\(\int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^3} \, dx\) [715]

Optimal result

Integrand size = 25, antiderivative size = 289 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^3} \, dx=\frac {(4 b c-9 d) d^3 x}{b^4}+\frac {(b c-3 d)^2 \left (108 b c d-30 b^3 c d+486 d^2+9 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 d^2\right )\right ) \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{b^4 \left (9-b^2\right )^{5/2} f}+\frac {d^2 \left (6 b c d-27 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{2 b^3 \left (9-b^2\right ) f}+\frac {3 (b c-3 d)^3 \left (3 b c+9 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \left (9-b^2\right )^2 f (3+b \sin (e+f x))}+\frac {(b c-3 d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b \left (9-b^2\right ) f (3+b \sin (e+f x))^2} \]

[Out]

d^3*(-3*a*d+4*b*c)*x/b^4+(-a*d+b*c)^2*(4*a^3*b*c*d-10*a*b^3*c*d+6*a^4*d^2+a^2*b^2*(2*c^2-15*d^2)+b^4*(c^2+12*d
^2))*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/b^4/(a^2-b^2)^(5/2)/f+1/2*d^2*(2*a*b*c*d-3*a^2*d^2-b^2*(
c^2-2*d^2))*cos(f*x+e)/b^3/(a^2-b^2)/f+3/2*(-a*d+b*c)^3*(a^2*d+a*b*c-2*b^2*d)*cos(f*x+e)/b^3/(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))^2/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^2

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2871, 3110, 3102, 2814, 2739, 632, 210} \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^3} \, dx=\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac {d^2 \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{2 b^3 f \left (a^2-b^2\right )}+\frac {3 (b c-a d)^3 \left (a^2 d+a b c-2 b^2 d\right ) \cos (e+f x)}{2 b^3 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \left (6 a^4 d^2+4 a^3 b c d+a^2 b^2 \left (2 c^2-15 d^2\right )-10 a b^3 c d+b^4 \left (c^2+12 d^2\right )\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^4 f \left (a^2-b^2\right )^{5/2}}+\frac {d^3 x (4 b c-3 a d)}{b^4} \]

[In]

Int[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^3,x]

[Out]

(d^3*(4*b*c - 3*a*d)*x)/b^4 + ((b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2
) + b^4*(c^2 + 12*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(5/2)*f) + (d^2*(2*
a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*Cos[e + f*x])/(2*b^3*(a^2 - b^2)*f) + (3*(b*c - a*d)^3*(a*b*c + a^2*d
- 2*b^2*d)*Cos[e + f*x])/(2*b^3*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Si
n[e + f*x])^2)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)

Rule 210

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])

Rule 632

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]

Rule 2739

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] &&
NeQ[a^2 - b^2, 0]

Rule 2814

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
, 0]

Rule 2871

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])

Rule 3102

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]

Rule 3110

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)
*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)
), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d +
b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m
+ 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] &&
NeQ[a^2 - b^2, 0] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{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 (2 \left (3 b^2 c^2 d+a^2 d^3-a b c \left (c^2+3 d^2\right )\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)+d \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 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 {3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \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))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {-b \left (4 a^3 b c d^3-3 a^4 d^4-4 a b^3 c d \left (3 c^2+4 d^2\right )+b^4 c^2 \left (c^2+12 d^2\right )+2 a^2 b^2 \left (c^4+3 c^2 d^2+3 d^4\right )\right )-\left (a^2-b^2\right ) d^2 \left (6 a^2 b c d-8 b^3 c d-3 a^3 d^2+a b^2 \left (c^2+4 d^2\right )\right ) \sin (e+f x)+b \left (a^2-b^2\right ) d^2 \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2} \\ & = \frac {d^2 \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right ) f}+\frac {3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \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))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {-b^2 \left (4 a^3 b c d^3-3 a^4 d^4-4 a b^3 c d \left (3 c^2+4 d^2\right )+b^4 c^2 \left (c^2+12 d^2\right )+2 a^2 b^2 \left (c^4+3 c^2 d^2+3 d^4\right )\right )-2 b \left (a^2-b^2\right )^2 d^3 (4 b c-3 a d) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2} \\ & = \frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^2 \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right ) f}+\frac {3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \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))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left ((b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 d^2\right )\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2} \\ & = \frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^2 \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right ) f}+\frac {3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \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))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left ((b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 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^4 \left (a^2-b^2\right )^2 f} \\ & = \frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^2 \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right ) f}+\frac {3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \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))^2}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\left (2 (b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 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^4 \left (a^2-b^2\right )^2 f} \\ & = \frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {(b c-a d)^2 \left (4 a^3 b c d-10 a b^3 c d+6 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 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^4 \left (a^2-b^2\right )^{5/2} f}+\frac {d^2 \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{2 b^3 \left (a^2-b^2\right ) f}+\frac {3 (b c-a d)^3 \left (a b c+a^2 d-2 b^2 d\right ) \cos (e+f x)}{2 b^3 \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))^2}{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. \(774\) vs. \(2(289)=578\).

Time = 10.07 (sec) , antiderivative size = 774, normalized size of antiderivative = 2.68 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^3} \, dx=\frac {\frac {4 (b c-3 d)^2 \left (108 b c d-30 b^3 c d+486 d^2+9 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+12 d^2\right )\right ) \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{\sqrt {9-b^2}}-\frac {-11664 b c d^3 e+1944 b^3 c d^3 e-8 b^7 c d^3 e+26244 d^4 e-4374 b^2 d^4 e+18 b^6 d^4 e-11664 b c d^3 f x+1944 b^3 c d^3 f x-8 b^7 c d^3 f x+26244 d^4 f x-4374 b^2 d^4 f x+18 b^6 d^4 f x+b \left (24 b^5 c^3 d-3888 b c d^3+8748 d^4-1701 b^2 d^4+216 b^3 \left (2 c^3 d+5 c d^3\right )-18 b^4 \left (4 c^4+18 c^2 d^2-d^4\right )+b^6 \left (2 c^4+d^4\right )\right ) \cos (e+f x)+2 b^2 \left (-9+b^2\right )^2 (4 b c-9 d) d^3 (e+f x) \cos (2 (e+f x))-81 b^3 d^4 \cos (3 (e+f x))+18 b^5 d^4 \cos (3 (e+f x))-b^7 d^4 \cos (3 (e+f x))-7776 b^2 c d^3 e \sin (e+f x)+1728 b^4 c d^3 e \sin (e+f x)-96 b^6 c d^3 e \sin (e+f x)+17496 b d^4 e \sin (e+f x)-3888 b^3 d^4 e \sin (e+f x)+216 b^5 d^4 e \sin (e+f x)-7776 b^2 c d^3 f x \sin (e+f x)+1728 b^4 c d^3 f x \sin (e+f x)-96 b^6 c d^3 f x \sin (e+f x)+17496 b d^4 f x \sin (e+f x)-3888 b^3 d^4 f x \sin (e+f x)+216 b^5 d^4 f x \sin (e+f x)-9 b^6 c^4 \sin (2 (e+f x))+36 b^5 c^3 d \sin (2 (e+f x))+8 b^7 c^3 d \sin (2 (e+f x))+162 b^4 c^2 d^2 \sin (2 (e+f x))-72 b^6 c^2 d^2 \sin (2 (e+f x))-972 b^3 c d^3 \sin (2 (e+f x))+216 b^5 c d^3 \sin (2 (e+f x))+2187 b^2 d^4 \sin (2 (e+f x))-432 b^4 d^4 \sin (2 (e+f x))+12 b^6 d^4 \sin (2 (e+f x))}{(3+b \sin (e+f x))^2}}{4 b^4 \left (-9+b^2\right )^2 f} \]

[In]

Integrate[(c + d*Sin[e + f*x])^4/(3 + b*Sin[e + f*x])^3,x]

[Out]

((4*(b*c - 3*d)^2*(108*b*c*d - 30*b^3*c*d + 486*d^2 + 9*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 12*d^2))*ArcTan[(b +
3*Tan[(e + f*x)/2])/Sqrt[9 - b^2]])/Sqrt[9 - b^2] - (-11664*b*c*d^3*e + 1944*b^3*c*d^3*e - 8*b^7*c*d^3*e + 26
244*d^4*e - 4374*b^2*d^4*e + 18*b^6*d^4*e - 11664*b*c*d^3*f*x + 1944*b^3*c*d^3*f*x - 8*b^7*c*d^3*f*x + 26244*d
^4*f*x - 4374*b^2*d^4*f*x + 18*b^6*d^4*f*x + b*(24*b^5*c^3*d - 3888*b*c*d^3 + 8748*d^4 - 1701*b^2*d^4 + 216*b^
3*(2*c^3*d + 5*c*d^3) - 18*b^4*(4*c^4 + 18*c^2*d^2 - d^4) + b^6*(2*c^4 + d^4))*Cos[e + f*x] + 2*b^2*(-9 + b^2)
^2*(4*b*c - 9*d)*d^3*(e + f*x)*Cos[2*(e + f*x)] - 81*b^3*d^4*Cos[3*(e + f*x)] + 18*b^5*d^4*Cos[3*(e + f*x)] -
b^7*d^4*Cos[3*(e + f*x)] - 7776*b^2*c*d^3*e*Sin[e + f*x] + 1728*b^4*c*d^3*e*Sin[e + f*x] - 96*b^6*c*d^3*e*Sin[
e + f*x] + 17496*b*d^4*e*Sin[e + f*x] - 3888*b^3*d^4*e*Sin[e + f*x] + 216*b^5*d^4*e*Sin[e + f*x] - 7776*b^2*c*
d^3*f*x*Sin[e + f*x] + 1728*b^4*c*d^3*f*x*Sin[e + f*x] - 96*b^6*c*d^3*f*x*Sin[e + f*x] + 17496*b*d^4*f*x*Sin[e
+ f*x] - 3888*b^3*d^4*f*x*Sin[e + f*x] + 216*b^5*d^4*f*x*Sin[e + f*x] - 9*b^6*c^4*Sin[2*(e + f*x)] + 36*b^5*c
^3*d*Sin[2*(e + f*x)] + 8*b^7*c^3*d*Sin[2*(e + f*x)] + 162*b^4*c^2*d^2*Sin[2*(e + f*x)] - 72*b^6*c^2*d^2*Sin[2
*(e + f*x)] - 972*b^3*c*d^3*Sin[2*(e + f*x)] + 216*b^5*c*d^3*Sin[2*(e + f*x)] + 2187*b^2*d^4*Sin[2*(e + f*x)]
- 432*b^4*d^4*Sin[2*(e + f*x)] + 12*b^6*d^4*Sin[2*(e + f*x)])/(3 + b*Sin[e + f*x])^2)/(4*b^4*(-9 + b^2)^2*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(853\) vs. \(2(307)=614\).

