\(\int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^2} \, dx\) [706]

Optimal result

Integrand size = 25, antiderivative size = 278 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^2} \, dx=-\frac {d^2 \left (48 b c d-54 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {2 (b c-3 d)^3 \left (3 b c+27 d-4 b^2 d\right ) \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{b^4 \left (9-b^2\right )^{3/2} f}+\frac {(2 b c-3 d) d \left (6 b c d-27 d^2-b^2 \left (c^2-2 d^2\right )\right ) \cos (e+f x)}{b^3 \left (9-b^2\right ) f}+\frac {d^2 \left (12 b c d-27 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (9-b^2\right ) f}+\frac {(b c-3 d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (9-b^2\right ) f (3+b \sin (e+f x))} \]

[Out]

-1/2*d^2*(16*a*b*c*d-6*a^2*d^2-b^2*(12*c^2+d^2))*x/b^4+2*(-a*d+b*c)^3*(3*a^2*d+a*b*c-4*b^2*d)*arctan((b+a*tan(
1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/b^4/(a^2-b^2)^(3/2)/f+d*(-a*d+2*b*c)*(2*a*b*c*d-3*a^2*d^2-b^2*(c^2-2*d^2))*co
s(f*x+e)/b^3/(a^2-b^2)/f+1/2*d^2*(4*a*b*c*d-3*a^2*d^2-b^2*(2*c^2-d^2))*cos(f*x+e)*sin(f*x+e)/b^2/(a^2-b^2)/f+(
-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))

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 306, 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, 3112, 3102, 2814, 2739, 632, 210} \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^2} \, dx=\frac {2 \left (3 a^2 d+a b c-4 b^2 d\right ) (b c-a d)^3 \arctan \left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^4 f \left (a^2-b^2\right )^{3/2}}+\frac {d^2 \left (-3 a^2 d^2+4 a b c d-\left (b^2 \left (2 c^2-d^2\right )\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b^2 f \left (a^2-b^2\right )}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {d^2 x \left (-6 a^2 d^2+16 a b c d-\left (b^2 \left (12 c^2+d^2\right )\right )\right )}{2 b^4}+\frac {d (2 b c-a d) \left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \cos (e+f x)}{b^3 f \left (a^2-b^2\right )} \]

[In]

Int[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x])^2,x]

[Out]

-1/2*(d^2*(16*a*b*c*d - 6*a^2*d^2 - b^2*(12*c^2 + d^2))*x)/b^4 + (2*(b*c - a*d)^3*(a*b*c + 3*a^2*d - 4*b^2*d)*
ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)*f) + (d*(2*b*c - a*d)*(2*a*b*c*d - 3*
a^2*d^2 - b^2*(c^2 - 2*d^2))*Cos[e + f*x])/(b^3*(a^2 - b^2)*f) + (d^2*(4*a*b*c*d - 3*a^2*d^2 - b^2*(2*c^2 - d^
2))*Cos[e + f*x]*Sin[e + f*x])/(2*b^2*(a^2 - b^2)*f) + ((b*c - a*d)^2*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(b*
(a^2 - b^2)*f*(a + b*Sin[e + f*x]))

