^10 + 2760*a^9*b^7*c^7*d^8 - 310560

*a^10*b^6*c^2*d^13 + 297240*a^10*b^6*c^4*d^11 - 9440*a^10*b^6*c^6*d^9 - 310860*a^11*b^5*c^3*d^12 + 36288*a^11*

b^5*c^5*d^10 + 180540*a^12*b^4*c^2*d^13 - 68400*a^12*b^4*c^4*d^11 + 70200*a^13*b^3*c^3*d^12 - 41040*a^14*b^2*c

^2*d^13 + 12960*a^15*b*c*d^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + ((a*d - b*c)^3*(-(a

+ b)^5*(a - b)^5)^(1/2)*((4*(2*a^2*b^16*d^10 + 40*a^4*b^14*d^10 + 108*a^6*b^12*d^10 - 872*a^8*b^10*d^10 + 153

8*a^10*b^8*d^10 - 1104*a^12*b^6*d^10 + 288*a^14*b^4*d^10 - 120*a^3*b^15*c*d^9 - 960*a^5*b^13*c*d^9 + 5040*a^7*

b^11*c*d^9 - 8160*a^9*b^9*c*d^9 + 5640*a^11*b^7*c*d^9 - 1440*a^13*b^5*c*d^9 + 80*a^2*b^16*c^2*d^8 + 800*a^2*b^

16*c^4*d^6 - 2400*a^3*b^15*c^3*d^7 + 2440*a^4*b^14*c^2*d^8 - 3200*a^4*b^14*c^4*d^6 + 9600*a^5*b^13*c^3*d^7 - 1

0560*a^6*b^12*c^2*d^8 + 4800*a^6*b^12*c^4*d^6 - 14400*a^7*b^11*c^3*d^7 + 16240*a^8*b^10*c^2*d^8 - 3200*a^8*b^1

0*c^4*d^6 + 9600*a^9*b^9*c^3*d^7 - 10960*a^10*b^8*c^2*d^8 + 800*a^10*b^8*c^4*d^6 - 2400*a^11*b^7*c^3*d^7 + 276

0*a^12*b^6*c^2*d^8))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/2 + (f*x)/2)*(a*b^18*

c^10 - 2*a*b^18*d^10 + 4*a^3*b^16*c^10 + 4*a^5*b^14*c^10 - 39*a^3*b^16*d^10 - 88*a^5*b^14*d^10 + 1326*a^7*b^12

*d^10 - 3134*a^9*b^10*d^10 + 3194*a^11*b^8*d^10 - 1536*a^13*b^6*d^10 + 288*a^15*b^4*d^10 - 80*a*b^18*c^2*d^8 -

800*a*b^18*c^4*d^6 + 400*a*b^18*c^6*d^4 + 40*a*b^18*c^8*d^2 + 120*a^2*b^17*c*d^9 - 30*a^2*b^17*c^9*d + 900*a^

4*b^15*c*d^9 - 60*a^4*b^15*c^9*d - 7920*a^6*b^13*c*d^9 + 17160*a^8*b^11*c*d^9 - 16710*a^10*b^9*c*d^9 + 7800*a^

12*b^7*c*d^9 - 1440*a^14*b^5*c*d^9 + 2400*a^2*b^17*c^3*d^7 - 2400*a^2*b^17*c^5*d^5 - 720*a^2*b^17*c^7*d^3 - 24

00*a^3*b^16*c^2*d^8 + 9600*a^3*b^16*c^4*d^6 + 2320*a^3*b^16*c^6*d^4 + 325*a^3*b^16*c^8*d^2 - 18800*a^4*b^15*c^

3*d^7 - 1040*a^4*b^15*c^5*d^5 - 440*a^4*b^15*c^7*d^3 + 17780*a^5*b^14*c^2*d^8 - 13600*a^5*b^14*c^4*d^6 - 1310*

a^5*b^14*c^6*d^4 + 40*a^5*b^14*c^8*d^2 + 34960*a^6*b^13*c^3*d^7 + 2428*a^6*b^13*c^5*d^5 + 160*a^6*b^13*c^7*d^3