Time = 3.22 (sec) , antiderivative size = 854, normalized size of antiderivative = 2.96

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (-\frac {b^{2} \left (3 a^{6} d^{4}-4 a^{5} b c \,d^{3}-6 b^{2} a^{4} c^{2} d^{2}-6 a^{4} b^{2} d^{4}+12 a^{3} b^{3} c^{3} d +16 a^{3} b^{3} c \,d^{3}-5 b^{4} a^{2} c^{4}-12 a^{2} b^{4} c^{2} d^{2}+2 b^{6} c^{4}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (4 a^{8} d^{4}-8 a^{7} b c \,d^{3}+a^{6} b^{2} d^{4}+8 a^{5} b^{3} c^{3} d +4 a^{5} b^{3} c \,d^{3}-4 a^{4} b^{4} c^{4}-18 a^{4} b^{4} c^{2} d^{2}-14 a^{4} b^{4} d^{4}+20 a^{3} b^{5} c^{3} d +40 a^{3} b^{5} c \,d^{3}-7 a^{2} b^{6} c^{4}-36 a^{2} b^{6} c^{2} d^{2}+8 a \,b^{7} c^{3} d +2 b^{8} c^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}-\frac {b^{2} \left (13 a^{6} d^{4}-28 a^{5} b c \,d^{3}+6 b^{2} a^{4} c^{2} d^{2}-22 a^{4} b^{2} d^{4}+20 a^{3} b^{3} c^{3} d +64 a^{3} b^{3} c \,d^{3}-11 b^{4} a^{2} c^{4}-60 a^{2} b^{4} c^{2} d^{2}+16 a \,b^{5} c^{3} d +2 b^{6} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}-\frac {b \left (4 a^{6} d^{4}-8 a^{5} b c \,d^{3}-7 a^{4} b^{2} d^{4}+8 a^{3} b^{3} c^{3} d +20 a^{3} b^{3} c \,d^{3}-4 b^{4} a^{2} c^{4}-18 a^{2} b^{4} c^{2} d^{2}+4 a \,b^{5} c^{3} d +b^{6} c^{4}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}\right )}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a \right )}^{2}}+\frac {\left (6 a^{6} d^{4}-8 a^{5} b c \,d^{3}-15 a^{4} b^{2} d^{4}+20 a^{3} b^{3} c \,d^{3}+2 b^{4} a^{2} c^{4}+6 a^{2} b^{4} c^{2} d^{2}+12 a^{2} b^{4} d^{4}-12 a \,b^{5} c^{3} d -24 a \,b^{5} c \,d^{3}+b^{6} c^{4}+12 b^{6} c^{2} d^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}}{b^{4}}-\frac {2 d^{3} \left (\frac {b d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (3 d a -4 c b \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{b^{4}}}{f}\)	\(854\)
default	\(\frac {\frac {\frac {2 \left (-\frac {b^{2} \left (3 a^{6} d^{4}-4 a^{5} b c \,d^{3}-6 b^{2} a^{4} c^{2} d^{2}-6 a^{4} b^{2} d^{4}+12 a^{3} b^{3} c^{3} d +16 a^{3} b^{3} c \,d^{3}-5 b^{4} a^{2} c^{4}-12 a^{2} b^{4} c^{2} d^{2}+2 b^{6} c^{4}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (4 a^{8} d^{4}-8 a^{7} b c \,d^{3}+a^{6} b^{2} d^{4}+8 a^{5} b^{3} c^{3} d +4 a^{5} b^{3} c \,d^{3}-4 a^{4} b^{4} c^{4}-18 a^{4} b^{4} c^{2} d^{2}-14 a^{4} b^{4} d^{4}+20 a^{3} b^{5} c^{3} d +40 a^{3} b^{5} c \,d^{3}-7 a^{2} b^{6} c^{4}-36 a^{2} b^{6} c^{2} d^{2}+8 a \,b^{7} c^{3} d +2 b^{8} c^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}-\frac {b^{2} \left (13 a^{6} d^{4}-28 a^{5} b c \,d^{3}+6 b^{2} a^{4} c^{2} d^{2}-22 a^{4} b^{2} d^{4}+20 a^{3} b^{3} c^{3} d +64 a^{3} b^{3} c \,d^{3}-11 b^{4} a^{2} c^{4}-60 a^{2} b^{4} c^{2} d^{2}+16 a \,b^{5} c^{3} d +2 b^{6} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}-\frac {b \left (4 a^{6} d^{4}-8 a^{5} b c \,d^{3}-7 a^{4} b^{2} d^{4}+8 a^{3} b^{3} c^{3} d +20 a^{3} b^{3} c \,d^{3}-4 b^{4} a^{2} c^{4}-18 a^{2} b^{4} c^{2} d^{2}+4 a \,b^{5} c^{3} d +b^{6} c^{4}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}\right )}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a \right )}^{2}}+\frac {\left (6 a^{6} d^{4}-8 a^{5} b c \,d^{3}-15 a^{4} b^{2} d^{4}+20 a^{3} b^{3} c \,d^{3}+2 b^{4} a^{2} c^{4}+6 a^{2} b^{4} c^{2} d^{2}+12 a^{2} b^{4} d^{4}-12 a \,b^{5} c^{3} d -24 a \,b^{5} c \,d^{3}+b^{6} c^{4}+12 b^{6} c^{2} d^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}}{b^{4}}-\frac {2 d^{3} \left (\frac {b d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (3 d a -4 c b \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{b^{4}}}{f}\)	\(854\)
risch	\(\text {Expression too large to display}\)	\(2791\)

[In]

int((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(2/b^4*((-1/2*b^2*(3*a^6*d^4-4*a^5*b*c*d^3-6*a^4*b^2*c^2*d^2-6*a^4*b^2*d^4+12*a^3*b^3*c^3*d+16*a^3*b^3*c*d
^3-5*a^2*b^4*c^4-12*a^2*b^4*c^2*d^2+2*b^6*c^4)/a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)^3-1/2*b*(4*a^8*d^4-8*a
^7*b*c*d^3+a^6*b^2*d^4+8*a^5*b^3*c^3*d+4*a^5*b^3*c*d^3-4*a^4*b^4*c^4-18*a^4*b^4*c^2*d^2-14*a^4*b^4*d^4+20*a^3*
b^5*c^3*d+40*a^3*b^5*c*d^3-7*a^2*b^6*c^4-36*a^2*b^6*c^2*d^2+8*a*b^7*c^3*d+2*b^8*c^4)/(a^4-2*a^2*b^2+b^4)/a^2*t
an(1/2*f*x+1/2*e)^2-1/2*b^2*(13*a^6*d^4-28*a^5*b*c*d^3+6*a^4*b^2*c^2*d^2-22*a^4*b^2*d^4+20*a^3*b^3*c^3*d+64*a^
3*b^3*c*d^3-11*a^2*b^4*c^4-60*a^2*b^4*c^2*d^2+16*a*b^5*c^3*d+2*b^6*c^4)/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*
e)-1/2*b*(4*a^6*d^4-8*a^5*b*c*d^3-7*a^4*b^2*d^4+8*a^3*b^3*c^3*d+20*a^3*b^3*c*d^3-4*a^2*b^4*c^4-18*a^2*b^4*c^2*
d^2+4*a*b^5*c^3*d+b^6*c^4)/(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*(6*a^6
*d^4-8*a^5*b*c*d^3-15*a^4*b^2*d^4+20*a^3*b^3*c*d^3+2*a^2*b^4*c^4+6*a^2*b^4*c^2*d^2+12*a^2*b^4*d^4-12*a*b^5*c^3
*d-24*a*b^5*c*d^3+b^6*c^4+12*b^6*c^2*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^4*(b*d/(1+tan(1/2*f*x+1/2*e)^2)+(3*a*d-4*b*c)*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. 1125 vs. \(2 (307) = 614\).