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 3112

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] &&
!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}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\int \frac {(c+d \sin (e+f x)) \left (4 b^2 c^2 d+2 a^2 d^3-a b c \left (c^2+5 d^2\right )-d \left (a^2 c d-3 b^2 c d+a b \left (c^2+d^2\right )\right ) \sin (e+f x)+d \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{b \left (a^2-b^2\right )} \\ & = \frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\int \frac {8 b^3 c^3 d+8 a^2 b c d^3-3 a^3 d^4-a b^2 \left (2 c^4+12 c^2 d^2-d^4\right )+b d \left (b^2 d \left (12 c^2+d^2\right )-4 a b c \left (c^2+2 d^2\right )-a^2 \left (2 c^2 d-d^3\right )\right ) \sin (e+f x)-2 d (2 b c-a d) \left (b^2 c^2-2 a b c d+3 a^2 d^2-2 b^2 d^2\right ) \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )} \\ & = \frac {d (2 b c-a d) \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)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\int \frac {b \left (8 b^3 c^3 d+8 a^2 b c d^3-3 a^3 d^4-a b^2 \left (2 c^4+12 c^2 d^2-d^4\right )\right )+\left (a^2-b^2\right ) d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{2 b^3 \left (a^2-b^2\right )} \\ & = -\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d (2 b c-a d) \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)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left ((b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{b^4 \left (a^2-b^2\right )} \\ & = -\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d (2 b c-a d) \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)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left (2 (b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\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 ) f} \\ & = -\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d (2 b c-a d) \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)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\left (4 (b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\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 ) f} \\ & = -\frac {d^2 \left (16 a b c d-6 a^2 d^2-b^2 \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {2 (b c-a d)^3 \left (a b c+3 a^2 d-4 b^2 d\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 )^{3/2} f}+\frac {d (2 b c-a d) \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)}{b^3 \left (a^2-b^2\right ) f}+\frac {d^2 \left (4 a b c d-3 a^2 d^2-b^2 \left (2 c^2-d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^2 \left (a^2-b^2\right ) f}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^2}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.85 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.66 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^2} \, dx=-\frac {-2 d^2 \left (-48 b c d+54 d^2+b^2 \left (12 c^2+d^2\right )\right ) (e+f x)+\frac {8 (b c-3 d)^3 \left (-3 b c-27 d+4 b^2 d\right ) \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{\left (9-b^2\right )^{3/2}}+8 b (2 b c-3 d) d^3 \cos (e+f x)+\frac {4 b (b c-3 d)^4 \cos (e+f x)}{(-3+b) (3+b) (3+b \sin (e+f x))}+b^2 d^4 \sin (2 (e+f x))}{4 b^4 f} \]

[In]

Integrate[(c + d*Sin[e + f*x])^4/(3 + b*Sin[e + f*x])^2,x]

[Out]

-1/4*(-2*d^2*(-48*b*c*d + 54*d^2 + b^2*(12*c^2 + d^2))*(e + f*x) + (8*(b*c - 3*d)^3*(-3*b*c - 27*d + 4*b^2*d)*
ArcTan[(b + 3*Tan[(e + f*x)/2])/Sqrt[9 - b^2]])/(9 - b^2)^(3/2) + 8*b*(2*b*c - 3*d)*d^3*Cos[e + f*x] + (4*b*(b
*c - 3*d)^4*Cos[e + f*x])/((-3 + b)*(3 + b)*(3 + b*Sin[e + f*x])) + b^2*d^4*Sin[2*(e + f*x)])/(b^4*f)

Maple [A] (verified)

Time = 2.19 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.64

method	result	size
derivativedivides	\(\frac {\frac {2 d^{2} \left (\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) b^{2} d^{2}}{2}+\left (2 a b \,d^{2}-4 b^{2} c d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{2} d^{2}}{2}+2 a b \,d^{2}-4 b^{2} c d}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (6 d^{2} a^{2}-16 a b c d +12 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) a}-\frac {b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{a^{2}-b^{2}}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (3 a^{5} d^{4}-8 a^{4} b c \,d^{3}+6 a^{3} b^{2} c^{2} d^{2}-4 a^{3} b^{2} d^{4}+12 a^{2} b^{3} c \,d^{3}-a \,b^{4} c^{4}-12 a \,b^{4} c^{2} d^{2}+4 b^{5} c^{3} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{4}}}{f}\)	\(456\)
default	\(\frac {\frac {2 d^{2} \left (\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) b^{2} d^{2}}{2}+\left (2 a b \,d^{2}-4 b^{2} c d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{2} d^{2}}{2}+2 a b \,d^{2}-4 b^{2} c d}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (6 d^{2} a^{2}-16 a b c d +12 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) a}-\frac {b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{a^{2}-b^{2}}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (3 a^{5} d^{4}-8 a^{4} b c \,d^{3}+6 a^{3} b^{2} c^{2} d^{2}-4 a^{3} b^{2} d^{4}+12 a^{2} b^{3} c \,d^{3}-a \,b^{4} c^{4}-12 a \,b^{4} c^{2} d^{2}+4 b^{5} c^{3} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{4}}}{f}\)	\(456\)
risch	\(\text {Expression too large to display}\)	\(1666\)