- 36000*a^7*b^12*c^2*d^8 + 9330*a^7*b^12*c^4*d^6 + 360*a^7*b^12*c^6*d^4 - 30200*a^8*b^11*c^3*d^7 - 1208*a^8*b

^11*c^5*d^5 - 80*a^8*b^11*c^7*d^3 + 33445*a^9*b^10*c^2*d^8 - 3440*a^9*b^10*c^4*d^6 + 120*a^9*b^10*c^6*d^4 + 12

960*a^10*b^9*c^3*d^7 - 48*a^10*b^9*c^5*d^5 - 15100*a^11*b^8*c^2*d^8 + 800*a^11*b^8*c^4*d^6 - 2400*a^12*b^7*c^3

*d^7 + 2760*a^13*b^6*c^2*d^8))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + ((a*d - b*c)^3*(-(a

+ b)^5*(a - b)^5)^(1/2)*((8*tan(e/2 + (f*x)/2)*(4*a*b^21*c^5 - 12*a^5*b^17*c^5 + 8*a^7*b^15*c^5 - 80*a^4*b^18*

d^5 + 276*a^6*b^16*d^5 - 360*a^8*b^14*d^5 + 212*a^10*b^12*d^5 - 48*a^12*b^10*d^5 + 80*a*b^21*c^3*d^2 - 60*a^2*

b^20*c^4*d + 240*a^3*b^19*c*d^4 + 120*a^4*b^18*c^4*d - 780*a^5*b^17*c*d^4 - 60*a^6*b^16*c^4*d + 960*a^7*b^15*c

*d^4 - 540*a^9*b^13*c*d^4 + 120*a^11*b^11*c*d^4 - 240*a^2*b^20*c^2*d^3 - 120*a^3*b^19*c^3*d^2 + 680*a^4*b^18*c

^2*d^3 - 720*a^6*b^16*c^2*d^3 + 40*a^7*b^15*c^3*d^2 + 360*a^8*b^14*c^2*d^3 - 80*a^10*b^12*c^2*d^3))/(b^20 - 4*

a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) - (4*(4*a*b^20*d^5 - 4*a^2*b^19*c^5 + 12*a^6*b^15*c^5 - 8*a^8*b

^13*c^5 + 28*a^3*b^18*d^5 - 120*a^5*b^16*d^5 + 164*a^7*b^14*d^5 - 100*a^9*b^12*d^5 + 24*a^11*b^10*d^5 + 80*a*b

^20*c^2*d^3 - 120*a^2*b^19*c*d^4 + 60*a^3*b^18*c^4*d + 360*a^4*b^17*c*d^4 - 120*a^5*b^16*c^4*d - 420*a^6*b^15*

c*d^4 + 60*a^7*b^14*c^4*d + 240*a^8*b^13*c*d^4 - 60*a^10*b^11*c*d^4 - 80*a^2*b^19*c^3*d^2 - 160*a^3*b^18*c^2*d

^3 + 120*a^4*b^17*c^3*d^2 + 120*a^5*b^16*c^2*d^3 - 80*a^7*b^14*c^2*d^3 - 40*a^8*b^13*c^3*d^2 + 40*a^9*b^12*c^2

*d^3))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18

- 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(e/2 + (f*x)/2

)*(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20 - 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a

^4*b^16 - 4*a^6*b^14 + a^8*b^12))*(a*d - b*c)^3*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4*d^2 + b^4*c^2 + 20*b^4*d^

2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^2 - 12*a*b^3*c*d + 6*a^3*b*c*d))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6

*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*a^4*d^2 + b^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^2 - 12*a*b^3*

c*d + 6*a^3*b*c*d))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*a^4*d^2 + b

^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^2 - 12*a*b^3*c*d + 6*a^3*b*c*d))/(2*(b^15 - 5*a^2*b^13 + 10