Time = 0.43 (sec) , antiderivative size = 2335, normalized size of antiderivative = 8.08 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[-1/4*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*d^4*cos(f*x + e)^3 - 4*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 -
b^9)*c*d^3 - 3*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*d^4)*f*x*cos(f*x + e)^2 + 4*(4*(a^8*b - 2*a^6*b^3 + 2
*a^2*b^7 - b^9)*c*d^3 - 3*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8)*d^4)*f*x - ((2*a^4*b^4 + 3*a^2*b^6 + b^8)*c^4
- 12*(a^3*b^5 + a*b^7)*c^3*d + 6*(a^4*b^4 + 3*a^2*b^6 + 2*b^8)*c^2*d^2 - 4*(2*a^7*b - 3*a^5*b^3 + a^3*b^5 + 6*
a*b^7)*c*d^3 + 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 + 4*a^2*b^6)*d^4 + (12*a*b^7*c^3*d - (2*a^2*b^6 + b^8)*c^4 - 6*(
a^2*b^6 + 2*b^8)*c^2*d^2 + 4*(2*a^5*b^3 - 5*a^3*b^5 + 6*a*b^7)*c*d^3 - 3*(2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*d
^4)*cos(f*x + e)^2 - 2*(12*a^2*b^6*c^3*d - (2*a^3*b^5 + a*b^7)*c^4 - 6*(a^3*b^5 + 2*a*b^7)*c^2*d^2 + 4*(2*a^6*
b^2 - 5*a^4*b^4 + 6*a^2*b^6)*c*d^3 - 3*(2*a^7*b - 5*a^5*b^3 + 4*a^3*b^5)*d^4)*sin(f*x + e))*sqrt(-a^2 + b^2)*l
og(((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^5 - 5*a^2*b^7 +
b^9)*c^4 - 4*(2*a^5*b^4 - a^3*b^6 - a*b^8)*c^3*d + 18*(a^4*b^5 - a^2*b^7)*c^2*d^2 + 4*(2*a^7*b^2 - 7*a^5*b^4 +
5*a^3*b^6)*c*d^3 - (6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*b^9)*d^4)*cos(f*x + e) + 2*(4*(4*(a^7*b^
2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*c*d^3 - 3*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*d^4)*f*x + (3*(a^3*b^6
- a*b^8)*c^4 - 4*(a^4*b^5 + a^2*b^7 - 2*b^9)*c^3*d - 6*(a^5*b^4 - 5*a^3*b^6 + 4*a*b^8)*c^2*d^2 + 12*(a^6*b^3 -
3*a^4*b^5 + 2*a^2*b^7)*c*d^3 - (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*a*b^8)*d^4)*cos(f*x + e))*sin(f*x + e
))/((a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*f*cos(f*x + e)^2 - 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*
f*sin(f*x + e) - (a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12)*f), -1/2*(2*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)
*d^4*cos(f*x + e)^3 - 2*(4*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*c*d^3 - 3*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6
- a*b^8)*d^4)*f*x*cos(f*x + e)^2 + 2*(4*(a^8*b - 2*a^6*b^3 + 2*a^2*b^7 - b^9)*c*d^3 - 3*(a^9 - 2*a^7*b^2 + 2*a
^3*b^6 - a*b^8)*d^4)*f*x - ((2*a^4*b^4 + 3*a^2*b^6 + b^8)*c^4 - 12*(a^3*b^5 + a*b^7)*c^3*d + 6*(a^4*b^4 + 3*a^
2*b^6 + 2*b^8)*c^2*d^2 - 4*(2*a^7*b - 3*a^5*b^3 + a^3*b^5 + 6*a*b^7)*c*d^3 + 3*(2*a^8 - 3*a^6*b^2 - a^4*b^4 +
4*a^2*b^6)*d^4 + (12*a*b^7*c^3*d - (2*a^2*b^6 + b^8)*c^4 - 6*(a^2*b^6 + 2*b^8)*c^2*d^2 + 4*(2*a^5*b^3 - 5*a^3*
b^5 + 6*a*b^7)*c*d^3 - 3*(2*a^6*b^2 - 5*a^4*b^4 + 4*a^2*b^6)*d^4)*cos(f*x + e)^2 - 2*(12*a^2*b^6*c^3*d - (2*a^
3*b^5 + a*b^7)*c^4 - 6*(a^3*b^5 + 2*a*b^7)*c^2*d^2 + 4*(2*a^6*b^2 - 5*a^4*b^4 + 6*a^2*b^6)*c*d^3 - 3*(2*a^7*b
- 5*a^5*b^3 + 4*a^3*b^5)*d^4)*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^5 - 5*a^2*b^7 + b^9)*c^4 - 4*(2*a^5*b^4 - a^3*b^6 - a*b^8)*c^3*d + 18*(a^4*b^5 - a^2*b^
7)*c^2*d^2 + 4*(2*a^7*b^2 - 7*a^5*b^4 + 5*a^3*b^6)*c*d^3 - (6*a^8*b - 15*a^6*b^3 + 7*a^4*b^5 + 4*a^2*b^7 - 2*b
^9)*d^4)*cos(f*x + e) + (4*(4*(a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*c*d^3 - 3*(a^8*b - 3*a^6*b^3 + 3*a^4*b
^5 - a^2*b^7)*d^4)*f*x + (3*(a^3*b^6 - a*b^8)*c^4 - 4*(a^4*b^5 + a^2*b^7 - 2*b^9)*c^3*d - 6*(a^5*b^4 - 5*a^3*b
^6 + 4*a*b^8)*c^2*d^2 + 12*(a^6*b^3 - 3*a^4*b^5 + 2*a^2*b^7)*c*d^3 - (9*a^7*b^2 - 25*a^5*b^4 + 20*a^3*b^6 - 4*
a*b^8)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*f*cos(f*x + e)^2 - 2*(a^7*b
^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*f*sin(f*x + e) - (a^8*b^4 - 2*a^6*b^6 + 2*a^2*b^10 - b^12)*f)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^3} \, dx=\text {Timed out} \]

[In]

integrate((c+d*sin(f*x+e))**4/(a+b*sin(f*x+e))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

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?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1120 vs. \(2 (307) = 614\).

Time = 0.34 (sec) , antiderivative size = 1120, normalized size of antiderivative = 3.88 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^3,x, algorithm="giac")

[Out]

((2*a^2*b^4*c^4 + b^6*c^4 - 12*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 + 12*b^6*c^2*d^2 - 8*a^5*b*c*d^3 + 20*a^3*b^3*c
*d^3 - 24*a*b^5*c*d^3 + 6*a^6*d^4 - 15*a^4*b^2*d^4 + 12*a^2*b^4*d^4)*(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^4 - 2*a^2*b^6 + b^8)*sqrt(a^2 - b^2)) - 2*d^4/
((tan(1/2*f*x + 1/2*e)^2 + 1)*b^3) + (5*a^3*b^5*c^4*tan(1/2*f*x + 1/2*e)^3 - 2*a*b^7*c^4*tan(1/2*f*x + 1/2*e)^
3 - 12*a^4*b^4*c^3*d*tan(1/2*f*x + 1/2*e)^3 + 6*a^5*b^3*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 + 12*a^3*b^5*c^2*d^2*ta
n(1/2*f*x + 1/2*e)^3 + 4*a^6*b^2*c*d^3*tan(1/2*f*x + 1/2*e)^3 - 16*a^4*b^4*c*d^3*tan(1/2*f*x + 1/2*e)^3 - 3*a^
7*b*d^4*tan(1/2*f*x + 1/2*e)^3 + 6*a^5*b^3*d^4*tan(1/2*f*x + 1/2*e)^3 + 4*a^4*b^4*c^4*tan(1/2*f*x + 1/2*e)^2 +
7*a^2*b^6*c^4*tan(1/2*f*x + 1/2*e)^2 - 2*b^8*c^4*tan(1/2*f*x + 1/2*e)^2 - 8*a^5*b^3*c^3*d*tan(1/2*f*x + 1/2*e
)^2 - 20*a^3*b^5*c^3*d*tan(1/2*f*x + 1/2*e)^2 - 8*a*b^7*c^3*d*tan(1/2*f*x + 1/2*e)^2 + 18*a^4*b^4*c^2*d^2*tan(
1/2*f*x + 1/2*e)^2 + 36*a^2*b^6*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 + 8*a^7*b*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 4*a^5*
b^3*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 40*a^3*b^5*c*d^3*tan(1/2*f*x + 1/2*e)^2 - 4*a^8*d^4*tan(1/2*f*x + 1/2*e)^2
- a^6*b^2*d^4*tan(1/2*f*x + 1/2*e)^2 + 14*a^4*b^4*d^4*tan(1/2*f*x + 1/2*e)^2 + 11*a^3*b^5*c^4*tan(1/2*f*x + 1/
2*e) - 2*a*b^7*c^4*tan(1/2*f*x + 1/2*e) - 20*a^4*b^4*c^3*d*tan(1/2*f*x + 1/2*e) - 16*a^2*b^6*c^3*d*tan(1/2*f*x
+ 1/2*e) - 6*a^5*b^3*c^2*d^2*tan(1/2*f*x + 1/2*e) + 60*a^3*b^5*c^2*d^2*tan(1/2*f*x + 1/2*e) + 28*a^6*b^2*c*d^
3*tan(1/2*f*x + 1/2*e) - 64*a^4*b^4*c*d^3*tan(1/2*f*x + 1/2*e) - 13*a^7*b*d^4*tan(1/2*f*x + 1/2*e) + 22*a^5*b^
3*d^4*tan(1/2*f*x + 1/2*e) + 4*a^4*b^4*c^4 - a^2*b^6*c^4 - 8*a^5*b^3*c^3*d - 4*a^3*b^5*c^3*d + 18*a^4*b^4*c^2*
d^2 + 8*a^7*b*c*d^3 - 20*a^5*b^3*c*d^3 - 4*a^8*d^4 + 7*a^6*b^2*d^4)/((a^6*b^3 - 2*a^4*b^5 + a^2*b^7)*(a*tan(1/
2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)^2) + (4*b*c*d^3 - 3*a*d^4)*(f*x + e)/b^4)/f

Mupad [B] (verification not implemented)

Time = 22.34 (sec) , antiderivative size = 16958, normalized size of antiderivative = 58.68 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

int((c + d*sin(e + f*x))^4/(a + b*sin(e + f*x))^3,x)

[Out]