[In]

int((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/f*(2*d^2/b^4*((1/2*tan(1/2*f*x+1/2*e)^3*b^2*d^2+(2*a*b*d^2-4*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+2*a*b*d^2-4*b^2*c*d)/(1+tan(1/2*f*x+1/2*e)^2)^2+1/2*(6*a^2*d^2-16*a*b*c*d+12*b^2*c^2+b^2*d^2)*ar
ctan(tan(1/2*f*x+1/2*e)))-2/b^4*((-b^2*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/(a^2-b^
2)/a*tan(1/2*f*x+1/2*e)-b*(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/(a^2-b^2))/(tan(1/2*
f*x+1/2*e)^2*a+2*b*tan(1/2*f*x+1/2*e)+a)+(3*a^5*d^4-8*a^4*b*c*d^3+6*a^3*b^2*c^2*d^2-4*a^3*b^2*d^4+12*a^2*b^3*c
*d^3-a*b^4*c^4-12*a*b^4*c^2*d^2+4*b^5*c^3*d)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)
^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 684 vs. \(2 (297) = 594\).

Time = 0.37 (sec) , antiderivative size = 1451, normalized size of antiderivative = 5.22 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

[1/2*((a^4*b^3 - 2*a^2*b^5 + b^7)*d^4*cos(f*x + e)^3 + (12*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^2*d^2 - 16*(a^6*b -
2*a^4*b^3 + a^2*b^5)*c*d^3 + (6*a^7 - 11*a^5*b^2 + 4*a^3*b^4 + a*b^6)*d^4)*f*x + (a^2*b^4*c^4 - 4*a*b^5*c^3*d
- 6*(a^4*b^2 - 2*a^2*b^4)*c^2*d^2 + 4*(2*a^5*b - 3*a^3*b^3)*c*d^3 - (3*a^6 - 4*a^4*b^2)*d^4 + (a*b^5*c^4 - 4*
b^6*c^3*d - 6*(a^3*b^3 - 2*a*b^5)*c^2*d^2 + 4*(2*a^4*b^2 - 3*a^2*b^4)*c*d^3 - (3*a^5*b - 4*a^3*b^3)*d^4)*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*(a^2*b^5 - b^7)*c^4 - 8*(a^3*b^4 - a*b^6)*c^3*d + 12*(a^4*b^3 - a^2*b^5)*c^2*d^2 - 8*(2*a^5*b^2 - 3*a^3*b^4
+ a*b^6)*c*d^3 + (6*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 - b^7)*d^4)*cos(f*x + e) + ((12*(a^4*b^3 - 2*a^2*b^5 + b^7
)*c^2*d^2 - 16*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 + (6*a^6*b - 11*a^4*b^3 + 4*a^2*b^5 + b^7)*d^4)*f*x - (8*(a
^4*b^3 - 2*a^2*b^5 + b^7)*c*d^3 - 3*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^4*b^5 -
2*a^2*b^7 + b^9)*f*sin(f*x + e) + (a^5*b^4 - 2*a^3*b^6 + a*b^8)*f), 1/2*((a^4*b^3 - 2*a^2*b^5 + b^7)*d^4*cos(
f*x + e)^3 + (12*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^2*d^2 - 16*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 + (6*a^7 - 11*
a^5*b^2 + 4*a^3*b^4 + a*b^6)*d^4)*f*x - 2*(a^2*b^4*c^4 - 4*a*b^5*c^3*d - 6*(a^4*b^2 - 2*a^2*b^4)*c^2*d^2 + 4*(
2*a^5*b - 3*a^3*b^3)*c*d^3 - (3*a^6 - 4*a^4*b^2)*d^4 + (a*b^5*c^4 - 4*b^6*c^3*d - 6*(a^3*b^3 - 2*a*b^5)*c^2*d^
2 + 4*(2*a^4*b^2 - 3*a^2*b^4)*c*d^3 - (3*a^5*b - 4*a^3*b^3)*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))) + (2*(a^2*b^5 - b^7)*c^4 - 8*(a^3*b^4 - a*b^6)*c^3*d + 12*(a^4*b
^3 - a^2*b^5)*c^2*d^2 - 8*(2*a^5*b^2 - 3*a^3*b^4 + a*b^6)*c*d^3 + (6*a^6*b - 11*a^4*b^3 + 6*a^2*b^5 - b^7)*d^4
)*cos(f*x + e) + ((12*(a^4*b^3 - 2*a^2*b^5 + b^7)*c^2*d^2 - 16*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 + (6*a^6*b
- 11*a^4*b^3 + 4*a^2*b^5 + b^7)*d^4)*f*x - (8*(a^4*b^3 - 2*a^2*b^5 + b^7)*c*d^3 - 3*(a^5*b^2 - 2*a^3*b^4 + a*b
^6)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^4*b^5 - 2*a^2*b^7 + b^9)*f*sin(f*x + e) + (a^5*b^4 - 2*a^3*b^6 + a*b^
8)*f)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^2} \, dx=\text {Timed out} \]