*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)) - ((a*d - b*c)^3*(-(a + b)^5*(a - b)^5)^(1/2)*((4*(2*a^2*b^16*

d^10 + 40*a^4*b^14*d^10 + 108*a^6*b^12*d^10 - 872*a^8*b^10*d^10 + 1538*a^10*b^8*d^10 - 1104*a^12*b^6*d^10 + 28

8*a^14*b^4*d^10 - 120*a^3*b^15*c*d^9 - 960*a^5*b^13*c*d^9 + 5040*a^7*b^11*c*d^9 - 8160*a^9*b^9*c*d^9 + 5640*a^

11*b^7*c*d^9 - 1440*a^13*b^5*c*d^9 + 80*a^2*b^16*c^2*d^8 + 800*a^2*b^16*c^4*d^6 - 2400*a^3*b^15*c^3*d^7 + 2440

*a^4*b^14*c^2*d^8 - 3200*a^4*b^14*c^4*d^6 + 9600*a^5*b^13*c^3*d^7 - 10560*a^6*b^12*c^2*d^8 + 4800*a^6*b^12*c^4

*d^6 - 14400*a^7*b^11*c^3*d^7 + 16240*a^8*b^10*c^2*d^8 - 3200*a^8*b^10*c^4*d^6 + 9600*a^9*b^9*c^3*d^7 - 10960*

a^10*b^8*c^2*d^8 + 800*a^10*b^8*c^4*d^6 - 2400*a^11*b^7*c^3*d^7 + 2760*a^12*b^6*c^2*d^8))/(b^19 - 4*a^2*b^17 +

6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/2 + (f*x)/2)*(a*b^18*c^10 - 2*a*b^18*d^10 + 4*a^3*b^16*c^10 +

4*a^5*b^14*c^10 - 39*a^3*b^16*d^10 - 88*a^5*b^14*d^10 + 1326*a^7*b^12*d^10 - 3134*a^9*b^10*d^10 + 3194*a^11*b^

8*d^10 - 1536*a^13*b^6*d^10 + 288*a^15*b^4*d^10 - 80*a*b^18*c^2*d^8 - 800*a*b^18*c^4*d^6 + 400*a*b^18*c^6*d^4

+ 40*a*b^18*c^8*d^2 + 120*a^2*b^17*c*d^9 - 30*a^2*b^17*c^9*d + 900*a^4*b^15*c*d^9 - 60*a^4*b^15*c^9*d - 7920*a

^6*b^13*c*d^9 + 17160*a^8*b^11*c*d^9 - 16710*a^10*b^9*c*d^9 + 7800*a^12*b^7*c*d^9 - 1440*a^14*b^5*c*d^9 + 2400

*a^2*b^17*c^3*d^7 - 2400*a^2*b^17*c^5*d^5 - 720*a^2*b^17*c^7*d^3 - 2400*a^3*b^16*c^2*d^8 + 9600*a^3*b^16*c^4*d

^6 + 2320*a^3*b^16*c^6*d^4 + 325*a^3*b^16*c^8*d^2 - 18800*a^4*b^15*c^3*d^7 - 1040*a^4*b^15*c^5*d^5 - 440*a^4*b

^15*c^7*d^3 + 17780*a^5*b^14*c^2*d^8 - 13600*a^5*b^14*c^4*d^6 - 1310*a^5*b^14*c^6*d^4 + 40*a^5*b^14*c^8*d^2 +

34960*a^6*b^13*c^3*d^7 + 2428*a^6*b^13*c^5*d^5 + 160*a^6*b^13*c^7*d^3 - 36000*a^7*b^12*c^2*d^8 + 9330*a^7*b^12

*c^4*d^6 + 360*a^7*b^12*c^6*d^4 - 30200*a^8*b^11*c^3*d^7 - 1208*a^8*b^11*c^5*d^5 - 80*a^8*b^11*c^7*d^3 + 33445

*a^9*b^10*c^2*d^8 - 3440*a^9*b^10*c^4*d^6 + 120*a^9*b^10*c^6*d^4 + 12960*a^10*b^9*c^3*d^7 - 48*a^10*b^9*c^5*d^