(2*d^3*atan(((d^3*(3*a*d - 4*b*c)*((8*tan(e/2 + (f*x)/2)*(a*b^15*c^8 + 4*a^3*b^13*c^8 + 4*a^5*b^11*c^8 - 72*a^
3*b^13*d^8 + 468*a^5*b^11*d^8 - 936*a^7*b^9*d^8 + 873*a^9*b^7*d^8 - 396*a^11*b^5*d^8 + 72*a^13*b^3*d^8 - 128*a
*b^15*c^2*d^6 + 144*a*b^15*c^4*d^4 + 24*a*b^15*c^6*d^2 + 192*a^2*b^14*c*d^7 - 24*a^2*b^14*c^7*d - 1440*a^4*b^1
2*c*d^7 - 48*a^4*b^12*c^7*d + 2736*a^6*b^10*c*d^7 - 2424*a^8*b^8*c*d^7 + 1056*a^10*b^6*c*d^7 - 192*a^12*b^4*c*
d^7 - 576*a^2*b^14*c^3*d^5 - 336*a^2*b^14*c^5*d^3 + 1440*a^3*b^13*c^2*d^6 + 744*a^3*b^13*c^4*d^4 + 204*a^3*b^1
3*c^6*d^2 - 96*a^4*b^12*c^3*d^5 - 200*a^4*b^12*c^5*d^3 - 2200*a^5*b^11*c^2*d^6 - 426*a^5*b^11*c^4*d^4 + 24*a^5
*b^11*c^6*d^2 + 408*a^6*b^10*c^3*d^5 + 64*a^6*b^10*c^5*d^3 + 1644*a^7*b^9*c^2*d^6 + 144*a^7*b^9*c^4*d^4 - 240*
a^8*b^8*c^3*d^5 - 32*a^8*b^8*c^5*d^3 - 632*a^9*b^7*c^2*d^6 + 24*a^9*b^7*c^4*d^4 + 128*a^11*b^5*c^2*d^6))/(b^17
- 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (8*(36*a^4*b^11*d^8 - 144*a^6*b^9*d^8 + 216*a^8*b^7*d^8 -
144*a^10*b^5*d^8 + 36*a^12*b^3*d^8 - 96*a^3*b^12*c*d^7 + 384*a^5*b^10*c*d^7 - 576*a^7*b^8*c*d^7 + 384*a^9*b^6
*c*d^7 - 96*a^11*b^4*c*d^7 + 64*a^2*b^13*c^2*d^6 - 256*a^4*b^11*c^2*d^6 + 384*a^6*b^9*c^2*d^6 - 256*a^8*b^7*c^
2*d^6 + 64*a^10*b^5*c^2*d^6))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (d^3*(3*a*d - 4*b*c)*(
(8*(2*a^2*b^16*c^4 - 6*a^6*b^12*c^4 + 4*a^8*b^10*c^4 + 12*a^2*b^16*d^4 - 36*a^4*b^14*d^4 + 42*a^6*b^12*d^4 - 2
4*a^8*b^10*d^4 + 6*a^10*b^8*d^4 + 32*a^3*b^15*c*d^3 - 24*a^3*b^15*c^3*d - 24*a^5*b^13*c*d^3 + 48*a^5*b^13*c^3*
d + 16*a^7*b^11*c*d^3 - 24*a^7*b^11*c^3*d - 8*a^9*b^9*c*d^3 + 24*a^2*b^16*c^2*d^2 - 36*a^4*b^14*c^2*d^2 + 12*a
^8*b^10*c^2*d^2 - 16*a*b^17*c*d^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*
x)/2)*(4*a*b^18*c^4 - 12*a^5*b^14*c^4 + 8*a^7*b^12*c^4 + 48*a^3*b^16*d^4 - 156*a^5*b^14*d^4 + 192*a^7*b^12*d^4
- 108*a^9*b^10*d^4 + 24*a^11*b^8*d^4 + 48*a*b^18*c^2*d^2 - 96*a^2*b^17*c*d^3 - 48*a^2*b^17*c^3*d + 272*a^4*b^
15*c*d^3 + 96*a^4*b^15*c^3*d - 288*a^6*b^13*c*d^3 - 48*a^6*b^13*c^3*d + 144*a^8*b^11*c*d^3 - 32*a^10*b^9*c*d^3
- 72*a^3*b^16*c^2*d^2 + 24*a^7*b^12*c^2*d^2))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (d^3*
((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*
a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^1
3 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(3*a*d - 4*b*c)*1i)/b^4)*1i)/b^4))/
b^4 - (d^3*(3*a*d - 4*b*c)*((8*(36*a^4*b^11*d^8 - 144*a^6*b^9*d^8 + 216*a^8*b^7*d^8 - 144*a^10*b^5*d^8 + 36*a^
12*b^3*d^8 - 96*a^3*b^12*c*d^7 + 384*a^5*b^10*c*d^7 - 576*a^7*b^8*c*d^7 + 384*a^9*b^6*c*d^7 - 96*a^11*b^4*c*d^
7 + 64*a^2*b^13*c^2*d^6 - 256*a^4*b^11*c^2*d^6 + 384*a^6*b^9*c^2*d^6 - 256*a^8*b^7*c^2*d^6 + 64*a^10*b^5*c^2*d
^6))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) - (8*tan(e/2 + (f*x)/2)*(a*b^15*c^8 + 4*a^3*b^13*
c^8 + 4*a^5*b^11*c^8 - 72*a^3*b^13*d^8 + 468*a^5*b^11*d^8 - 936*a^7*b^9*d^8 + 873*a^9*b^7*d^8 - 396*a^11*b^5*d
^8 + 72*a^13*b^3*d^8 - 128*a*b^15*c^2*d^6 + 144*a*b^15*c^4*d^4 + 24*a*b^15*c^6*d^2 + 192*a^2*b^14*c*d^7 - 24*a
^2*b^14*c^7*d - 1440*a^4*b^12*c*d^7 - 48*a^4*b^12*c^7*d + 2736*a^6*b^10*c*d^7 - 2424*a^8*b^8*c*d^7 + 1056*a^10
*b^6*c*d^7 - 192*a^12*b^4*c*d^7 - 576*a^2*b^14*c^3*d^5 - 336*a^2*b^14*c^5*d^3 + 1440*a^3*b^13*c^2*d^6 + 744*a^
3*b^13*c^4*d^4 + 204*a^3*b^13*c^6*d^2 - 96*a^4*b^12*c^3*d^5 - 200*a^4*b^12*c^5*d^3 - 2200*a^5*b^11*c^2*d^6 - 4
26*a^5*b^11*c^4*d^4 + 24*a^5*b^11*c^6*d^2 + 408*a^6*b^10*c^3*d^5 + 64*a^6*b^10*c^5*d^3 + 1644*a^7*b^9*c^2*d^6
+ 144*a^7*b^9*c^4*d^4 - 240*a^8*b^8*c^3*d^5 - 32*a^8*b^8*c^5*d^3 - 632*a^9*b^7*c^2*d^6 + 24*a^9*b^7*c^4*d^4 +
128*a^11*b^5*c^2*d^6))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + (d^3*(3*a*d - 4*b*c)*((8*(2*a
^2*b^16*c^4 - 6*a^6*b^12*c^4 + 4*a^8*b^10*c^4 + 12*a^2*b^16*d^4 - 36*a^4*b^14*d^4 + 42*a^6*b^12*d^4 - 24*a^8*b
^10*d^4 + 6*a^10*b^8*d^4 + 32*a^3*b^15*c*d^3 - 24*a^3*b^15*c^3*d - 24*a^5*b^13*c*d^3 + 48*a^5*b^13*c^3*d + 16*
a^7*b^11*c*d^3 - 24*a^7*b^11*c^3*d - 8*a^9*b^9*c*d^3 + 24*a^2*b^16*c^2*d^2 - 36*a^4*b^14*c^2*d^2 + 12*a^8*b^10
*c^2*d^2 - 16*a*b^17*c*d^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(
4*a*b^18*c^4 - 12*a^5*b^14*c^4 + 8*a^7*b^12*c^4 + 48*a^3*b^16*d^4 - 156*a^5*b^14*d^4 + 192*a^7*b^12*d^4 - 108*
a^9*b^10*d^4 + 24*a^11*b^8*d^4 + 48*a*b^18*c^2*d^2 - 96*a^2*b^17*c*d^3 - 48*a^2*b^17*c^3*d + 272*a^4*b^15*c*d^
3 + 96*a^4*b^15*c^3*d - 288*a^6*b^13*c*d^3 - 48*a^6*b^13*c^3*d + 144*a^8*b^11*c*d^3 - 32*a^10*b^9*c*d^3 - 72*a
^3*b^16*c^2*d^2 + 24*a^7*b^12*c^2*d^2))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + (d^3*((8*(4*
a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^1
0 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a
^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(3*a*d - 4*b*c)*1i)/b^4)*1i)/b^4))/b^4)/((
16*(54*a^12*d^12 - 216*a^6*b^6*d^12 + 378*a^8*b^4*d^12 - 243*a^10*b^2*d^12 + 576*a*b^11*c^5*d^7 + 96*a*b^11*c^
7*d^5 + 4*a*b^11*c^9*d^3 + 1296*a^5*b^7*c*d^11 - 1944*a^7*b^5*c*d^11 + 1116*a^9*b^3*c*d^11 - 2352*a^2*b^10*c^4
*d^8 - 1384*a^2*b^10*c^6*d^6 - 99*a^2*b^10*c^8*d^4 + 3840*a^3*b^9*c^3*d^9 + 3552*a^3*b^9*c^5*d^7 + 888*a^3*b^9
*c^7*d^5 + 16*a^3*b^9*c^9*d^3 - 3144*a^4*b^8*c^2*d^10 - 2598*a^4*b^8*c^4*d^8 - 1412*a^4*b^8*c^6*d^6 - 204*a^4*
b^8*c^8*d^4 - 1592*a^5*b^7*c^3*d^9 - 336*a^5*b^7*c^5*d^7 + 240*a^5*b^7*c^7*d^5 + 16*a^5*b^7*c^9*d^3 + 3492*a^6
*b^6*c^2*d^10 + 1758*a^6*b^6*c^4*d^8 + 88*a^6*b^6*c^6*d^6 - 12*a^6*b^6*c^8*d^4 - 104*a^7*b^5*c^3*d^9 + 144*a^7