[In]

integrate((c+d*sin(f*x+e))**4/(a+b*sin(f*x+e))**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,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 [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.81 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^2} \, dx=\frac {\frac {4 \, {\left (a b^{4} c^{4} - 4 \, b^{5} c^{3} d - 6 \, a^{3} b^{2} c^{2} d^{2} + 12 \, a b^{4} c^{2} d^{2} + 8 \, a^{4} b c d^{3} - 12 \, a^{2} b^{3} c d^{3} - 3 \, a^{5} d^{4} + 4 \, a^{3} b^{2} d^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {4 \, {\left (b^{5} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, a b^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2} b^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 4 \, a^{3} b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{4} b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d + 6 \, a^{3} b^{2} c^{2} d^{2} - 4 \, a^{4} b c d^{3} + a^{5} d^{4}\right )}}{{\left (a^{3} b^{3} - a b^{5}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a\right )}} + \frac {{\left (12 \, b^{2} c^{2} d^{2} - 16 \, a b c d^{3} + 6 \, a^{2} d^{4} + b^{2} d^{4}\right )} {\left (f x + e\right )}}{b^{4}} + \frac {2 \, {\left (b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, a d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 8 \, b c d^{3} + 4 \, a d^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} b^{3}}}{2 \, f} \]

[In]

integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e))^2,x, algorithm="giac")

[Out]

1/2*(4*(a*b^4*c^4 - 4*b^5*c^3*d - 6*a^3*b^2*c^2*d^2 + 12*a*b^4*c^2*d^2 + 8*a^4*b*c*d^3 - 12*a^2*b^3*c*d^3 - 3*
a^5*d^4 + 4*a^3*b^2*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^2*b^4 - b^6)*sqrt(a^2 - b^2)) + 4*(b^5*c^4*tan(1/2*f*x + 1/2*e) - 4*a*b^4*c^3*d*tan(1/2*f*x +
1/2*e) + 6*a^2*b^3*c^2*d^2*tan(1/2*f*x + 1/2*e) - 4*a^3*b^2*c*d^3*tan(1/2*f*x + 1/2*e) + a^4*b*d^4*tan(1/2*f*x
+ 1/2*e) + a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + a^5*d^4)/((a^3*b^3 - a*b^5)*(a*t
an(1/2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a)) + (12*b^2*c^2*d^2 - 16*a*b*c*d^3 + 6*a^2*d^4 + b^2*d^4)
*(f*x + e)/b^4 + 2*(b*d^4*tan(1/2*f*x + 1/2*e)^3 - 8*b*c*d^3*tan(1/2*f*x + 1/2*e)^2 + 4*a*d^4*tan(1/2*f*x + 1/
2*e)^2 - b*d^4*tan(1/2*f*x + 1/2*e) - 8*b*c*d^3 + 4*a*d^4)/((tan(1/2*f*x + 1/2*e)^2 + 1)^2*b^3))/f