5 - 15100*a^11*b^8*c^2*d^8 + 800*a^11*b^8*c^4*d^6 - 2400*a^12*b^7*c^3*d^7 + 2760*a^13*b^6*c^2*d^8))/(b^20 - 4*

a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + ((a*d - b*c)^3*(-(a + b)^5*(a - b)^5)^(1/2)*((4*(4*a*b^20*d^5

- 4*a^2*b^19*c^5 + 12*a^6*b^15*c^5 - 8*a^8*b^13*c^5 + 28*a^3*b^18*d^5 - 120*a^5*b^16*d^5 + 164*a^7*b^14*d^5 -

100*a^9*b^12*d^5 + 24*a^11*b^10*d^5 + 80*a*b^20*c^2*d^3 - 120*a^2*b^19*c*d^4 + 60*a^3*b^18*c^4*d + 360*a^4*b^

17*c*d^4 - 120*a^5*b^16*c^4*d - 420*a^6*b^15*c*d^4 + 60*a^7*b^14*c^4*d + 240*a^8*b^13*c*d^4 - 60*a^10*b^11*c*d

^4 - 80*a^2*b^19*c^3*d^2 - 160*a^3*b^18*c^2*d^3 + 120*a^4*b^17*c^3*d^2 + 120*a^5*b^16*c^2*d^3 - 80*a^7*b^14*c^

2*d^3 - 40*a^8*b^13*c^3*d^2 + 40*a^9*b^12*c^2*d^3))/(b^19 - 4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) -

(8*tan(e/2 + (f*x)/2)*(4*a*b^21*c^5 - 12*a^5*b^17*c^5 + 8*a^7*b^15*c^5 - 80*a^4*b^18*d^5 + 276*a^6*b^16*d^5 -

360*a^8*b^14*d^5 + 212*a^10*b^12*d^5 - 48*a^12*b^10*d^5 + 80*a*b^21*c^3*d^2 - 60*a^2*b^20*c^4*d + 240*a^3*b^1

9*c*d^4 + 120*a^4*b^18*c^4*d - 780*a^5*b^17*c*d^4 - 60*a^6*b^16*c^4*d + 960*a^7*b^15*c*d^4 - 540*a^9*b^13*c*d^

4 + 120*a^11*b^11*c*d^4 - 240*a^2*b^20*c^2*d^3 - 120*a^3*b^19*c^3*d^2 + 680*a^4*b^18*c^2*d^3 - 720*a^6*b^16*c^

2*d^3 + 40*a^7*b^15*c^3*d^2 + 360*a^8*b^14*c^2*d^3 - 80*a^10*b^12*c^2*d^3))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 -

4*a^6*b^14 + a^8*b^12) + (((4*(8*a^2*b^22 - 32*a^4*b^20 + 48*a^6*b^18 - 32*a^8*b^16 + 8*a^10*b^14))/(b^19 - 4*

a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) + (8*tan(e/2 + (f*x)/2)*(12*a*b^24 - 56*a^3*b^22 + 104*a^5*b^20

- 96*a^7*b^18 + 44*a^9*b^16 - 8*a^11*b^14))/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12))*(a*d -

b*c)^3*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4*d^2 + b^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^2 - 12*a

*b^3*c*d + 6*a^3*b*c*d))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*a^4*d^

2 + b^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*a^2*b^2*d^2 - 12*a*b^3*c*d + 6*a^3*b*c*d))/(2*(b^15 - 5*a^2*b^13

+ 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*a^4*d^2 + b^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*

a^2*b^2*d^2 - 12*a*b^3*c*d + 6*a^3*b*c*d))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10

*b^5))))*(a*d - b*c)^3*(-(a + b)^5*(a - b)^5)^(1/2)*(12*a^4*d^2 + b^4*c^2 + 20*b^4*d^2 + 2*a^2*b^2*c^2 - 29*a^

2*b^2*d^2 - 12*a*b^3*c*d + 6*a^3*b*c*d)*1i)/(f*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^1

\(\int \frac {(c+d \sin (e+f x))^5}{(3+b \sin (e+f x))^3} \, dx\) [714]

Optimal result