*b^5*c^5*d^7 - 1572*a^8*b^4*c^2*d^10 - 678*a^8*b^4*c^4*d^8 - 64*a^8*b^4*c^6*d^6 + 376*a^9*b^3*c^3*d^9 + 96*a^9
*b^3*c^5*d^7 + 180*a^10*b^2*c^2*d^10 - 36*a^10*b^2*c^4*d^8 - 216*a^11*b*c*d^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^
12 - 4*a^6*b^10 + a^8*b^8) + (16*tan(e/2 + (f*x)/2)*(216*a^13*d^12 + 432*a^5*b^8*d^12 - 1404*a^7*b^6*d^12 + 17
28*a^9*b^4*d^12 - 972*a^11*b^2*d^12 + 768*a*b^12*c^4*d^8 + 64*a*b^12*c^6*d^6 - 2016*a^4*b^9*c*d^11 + 6192*a^6*
b^7*c*d^11 - 7200*a^8*b^5*c*d^11 + 3888*a^10*b^3*c*d^11 - 2688*a^2*b^11*c^3*d^9 - 864*a^2*b^11*c^5*d^7 + 3504*
a^3*b^10*c^2*d^10 + 36*a^3*b^10*c^4*d^8 + 5648*a^4*b^9*c^3*d^9 + 1536*a^4*b^9*c^5*d^7 - 9672*a^5*b^8*c^2*d^10
- 2304*a^5*b^8*c^4*d^8 - 192*a^5*b^8*c^6*d^6 - 3744*a^6*b^7*c^3*d^9 - 480*a^6*b^7*c^5*d^7 + 9984*a^7*b^6*c^2*d
^10 + 1428*a^7*b^6*c^4*d^8 + 128*a^7*b^6*c^6*d^6 + 1296*a^8*b^5*c^3*d^9 - 192*a^8*b^5*c^5*d^7 - 4968*a^9*b^4*c
^2*d^10 + 72*a^9*b^4*c^4*d^8 - 512*a^10*b^3*c^3*d^9 + 1152*a^11*b^2*c^2*d^10 - 864*a^12*b*c*d^11))/(b^17 - 4*a
^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + (d^3*(3*a*d - 4*b*c)*((8*tan(e/2 + (f*x)/2)*(a*b^15*c^8 + 4*a^3
*b^13*c^8 + 4*a^5*b^11*c^8 - 72*a^3*b^13*d^8 + 468*a^5*b^11*d^8 - 936*a^7*b^9*d^8 + 873*a^9*b^7*d^8 - 396*a^11
*b^5*d^8 + 72*a^13*b^3*d^8 - 128*a*b^15*c^2*d^6 + 144*a*b^15*c^4*d^4 + 24*a*b^15*c^6*d^2 + 192*a^2*b^14*c*d^7
- 24*a^2*b^14*c^7*d - 1440*a^4*b^12*c*d^7 - 48*a^4*b^12*c^7*d + 2736*a^6*b^10*c*d^7 - 2424*a^8*b^8*c*d^7 + 105
6*a^10*b^6*c*d^7 - 192*a^12*b^4*c*d^7 - 576*a^2*b^14*c^3*d^5 - 336*a^2*b^14*c^5*d^3 + 1440*a^3*b^13*c^2*d^6 +
744*a^3*b^13*c^4*d^4 + 204*a^3*b^13*c^6*d^2 - 96*a^4*b^12*c^3*d^5 - 200*a^4*b^12*c^5*d^3 - 2200*a^5*b^11*c^2*d
^6 - 426*a^5*b^11*c^4*d^4 + 24*a^5*b^11*c^6*d^2 + 408*a^6*b^10*c^3*d^5 + 64*a^6*b^10*c^5*d^3 + 1644*a^7*b^9*c^
2*d^6 + 144*a^7*b^9*c^4*d^4 - 240*a^8*b^8*c^3*d^5 - 32*a^8*b^8*c^5*d^3 - 632*a^9*b^7*c^2*d^6 + 24*a^9*b^7*c^4*
d^4 + 128*a^11*b^5*c^2*d^6))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (8*(36*a^4*b^11*d^8 - 1
44*a^6*b^9*d^8 + 216*a^8*b^7*d^8 - 144*a^10*b^5*d^8 + 36*a^12*b^3*d^8 - 96*a^3*b^12*c*d^7 + 384*a^5*b^10*c*d^7
- 576*a^7*b^8*c*d^7 + 384*a^9*b^6*c*d^7 - 96*a^11*b^4*c*d^7 + 64*a^2*b^13*c^2*d^6 - 256*a^4*b^11*c^2*d^6 + 38
4*a^6*b^9*c^2*d^6 - 256*a^8*b^7*c^2*d^6 + 64*a^10*b^5*c^2*d^6))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 +
a^8*b^8) + (d^3*(3*a*d - 4*b*c)*((8*(2*a^2*b^16*c^4 - 6*a^6*b^12*c^4 + 4*a^8*b^10*c^4 + 12*a^2*b^16*d^4 - 36*
a^4*b^14*d^4 + 42*a^6*b^12*d^4 - 24*a^8*b^10*d^4 + 6*a^10*b^8*d^4 + 32*a^3*b^15*c*d^3 - 24*a^3*b^15*c^3*d - 24
*a^5*b^13*c*d^3 + 48*a^5*b^13*c^3*d + 16*a^7*b^11*c*d^3 - 24*a^7*b^11*c^3*d - 8*a^9*b^9*c*d^3 + 24*a^2*b^16*c^
2*d^2 - 36*a^4*b^14*c^2*d^2 + 12*a^8*b^10*c^2*d^2 - 16*a*b^17*c*d^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*
b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(4*a*b^18*c^4 - 12*a^5*b^14*c^4 + 8*a^7*b^12*c^4 + 48*a^3*b^16*d^4 - 1
56*a^5*b^14*d^4 + 192*a^7*b^12*d^4 - 108*a^9*b^10*d^4 + 24*a^11*b^8*d^4 + 48*a*b^18*c^2*d^2 - 96*a^2*b^17*c*d^
3 - 48*a^2*b^17*c^3*d + 272*a^4*b^15*c*d^3 + 96*a^4*b^15*c^3*d - 288*a^6*b^13*c*d^3 - 48*a^6*b^13*c^3*d + 144*
a^8*b^11*c*d^3 - 32*a^10*b^9*c*d^3 - 72*a^3*b^16*c^2*d^2 + 24*a^7*b^12*c^2*d^2))/(b^17 - 4*a^2*b^15 + 6*a^4*b^
13 - 4*a^6*b^11 + a^8*b^9) - (d^3*((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b
^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a
^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(
3*a*d - 4*b*c)*1i)/b^4)*1i)/b^4)*1i)/b^4 + (d^3*(3*a*d - 4*b*c)*((8*(36*a^4*b^11*d^8 - 144*a^6*b^9*d^8 + 216*a
^8*b^7*d^8 - 144*a^10*b^5*d^8 + 36*a^12*b^3*d^8 - 96*a^3*b^12*c*d^7 + 384*a^5*b^10*c*d^7 - 576*a^7*b^8*c*d^7 +
384*a^9*b^6*c*d^7 - 96*a^11*b^4*c*d^7 + 64*a^2*b^13*c^2*d^6 - 256*a^4*b^11*c^2*d^6 + 384*a^6*b^9*c^2*d^6 - 25
6*a^8*b^7*c^2*d^6 + 64*a^10*b^5*c^2*d^6))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) - (8*tan(e/2
+ (f*x)/2)*(a*b^15*c^8 + 4*a^3*b^13*c^8 + 4*a^5*b^11*c^8 - 72*a^3*b^13*d^8 + 468*a^5*b^11*d^8 - 936*a^7*b^9*d
^8 + 873*a^9*b^7*d^8 - 396*a^11*b^5*d^8 + 72*a^13*b^3*d^8 - 128*a*b^15*c^2*d^6 + 144*a*b^15*c^4*d^4 + 24*a*b^1
5*c^6*d^2 + 192*a^2*b^14*c*d^7 - 24*a^2*b^14*c^7*d - 1440*a^4*b^12*c*d^7 - 48*a^4*b^12*c^7*d + 2736*a^6*b^10*c
*d^7 - 2424*a^8*b^8*c*d^7 + 1056*a^10*b^6*c*d^7 - 192*a^12*b^4*c*d^7 - 576*a^2*b^14*c^3*d^5 - 336*a^2*b^14*c^5
*d^3 + 1440*a^3*b^13*c^2*d^6 + 744*a^3*b^13*c^4*d^4 + 204*a^3*b^13*c^6*d^2 - 96*a^4*b^12*c^3*d^5 - 200*a^4*b^1
2*c^5*d^3 - 2200*a^5*b^11*c^2*d^6 - 426*a^5*b^11*c^4*d^4 + 24*a^5*b^11*c^6*d^2 + 408*a^6*b^10*c^3*d^5 + 64*a^6
*b^10*c^5*d^3 + 1644*a^7*b^9*c^2*d^6 + 144*a^7*b^9*c^4*d^4 - 240*a^8*b^8*c^3*d^5 - 32*a^8*b^8*c^5*d^3 - 632*a^
9*b^7*c^2*d^6 + 24*a^9*b^7*c^4*d^4 + 128*a^11*b^5*c^2*d^6))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8
*b^9) + (d^3*(3*a*d - 4*b*c)*((8*(2*a^2*b^16*c^4 - 6*a^6*b^12*c^4 + 4*a^8*b^10*c^4 + 12*a^2*b^16*d^4 - 36*a^4*
b^14*d^4 + 42*a^6*b^12*d^4 - 24*a^8*b^10*d^4 + 6*a^10*b^8*d^4 + 32*a^3*b^15*c*d^3 - 24*a^3*b^15*c^3*d - 24*a^5
*b^13*c*d^3 + 48*a^5*b^13*c^3*d + 16*a^7*b^11*c*d^3 - 24*a^7*b^11*c^3*d - 8*a^9*b^9*c*d^3 + 24*a^2*b^16*c^2*d^
2 - 36*a^4*b^14*c^2*d^2 + 12*a^8*b^10*c^2*d^2 - 16*a*b^17*c*d^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10
+ a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(4*a*b^18*c^4 - 12*a^5*b^14*c^4 + 8*a^7*b^12*c^4 + 48*a^3*b^16*d^4 - 156*a
^5*b^14*d^4 + 192*a^7*b^12*d^4 - 108*a^9*b^10*d^4 + 24*a^11*b^8*d^4 + 48*a*b^18*c^2*d^2 - 96*a^2*b^17*c*d^3 -