Mupad [B] (verification not implemented)

Time = 23.03 (sec) , antiderivative size = 13700, normalized size of antiderivative = 49.28 \[ \int \frac {(c+d \sin (e+f x))^4}{(3+b \sin (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

int((c + d*sin(e + f*x))^4/(a + b*sin(e + f*x))^2,x)

[Out]

((2*(3*a^4*d^4 + b^4*c^4 - 2*a^2*b^2*d^4 + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d^3 - 4*a*b^3*c^3*d - 8*a^3*b*c*d^3))
/(b^3*(a^2 - b^2)) + (2*tan(e/2 + (f*x)/2)^4*(3*a^4*d^4 + b^4*c^4 - b^4*d^4 - a^2*b^2*d^4 + 6*a^2*b^2*c^2*d^2
+ 4*a*b^3*c*d^3 - 4*a*b^3*c^3*d - 8*a^3*b*c*d^3))/(b^3*(a^2 - b^2)) + (2*tan(e/2 + (f*x)/2)^2*(6*a^4*d^4 + 2*b
^4*c^4 + b^4*d^4 - 5*a^2*b^2*d^4 + 12*a^2*b^2*c^2*d^2 + 8*a*b^3*c*d^3 - 8*a*b^3*c^3*d - 16*a^3*b*c*d^3))/(b^3*
(a^2 - b^2)) + (tan(e/2 + (f*x)/2)^5*(3*a^4*d^4 + 2*b^4*c^4 - a^2*b^2*d^4 + 12*a^2*b^2*c^2*d^2 - 8*a*b^3*c^3*d
- 8*a^3*b*c*d^3))/(a*b^2*(a^2 - b^2)) + (tan(e/2 + (f*x)/2)*(9*a^4*d^4 + 2*b^4*c^4 - 7*a^2*b^2*d^4 + 12*a^2*b
^2*c^2*d^2 + 16*a*b^3*c*d^3 - 8*a*b^3*c^3*d - 24*a^3*b*c*d^3))/(a*b^2*(a^2 - b^2)) + (4*tan(e/2 + (f*x)/2)^3*(
3*a^4*d^4 + b^4*c^4 - 2*a^2*b^2*d^4 + 6*a^2*b^2*c^2*d^2 + 4*a*b^3*c*d^3 - 4*a*b^3*c^3*d - 8*a^3*b*c*d^3))/(a*b
^2*(a^2 - b^2)))/(f*(a + 2*b*tan(e/2 + (f*x)/2) + 3*a*tan(e/2 + (f*x)/2)^2 + 3*a*tan(e/2 + (f*x)/2)^4 + a*tan(
e/2 + (f*x)/2)^6 + 4*b*tan(e/2 + (f*x)/2)^3 + 2*b*tan(e/2 + (f*x)/2)^5)) + (atan(((((8*(a^2*b^11*d^8 + 10*a^4*
b^9*d^8 + 13*a^6*b^7*d^8 - 60*a^8*b^5*d^8 + 36*a^10*b^3*d^8 - 32*a^3*b^10*c*d^7 - 128*a^5*b^8*c*d^7 + 352*a^7*
b^6*c*d^7 - 192*a^9*b^4*c*d^7 + 24*a^2*b^11*c^2*d^6 + 144*a^2*b^11*c^4*d^4 - 384*a^3*b^10*c^3*d^5 + 352*a^4*b^
9*c^2*d^6 - 288*a^4*b^9*c^4*d^4 + 768*a^5*b^8*c^3*d^5 - 776*a^6*b^7*c^2*d^6 + 144*a^6*b^7*c^4*d^4 - 384*a^7*b^
6*c^3*d^5 + 400*a^8*b^5*c^2*d^6))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (((8*(2*a*b^14*d^4 + 4*a^3*b^12*c^4 - 4*a^5*
b^10*c^4 + 6*a^3*b^12*d^4 - 14*a^5*b^10*d^4 + 6*a^7*b^8*d^4 + 24*a*b^14*c^2*d^2 - 32*a^2*b^13*c*d^3 - 16*a^2*b