48*a^2*b^17*c^3*d + 272*a^4*b^15*c*d^3 + 96*a^4*b^15*c^3*d - 288*a^6*b^13*c*d^3 - 48*a^6*b^13*c^3*d + 144*a^8*
b^11*c*d^3 - 32*a^10*b^9*c*d^3 - 72*a^3*b^16*c^2*d^2 + 24*a^7*b^12*c^2*d^2))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 -
4*a^6*b^11 + a^8*b^9) + (d^3*((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^16
- 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b
^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(3*a*
d - 4*b*c)*1i)/b^4)*1i)/b^4)*1i)/b^4))*(3*a*d - 4*b*c))/(b^4*f) - ((6*a^6*d^4 + b^6*c^4 - 4*a^2*b^4*c^4 + 2*a^
2*b^4*d^4 - 11*a^4*b^2*d^4 + 20*a^3*b^3*c*d^3 + 8*a^3*b^3*c^3*d - 18*a^2*b^4*c^2*d^2 + 4*a*b^5*c^3*d - 8*a^5*b
*c*d^3)/(b^3*(a^2 - b^2)^2) + (2*tan(e/2 + (f*x)/2)^2*(6*a^8*d^4 + b^8*c^4 - 3*a^2*b^6*c^4 - 4*a^4*b^4*c^4 + 4
*a^2*b^6*d^4 - 13*a^4*b^4*d^4 - 3*a^6*b^2*d^4 + 20*a^3*b^5*c*d^3 + 12*a^3*b^5*c^3*d + 12*a^5*b^3*c*d^3 + 8*a^5
*b^3*c^3*d - 18*a^2*b^6*c^2*d^2 - 18*a^4*b^4*c^2*d^2 + 4*a*b^7*c^3*d - 8*a^7*b*c*d^3))/(a^2*b^3*(a^2 - b^2)^2)
+ (4*tan(e/2 + (f*x)/2)^3*(6*a^6*d^4 + b^6*c^4 - 4*a^2*b^4*c^4 + 2*a^2*b^4*d^4 - 11*a^4*b^2*d^4 + 20*a^3*b^3*
c*d^3 + 8*a^3*b^3*c^3*d - 18*a^2*b^4*c^2*d^2 + 4*a*b^5*c^3*d - 8*a^5*b*c*d^3))/(a*b^2*(a^2 - b^2)^2) + (tan(e/
2 + (f*x)/2)*(21*a^6*d^4 + 2*b^6*c^4 - 11*a^2*b^4*c^4 + 8*a^2*b^4*d^4 - 38*a^4*b^2*d^4 + 64*a^3*b^3*c*d^3 + 20
*a^3*b^3*c^3*d - 60*a^2*b^4*c^2*d^2 + 6*a^4*b^2*c^2*d^2 + 16*a*b^5*c^3*d - 28*a^5*b*c*d^3))/(a*b^2*(a^2 - b^2)
^2) - (tan(e/2 + (f*x)/2)^5*(5*a^2*b^4*c^4 - 2*b^6*c^4 - 3*a^6*d^4 + 6*a^4*b^2*d^4 - 16*a^3*b^3*c*d^3 - 12*a^3
*b^3*c^3*d + 12*a^2*b^4*c^2*d^2 + 6*a^4*b^2*c^2*d^2 + 4*a^5*b*c*d^3))/(a*b^2*(a^2 - b^2)^2) + (tan(e/2 + (f*x)
/2)^4*(6*a^8*d^4 + 2*b^8*c^4 - 7*a^2*b^6*c^4 - 4*a^4*b^4*c^4 - 12*a^4*b^4*d^4 - 3*a^6*b^2*d^4 + 40*a^3*b^5*c*d
^3 + 20*a^3*b^5*c^3*d + 4*a^5*b^3*c*d^3 + 8*a^5*b^3*c^3*d - 36*a^2*b^6*c^2*d^2 - 18*a^4*b^4*c^2*d^2 + 8*a*b^7*
c^3*d - 8*a^7*b*c*d^3))/(a^2*b^3*(a^2 - b^2)^2))/(f*(tan(e/2 + (f*x)/2)^2*(3*a^2 + 4*b^2) + tan(e/2 + (f*x)/2)
^4*(3*a^2 + 4*b^2) + a^2*tan(e/2 + (f*x)/2)^6 + a^2 + 8*a*b*tan(e/2 + (f*x)/2)^3 + 4*a*b*tan(e/2 + (f*x)/2)^5
+ 4*a*b*tan(e/2 + (f*x)/2))) + (atan((((a*d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*tan(e/2 + (f*x)/2)*(a*b^
15*c^8 + 4*a^3*b^13*c^8 + 4*a^5*b^11*c^8 - 72*a^3*b^13*d^8 + 468*a^5*b^11*d^8 - 936*a^7*b^9*d^8 + 873*a^9*b^7*
d^8 - 396*a^11*b^5*d^8 + 72*a^13*b^3*d^8 - 128*a*b^15*c^2*d^6 + 144*a*b^15*c^4*d^4 + 24*a*b^15*c^6*d^2 + 192*a
^2*b^14*c*d^7 - 24*a^2*b^14*c^7*d - 1440*a^4*b^12*c*d^7 - 48*a^4*b^12*c^7*d + 2736*a^6*b^10*c*d^7 - 2424*a^8*b
^8*c*d^7 + 1056*a^10*b^6*c*d^7 - 192*a^12*b^4*c*d^7 - 576*a^2*b^14*c^3*d^5 - 336*a^2*b^14*c^5*d^3 + 1440*a^3*b
^13*c^2*d^6 + 744*a^3*b^13*c^4*d^4 + 204*a^3*b^13*c^6*d^2 - 96*a^4*b^12*c^3*d^5 - 200*a^4*b^12*c^5*d^3 - 2200*
a^5*b^11*c^2*d^6 - 426*a^5*b^11*c^4*d^4 + 24*a^5*b^11*c^6*d^2 + 408*a^6*b^10*c^3*d^5 + 64*a^6*b^10*c^5*d^3 + 1
644*a^7*b^9*c^2*d^6 + 144*a^7*b^9*c^4*d^4 - 240*a^8*b^8*c^3*d^5 - 32*a^8*b^8*c^5*d^3 - 632*a^9*b^7*c^2*d^6 + 2
4*a^9*b^7*c^4*d^4 + 128*a^11*b^5*c^2*d^6))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (8*(36*a^
4*b^11*d^8 - 144*a^6*b^9*d^8 + 216*a^8*b^7*d^8 - 144*a^10*b^5*d^8 + 36*a^12*b^3*d^8 - 96*a^3*b^12*c*d^7 + 384*
a^5*b^10*c*d^7 - 576*a^7*b^8*c*d^7 + 384*a^9*b^6*c*d^7 - 96*a^11*b^4*c*d^7 + 64*a^2*b^13*c^2*d^6 - 256*a^4*b^1
1*c^2*d^6 + 384*a^6*b^9*c^2*d^6 - 256*a^8*b^7*c^2*d^6 + 64*a^10*b^5*c^2*d^6))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12
- 4*a^6*b^10 + a^8*b^8) + ((a*d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*(2*a^2*b^16*c^4 - 6*a^6*b^12*c^4 + 4
*a^8*b^10*c^4 + 12*a^2*b^16*d^4 - 36*a^4*b^14*d^4 + 42*a^6*b^12*d^4 - 24*a^8*b^10*d^4 + 6*a^10*b^8*d^4 + 32*a^
3*b^15*c*d^3 - 24*a^3*b^15*c^3*d - 24*a^5*b^13*c*d^3 + 48*a^5*b^13*c^3*d + 16*a^7*b^11*c*d^3 - 24*a^7*b^11*c^3
*d - 8*a^9*b^9*c*d^3 + 24*a^2*b^16*c^2*d^2 - 36*a^4*b^14*c^2*d^2 + 12*a^8*b^10*c^2*d^2 - 16*a*b^17*c*d^3))/(b^
16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(4*a*b^18*c^4 - 12*a^5*b^14*c^4 +
8*a^7*b^12*c^4 + 48*a^3*b^16*d^4 - 156*a^5*b^14*d^4 + 192*a^7*b^12*d^4 - 108*a^9*b^10*d^4 + 24*a^11*b^8*d^4 +
48*a*b^18*c^2*d^2 - 96*a^2*b^17*c*d^3 - 48*a^2*b^17*c^3*d + 272*a^4*b^15*c*d^3 + 96*a^4*b^15*c^3*d - 288*a^6*
b^13*c*d^3 - 48*a^6*b^13*c^3*d + 144*a^8*b^11*c*d^3 - 32*a^10*b^9*c*d^3 - 72*a^3*b^16*c^2*d^2 + 24*a^7*b^12*c^
2*d^2))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15
- 16*a^8*b^13 + 4*a^10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)
*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^
4*b^13 - 4*a^6*b^11 + a^8*b^9))*(a*d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*(6*a^4*d^2 + b^4*c^2 + 12*b^4*d^2 +
2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 10*a*b^3*c*d + 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^
8 + 5*a^8*b^6 - a^10*b^4)))*(6*a^4*d^2 + b^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 10*a*b^3*c*d
+ 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(6*a^4*d^2 + b^4*c^
2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 10*a*b^3*c*d + 4*a^3*b*c*d)*1i)/(2*(b^14 - 5*a^2*b^12 + 10*a
^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)) - ((a*d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*(36*a^4*b^11*d
^8 - 144*a^6*b^9*d^8 + 216*a^8*b^7*d^8 - 144*a^10*b^5*d^8 + 36*a^12*b^3*d^8 - 96*a^3*b^12*c*d^7 + 384*a^5*b^10
*c*d^7 - 576*a^7*b^8*c*d^7 + 384*a^9*b^6*c*d^7 - 96*a^11*b^4*c*d^7 + 64*a^2*b^13*c^2*d^6 - 256*a^4*b^11*c^2*d^