^13*c^3*d + 48*a^4*b^11*c*d^3 + 16*a^4*b^11*c^3*d - 16*a^6*b^9*c*d^3 - 24*a^3*b^12*c^2*d^2))/(b^12 - 2*a^2*b^1
0 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(8*a^2*b^14*c^4 - 8*a^4*b^12*c^4 + 32*a^4*b^12*d^4 - 56*a^6*b^10*d^4 + 24
*a^8*b^8*d^4 - 96*a^3*b^13*c*d^3 + 32*a^3*b^13*c^3*d + 160*a^5*b^11*c*d^3 - 64*a^7*b^9*c*d^3 + 96*a^2*b^14*c^2
*d^2 - 144*a^4*b^12*c^2*d^2 + 48*a^6*b^10*c^2*d^2 - 32*a*b^15*c^3*d))/(b^13 - 2*a^2*b^11 + a^4*b^9) + (((8*(4*
a^2*b^15 - 8*a^4*b^13 + 4*a^6*b^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^17 - 32*a^3
*b^15 + 28*a^5*b^13 - 8*a^7*b^11))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/2
- a*b*c*d^3*8i))/b^4)*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/2 - a*b*c*d^3*8i))/b^4 + (8*tan(e/2 + (f*x)/2)
*(2*a*b^13*d^8 - 4*a^3*b^11*c^8 + 19*a^3*b^11*d^8 + 16*a^5*b^9*d^8 - 197*a^7*b^7*d^8 + 228*a^9*b^5*d^8 - 72*a^
11*b^3*d^8 + 48*a*b^13*c^2*d^6 + 288*a*b^13*c^4*d^4 - 64*a*b^13*c^6*d^2 - 64*a^2*b^12*c*d^7 + 32*a^2*b^12*c^7*
d - 224*a^4*b^10*c*d^7 + 1216*a^6*b^8*c*d^7 - 1280*a^8*b^6*c*d^7 + 384*a^10*b^4*c*d^7 - 768*a^2*b^12*c^3*d^5 +
384*a^2*b^12*c^5*d^3 + 680*a^3*b^11*c^2*d^6 - 1680*a^3*b^11*c^4*d^4 - 96*a^3*b^11*c^6*d^2 + 3200*a^4*b^10*c^3
*d^5 - 96*a^4*b^10*c^5*d^3 - 2864*a^5*b^9*c^2*d^6 + 1376*a^5*b^9*c^4*d^4 + 48*a^5*b^9*c^6*d^2 - 2976*a^6*b^8*c
^3*d^5 - 64*a^6*b^8*c^5*d^3 + 2824*a^7*b^7*c^2*d^6 - 264*a^7*b^7*c^4*d^4 + 768*a^8*b^6*c^3*d^5 - 800*a^9*b^5*c
^2*d^6))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/2 - a*b*c*d^3*8i)*1i)/b^4 +
(((8*(a^2*b^11*d^8 + 10*a^4*b^9*d^8 + 13*a^6*b^7*d^8 - 60*a^8*b^5*d^8 + 36*a^10*b^3*d^8 - 32*a^3*b^10*c*d^7 -
128*a^5*b^8*c*d^7 + 352*a^7*b^6*c*d^7 - 192*a^9*b^4*c*d^7 + 24*a^2*b^11*c^2*d^6 + 144*a^2*b^11*c^4*d^4 - 384*a
^3*b^10*c^3*d^5 + 352*a^4*b^9*c^2*d^6 - 288*a^4*b^9*c^4*d^4 + 768*a^5*b^8*c^3*d^5 - 776*a^6*b^7*c^2*d^6 + 144*
a^6*b^7*c^4*d^4 - 384*a^7*b^6*c^3*d^5 + 400*a^8*b^5*c^2*d^6))/(b^12 - 2*a^2*b^10 + a^4*b^8) - (((8*(2*a*b^14*d