6 + 384*a^6*b^9*c^2*d^6 - 256*a^8*b^7*c^2*d^6 + 64*a^10*b^5*c^2*d^6))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*
b^10 + a^8*b^8) - (8*tan(e/2 + (f*x)/2)*(a*b^15*c^8 + 4*a^3*b^13*c^8 + 4*a^5*b^11*c^8 - 72*a^3*b^13*d^8 + 468*
a^5*b^11*d^8 - 936*a^7*b^9*d^8 + 873*a^9*b^7*d^8 - 396*a^11*b^5*d^8 + 72*a^13*b^3*d^8 - 128*a*b^15*c^2*d^6 + 1
44*a*b^15*c^4*d^4 + 24*a*b^15*c^6*d^2 + 192*a^2*b^14*c*d^7 - 24*a^2*b^14*c^7*d - 1440*a^4*b^12*c*d^7 - 48*a^4*
b^12*c^7*d + 2736*a^6*b^10*c*d^7 - 2424*a^8*b^8*c*d^7 + 1056*a^10*b^6*c*d^7 - 192*a^12*b^4*c*d^7 - 576*a^2*b^1
4*c^3*d^5 - 336*a^2*b^14*c^5*d^3 + 1440*a^3*b^13*c^2*d^6 + 744*a^3*b^13*c^4*d^4 + 204*a^3*b^13*c^6*d^2 - 96*a^
4*b^12*c^3*d^5 - 200*a^4*b^12*c^5*d^3 - 2200*a^5*b^11*c^2*d^6 - 426*a^5*b^11*c^4*d^4 + 24*a^5*b^11*c^6*d^2 + 4
08*a^6*b^10*c^3*d^5 + 64*a^6*b^10*c^5*d^3 + 1644*a^7*b^9*c^2*d^6 + 144*a^7*b^9*c^4*d^4 - 240*a^8*b^8*c^3*d^5 -
32*a^8*b^8*c^5*d^3 - 632*a^9*b^7*c^2*d^6 + 24*a^9*b^7*c^4*d^4 + 128*a^11*b^5*c^2*d^6))/(b^17 - 4*a^2*b^15 + 6
*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + ((a*d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*(2*a^2*b^16*c^4 - 6*a^6*b^
12*c^4 + 4*a^8*b^10*c^4 + 12*a^2*b^16*d^4 - 36*a^4*b^14*d^4 + 42*a^6*b^12*d^4 - 24*a^8*b^10*d^4 + 6*a^10*b^8*d
^4 + 32*a^3*b^15*c*d^3 - 24*a^3*b^15*c^3*d - 24*a^5*b^13*c*d^3 + 48*a^5*b^13*c^3*d + 16*a^7*b^11*c*d^3 - 24*a^
7*b^11*c^3*d - 8*a^9*b^9*c*d^3 + 24*a^2*b^16*c^2*d^2 - 36*a^4*b^14*c^2*d^2 + 12*a^8*b^10*c^2*d^2 - 16*a*b^17*c
*d^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(4*a*b^18*c^4 - 12*a^5*
b^14*c^4 + 8*a^7*b^12*c^4 + 48*a^3*b^16*d^4 - 156*a^5*b^14*d^4 + 192*a^7*b^12*d^4 - 108*a^9*b^10*d^4 + 24*a^11
*b^8*d^4 + 48*a*b^18*c^2*d^2 - 96*a^2*b^17*c*d^3 - 48*a^2*b^17*c^3*d + 272*a^4*b^15*c*d^3 + 96*a^4*b^15*c^3*d
- 288*a^6*b^13*c*d^3 - 48*a^6*b^13*c^3*d + 144*a^8*b^11*c*d^3 - 32*a^10*b^9*c*d^3 - 72*a^3*b^16*c^2*d^2 + 24*a
^7*b^12*c^2*d^2))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + (((8*(4*a^2*b^19 - 16*a^4*b^17 + 2
4*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2
+ (f*x)/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b
^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(a*d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*(6*a^4*d^2 + b^4*c^2 + 12
*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 10*a*b^3*c*d + 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 -
10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(6*a^4*d^2 + b^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 10*
a*b^3*c*d + 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(6*a^4*d^
2 + b^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 10*a*b^3*c*d + 4*a^3*b*c*d)*1i)/(2*(b^14 - 5*a^2*b
^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))/((16*(54*a^12*d^12 - 216*a^6*b^6*d^12 + 378*a^8*b^4*d
^12 - 243*a^10*b^2*d^12 + 576*a*b^11*c^5*d^7 + 96*a*b^11*c^7*d^5 + 4*a*b^11*c^9*d^3 + 1296*a^5*b^7*c*d^11 - 19
44*a^7*b^5*c*d^11 + 1116*a^9*b^3*c*d^11 - 2352*a^2*b^10*c^4*d^8 - 1384*a^2*b^10*c^6*d^6 - 99*a^2*b^10*c^8*d^4
+ 3840*a^3*b^9*c^3*d^9 + 3552*a^3*b^9*c^5*d^7 + 888*a^3*b^9*c^7*d^5 + 16*a^3*b^9*c^9*d^3 - 3144*a^4*b^8*c^2*d^
10 - 2598*a^4*b^8*c^4*d^8 - 1412*a^4*b^8*c^6*d^6 - 204*a^4*b^8*c^8*d^4 - 1592*a^5*b^7*c^3*d^9 - 336*a^5*b^7*c^
5*d^7 + 240*a^5*b^7*c^7*d^5 + 16*a^5*b^7*c^9*d^3 + 3492*a^6*b^6*c^2*d^10 + 1758*a^6*b^6*c^4*d^8 + 88*a^6*b^6*c
^6*d^6 - 12*a^6*b^6*c^8*d^4 - 104*a^7*b^5*c^3*d^9 + 144*a^7*b^5*c^5*d^7 - 1572*a^8*b^4*c^2*d^10 - 678*a^8*b^4*
c^4*d^8 - 64*a^8*b^4*c^6*d^6 + 376*a^9*b^3*c^3*d^9 + 96*a^9*b^3*c^5*d^7 + 180*a^10*b^2*c^2*d^10 - 36*a^10*b^2*
c^4*d^8 - 216*a^11*b*c*d^11))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (16*tan(e/2 + (f*x)/2)
*(216*a^13*d^12 + 432*a^5*b^8*d^12 - 1404*a^7*b^6*d^12 + 1728*a^9*b^4*d^12 - 972*a^11*b^2*d^12 + 768*a*b^12*c^
4*d^8 + 64*a*b^12*c^6*d^6 - 2016*a^4*b^9*c*d^11 + 6192*a^6*b^7*c*d^11 - 7200*a^8*b^5*c*d^11 + 3888*a^10*b^3*c*
d^11 - 2688*a^2*b^11*c^3*d^9 - 864*a^2*b^11*c^5*d^7 + 3504*a^3*b^10*c^2*d^10 + 36*a^3*b^10*c^4*d^8 + 5648*a^4*
b^9*c^3*d^9 + 1536*a^4*b^9*c^5*d^7 - 9672*a^5*b^8*c^2*d^10 - 2304*a^5*b^8*c^4*d^8 - 192*a^5*b^8*c^6*d^6 - 3744
*a^6*b^7*c^3*d^9 - 480*a^6*b^7*c^5*d^7 + 9984*a^7*b^6*c^2*d^10 + 1428*a^7*b^6*c^4*d^8 + 128*a^7*b^6*c^6*d^6 +
1296*a^8*b^5*c^3*d^9 - 192*a^8*b^5*c^5*d^7 - 4968*a^9*b^4*c^2*d^10 + 72*a^9*b^4*c^4*d^8 - 512*a^10*b^3*c^3*d^9
+ 1152*a^11*b^2*c^2*d^10 - 864*a^12*b*c*d^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + ((a*
d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*tan(e/2 + (f*x)/2)*(a*b^15*c^8 + 4*a^3*b^13*c^8 + 4*a^5*b^11*c^8 -
72*a^3*b^13*d^8 + 468*a^5*b^11*d^8 - 936*a^7*b^9*d^8 + 873*a^9*b^7*d^8 - 396*a^11*b^5*d^8 + 72*a^13*b^3*d^8 -
128*a*b^15*c^2*d^6 + 144*a*b^15*c^4*d^4 + 24*a*b^15*c^6*d^2 + 192*a^2*b^14*c*d^7 - 24*a^2*b^14*c^7*d - 1440*a
^4*b^12*c*d^7 - 48*a^4*b^12*c^7*d + 2736*a^6*b^10*c*d^7 - 2424*a^8*b^8*c*d^7 + 1056*a^10*b^6*c*d^7 - 192*a^12*
b^4*c*d^7 - 576*a^2*b^14*c^3*d^5 - 336*a^2*b^14*c^5*d^3 + 1440*a^3*b^13*c^2*d^6 + 744*a^3*b^13*c^4*d^4 + 204*a
^3*b^13*c^6*d^2 - 96*a^4*b^12*c^3*d^5 - 200*a^4*b^12*c^5*d^3 - 2200*a^5*b^11*c^2*d^6 - 426*a^5*b^11*c^4*d^4 +
24*a^5*b^11*c^6*d^2 + 408*a^6*b^10*c^3*d^5 + 64*a^6*b^10*c^5*d^3 + 1644*a^7*b^9*c^2*d^6 + 144*a^7*b^9*c^4*d^4
- 240*a^8*b^8*c^3*d^5 - 32*a^8*b^8*c^5*d^3 - 632*a^9*b^7*c^2*d^6 + 24*a^9*b^7*c^4*d^4 + 128*a^11*b^5*c^2*d^6))
/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) - (8*(36*a^4*b^11*d^8 - 144*a^6*b^9*d^8 + 216*a^8*b^7