^4 + 4*a^3*b^12*c^4 - 4*a^5*b^10*c^4 + 6*a^3*b^12*d^4 - 14*a^5*b^10*d^4 + 6*a^7*b^8*d^4 + 24*a*b^14*c^2*d^2 -
32*a^2*b^13*c*d^3 - 16*a^2*b^13*c^3*d + 48*a^4*b^11*c*d^3 + 16*a^4*b^11*c^3*d - 16*a^6*b^9*c*d^3 - 24*a^3*b^12
*c^2*d^2))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(8*a^2*b^14*c^4 - 8*a^4*b^12*c^4 + 32*a^4*b^1
2*d^4 - 56*a^6*b^10*d^4 + 24*a^8*b^8*d^4 - 96*a^3*b^13*c*d^3 + 32*a^3*b^13*c^3*d + 160*a^5*b^11*c*d^3 - 64*a^7
*b^9*c*d^3 + 96*a^2*b^14*c^2*d^2 - 144*a^4*b^12*c^2*d^2 + 48*a^6*b^10*c^2*d^2 - 32*a*b^15*c^3*d))/(b^13 - 2*a^
2*b^11 + a^4*b^9) - (((8*(4*a^2*b^15 - 8*a^4*b^13 + 4*a^6*b^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 +
(f*x)/2)*(12*a*b^17 - 32*a^3*b^15 + 28*a^5*b^13 - 8*a^7*b^11))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a^2*d^4*3i + (b
^2*d^2*(12*c^2 + d^2)*1i)/2 - a*b*c*d^3*8i))/b^4)*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/2 - a*b*c*d^3*8i))
/b^4 + (8*tan(e/2 + (f*x)/2)*(2*a*b^13*d^8 - 4*a^3*b^11*c^8 + 19*a^3*b^11*d^8 + 16*a^5*b^9*d^8 - 197*a^7*b^7*d
^8 + 228*a^9*b^5*d^8 - 72*a^11*b^3*d^8 + 48*a*b^13*c^2*d^6 + 288*a*b^13*c^4*d^4 - 64*a*b^13*c^6*d^2 - 64*a^2*b
^12*c*d^7 + 32*a^2*b^12*c^7*d - 224*a^4*b^10*c*d^7 + 1216*a^6*b^8*c*d^7 - 1280*a^8*b^6*c*d^7 + 384*a^10*b^4*c*
d^7 - 768*a^2*b^12*c^3*d^5 + 384*a^2*b^12*c^5*d^3 + 680*a^3*b^11*c^2*d^6 - 1680*a^3*b^11*c^4*d^4 - 96*a^3*b^11
*c^6*d^2 + 3200*a^4*b^10*c^3*d^5 - 96*a^4*b^10*c^5*d^3 - 2864*a^5*b^9*c^2*d^6 + 1376*a^5*b^9*c^4*d^4 + 48*a^5*
b^9*c^6*d^2 - 2976*a^6*b^8*c^3*d^5 - 64*a^6*b^8*c^5*d^3 + 2824*a^7*b^7*c^2*d^6 - 264*a^7*b^7*c^4*d^4 + 768*a^8
*b^6*c^3*d^5 - 800*a^9*b^5*c^2*d^6))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/
2 - a*b*c*d^3*8i)*1i)/b^4)/((16*(54*a^11*d^12 + 4*a^5*b^6*d^12 + 9*a^7*b^4*d^12 - 81*a^9*b^2*d^12 - 32*a*b^10*
c^6*d^6 - 384*a*b^10*c^8*d^4 - 12*a^4*b^7*c*d^11 - 60*a^6*b^5*c*d^11 + 648*a^8*b^3*c*d^11 - 4*a^2*b^9*c^3*d^9
+ 96*a^2*b^9*c^5*d^7 + 2256*a^2*b^9*c^7*d^5 + 192*a^2*b^9*c^9*d^3 + 12*a^3*b^8*c^2*d^10 - 63*a^3*b^8*c^4*d^8 -