*d^8 - 144*a^10*b^5*d^8 + 36*a^12*b^3*d^8 - 96*a^3*b^12*c*d^7 + 384*a^5*b^10*c*d^7 - 576*a^7*b^8*c*d^7 + 384*a
^9*b^6*c*d^7 - 96*a^11*b^4*c*d^7 + 64*a^2*b^13*c^2*d^6 - 256*a^4*b^11*c^2*d^6 + 384*a^6*b^9*c^2*d^6 - 256*a^8*
b^7*c^2*d^6 + 64*a^10*b^5*c^2*d^6))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + ((a*d - b*c)^2*(
-(a + b)^5*(a - b)^5)^(1/2)*((8*(2*a^2*b^16*c^4 - 6*a^6*b^12*c^4 + 4*a^8*b^10*c^4 + 12*a^2*b^16*d^4 - 36*a^4*b
^14*d^4 + 42*a^6*b^12*d^4 - 24*a^8*b^10*d^4 + 6*a^10*b^8*d^4 + 32*a^3*b^15*c*d^3 - 24*a^3*b^15*c^3*d - 24*a^5*
b^13*c*d^3 + 48*a^5*b^13*c^3*d + 16*a^7*b^11*c*d^3 - 24*a^7*b^11*c^3*d - 8*a^9*b^9*c*d^3 + 24*a^2*b^16*c^2*d^2
- 36*a^4*b^14*c^2*d^2 + 12*a^8*b^10*c^2*d^2 - 16*a*b^17*c*d^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10
+ a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(4*a*b^18*c^4 - 12*a^5*b^14*c^4 + 8*a^7*b^12*c^4 + 48*a^3*b^16*d^4 - 156*a^
5*b^14*d^4 + 192*a^7*b^12*d^4 - 108*a^9*b^10*d^4 + 24*a^11*b^8*d^4 + 48*a*b^18*c^2*d^2 - 96*a^2*b^17*c*d^3 - 4
8*a^2*b^17*c^3*d + 272*a^4*b^15*c*d^3 + 96*a^4*b^15*c^3*d - 288*a^6*b^13*c*d^3 - 48*a^6*b^13*c^3*d + 144*a^8*b
^11*c*d^3 - 32*a^10*b^9*c*d^3 - 72*a^3*b^16*c^2*d^2 + 24*a^7*b^12*c^2*d^2))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 -
4*a^6*b^11 + a^8*b^9) - (((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^16 - 4*a
^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5*b^17 -
96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(a*d - b*c
)^2*(-(a + b)^5*(a - b)^5)^(1/2)*(6*a^4*d^2 + b^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 10*a*b^3
*c*d + 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(6*a^4*d^2 + b
^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 10*a*b^3*c*d + 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^12 + 10
*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(6*a^4*d^2 + b^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^
2*d^2 - 10*a*b^3*c*d + 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4))
+ ((a*d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*(36*a^4*b^11*d^8 - 144*a^6*b^9*d^8 + 216*a^8*b^7*d^8 - 144*
a^10*b^5*d^8 + 36*a^12*b^3*d^8 - 96*a^3*b^12*c*d^7 + 384*a^5*b^10*c*d^7 - 576*a^7*b^8*c*d^7 + 384*a^9*b^6*c*d^
7 - 96*a^11*b^4*c*d^7 + 64*a^2*b^13*c^2*d^6 - 256*a^4*b^11*c^2*d^6 + 384*a^6*b^9*c^2*d^6 - 256*a^8*b^7*c^2*d^6
+ 64*a^10*b^5*c^2*d^6))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) - (8*tan(e/2 + (f*x)/2)*(a*b^
15*c^8 + 4*a^3*b^13*c^8 + 4*a^5*b^11*c^8 - 72*a^3*b^13*d^8 + 468*a^5*b^11*d^8 - 936*a^7*b^9*d^8 + 873*a^9*b^7*
d^8 - 396*a^11*b^5*d^8 + 72*a^13*b^3*d^8 - 128*a*b^15*c^2*d^6 + 144*a*b^15*c^4*d^4 + 24*a*b^15*c^6*d^2 + 192*a
^2*b^14*c*d^7 - 24*a^2*b^14*c^7*d - 1440*a^4*b^12*c*d^7 - 48*a^4*b^12*c^7*d + 2736*a^6*b^10*c*d^7 - 2424*a^8*b
^8*c*d^7 + 1056*a^10*b^6*c*d^7 - 192*a^12*b^4*c*d^7 - 576*a^2*b^14*c^3*d^5 - 336*a^2*b^14*c^5*d^3 + 1440*a^3*b
^13*c^2*d^6 + 744*a^3*b^13*c^4*d^4 + 204*a^3*b^13*c^6*d^2 - 96*a^4*b^12*c^3*d^5 - 200*a^4*b^12*c^5*d^3 - 2200*
a^5*b^11*c^2*d^6 - 426*a^5*b^11*c^4*d^4 + 24*a^5*b^11*c^6*d^2 + 408*a^6*b^10*c^3*d^5 + 64*a^6*b^10*c^5*d^3 + 1
644*a^7*b^9*c^2*d^6 + 144*a^7*b^9*c^4*d^4 - 240*a^8*b^8*c^3*d^5 - 32*a^8*b^8*c^5*d^3 - 632*a^9*b^7*c^2*d^6 + 2
4*a^9*b^7*c^4*d^4 + 128*a^11*b^5*c^2*d^6))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9) + ((a*d - b
*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*((8*(2*a^2*b^16*c^4 - 6*a^6*b^12*c^4 + 4*a^8*b^10*c^4 + 12*a^2*b^16*d^4 - 3
6*a^4*b^14*d^4 + 42*a^6*b^12*d^4 - 24*a^8*b^10*d^4 + 6*a^10*b^8*d^4 + 32*a^3*b^15*c*d^3 - 24*a^3*b^15*c^3*d -
24*a^5*b^13*c*d^3 + 48*a^5*b^13*c^3*d + 16*a^7*b^11*c*d^3 - 24*a^7*b^11*c^3*d - 8*a^9*b^9*c*d^3 + 24*a^2*b^16*
c^2*d^2 - 36*a^4*b^14*c^2*d^2 + 12*a^8*b^10*c^2*d^2 - 16*a*b^17*c*d^3))/(b^16 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^
6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(4*a*b^18*c^4 - 12*a^5*b^14*c^4 + 8*a^7*b^12*c^4 + 48*a^3*b^16*d^4 -
156*a^5*b^14*d^4 + 192*a^7*b^12*d^4 - 108*a^9*b^10*d^4 + 24*a^11*b^8*d^4 + 48*a*b^18*c^2*d^2 - 96*a^2*b^17*c*
d^3 - 48*a^2*b^17*c^3*d + 272*a^4*b^15*c*d^3 + 96*a^4*b^15*c^3*d - 288*a^6*b^13*c*d^3 - 48*a^6*b^13*c^3*d + 14
4*a^8*b^11*c*d^3 - 32*a^10*b^9*c*d^3 - 72*a^3*b^16*c^2*d^2 + 24*a^7*b^12*c^2*d^2))/(b^17 - 4*a^2*b^15 + 6*a^4*
b^13 - 4*a^6*b^11 + a^8*b^9) + (((8*(4*a^2*b^19 - 16*a^4*b^17 + 24*a^6*b^15 - 16*a^8*b^13 + 4*a^10*b^11))/(b^1
6 - 4*a^2*b^14 + 6*a^4*b^12 - 4*a^6*b^10 + a^8*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^21 - 56*a^3*b^19 + 104*a^5
*b^17 - 96*a^7*b^15 + 44*a^9*b^13 - 8*a^11*b^11))/(b^17 - 4*a^2*b^15 + 6*a^4*b^13 - 4*a^6*b^11 + a^8*b^9))*(a*
d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*(6*a^4*d^2 + b^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 1
0*a*b^3*c*d + 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(6*a^4*
d^2 + b^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^2*b^2*d^2 - 10*a*b^3*c*d + 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^
12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(6*a^4*d^2 + b^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15
*a^2*b^2*d^2 - 10*a*b^3*c*d + 4*a^3*b*c*d))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^1
0*b^4))))*(a*d - b*c)^2*(-(a + b)^5*(a - b)^5)^(1/2)*(6*a^4*d^2 + b^4*c^2 + 12*b^4*d^2 + 2*a^2*b^2*c^2 - 15*a^
2*b^2*d^2 - 10*a*b^3*c*d + 4*a^3*b*c*d)*1i)/(f*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^1
0*b^4))