5784*a^3*b^8*c^6*d^6 - 690*a^3*b^8*c^8*d^4 - 24*a^3*b^8*c^10*d^2 - 76*a^4*b^7*c^3*d^9 + 8592*a^4*b^7*c^5*d^7
+ 480*a^4*b^7*c^7*d^5 + 32*a^4*b^7*c^9*d^3 + 126*a^5*b^6*c^2*d^10 - 8277*a^5*b^6*c^4*d^8 + 1552*a^5*b^6*c^6*d^
6 + 132*a^5*b^6*c^8*d^4 + 5424*a^6*b^5*c^3*d^9 - 4128*a^6*b^5*c^5*d^7 - 384*a^6*b^5*c^7*d^5 - 2394*a^7*b^4*c^2
*d^10 + 4860*a^7*b^4*c^4*d^8 + 400*a^7*b^4*c^6*d^6 - 3472*a^8*b^3*c^3*d^9 - 192*a^8*b^3*c^5*d^7 + 1584*a^9*b^2
*c^2*d^10 + 36*a^9*b^2*c^4*d^8 - 432*a^10*b*c*d^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (((8*(a^2*b^11*d^8 + 10*a
^4*b^9*d^8 + 13*a^6*b^7*d^8 - 60*a^8*b^5*d^8 + 36*a^10*b^3*d^8 - 32*a^3*b^10*c*d^7 - 128*a^5*b^8*c*d^7 + 352*a
^7*b^6*c*d^7 - 192*a^9*b^4*c*d^7 + 24*a^2*b^11*c^2*d^6 + 144*a^2*b^11*c^4*d^4 - 384*a^3*b^10*c^3*d^5 + 352*a^4
*b^9*c^2*d^6 - 288*a^4*b^9*c^4*d^4 + 768*a^5*b^8*c^3*d^5 - 776*a^6*b^7*c^2*d^6 + 144*a^6*b^7*c^4*d^4 - 384*a^7
*b^6*c^3*d^5 + 400*a^8*b^5*c^2*d^6))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (((8*(2*a*b^14*d^4 + 4*a^3*b^12*c^4 - 4*a
^5*b^10*c^4 + 6*a^3*b^12*d^4 - 14*a^5*b^10*d^4 + 6*a^7*b^8*d^4 + 24*a*b^14*c^2*d^2 - 32*a^2*b^13*c*d^3 - 16*a^
2*b^13*c^3*d + 48*a^4*b^11*c*d^3 + 16*a^4*b^11*c^3*d - 16*a^6*b^9*c*d^3 - 24*a^3*b^12*c^2*d^2))/(b^12 - 2*a^2*
b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(8*a^2*b^14*c^4 - 8*a^4*b^12*c^4 + 32*a^4*b^12*d^4 - 56*a^6*b^10*d^4 +
24*a^8*b^8*d^4 - 96*a^3*b^13*c*d^3 + 32*a^3*b^13*c^3*d + 160*a^5*b^11*c*d^3 - 64*a^7*b^9*c*d^3 + 96*a^2*b^14*
c^2*d^2 - 144*a^4*b^12*c^2*d^2 + 48*a^6*b^10*c^2*d^2 - 32*a*b^15*c^3*d))/(b^13 - 2*a^2*b^11 + a^4*b^9) + (((8*
(4*a^2*b^15 - 8*a^4*b^13 + 4*a^6*b^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^17 - 32*
a^3*b^15 + 28*a^5*b^13 - 8*a^7*b^11))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)
/2 - a*b*c*d^3*8i))/b^4)*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/2 - a*b*c*d^3*8i))/b^4 + (8*tan(e/2 + (f*x)
/2)*(2*a*b^13*d^8 - 4*a^3*b^11*c^8 + 19*a^3*b^11*d^8 + 16*a^5*b^9*d^8 - 197*a^7*b^7*d^8 + 228*a^9*b^5*d^8 - 72
*a^11*b^3*d^8 + 48*a*b^13*c^2*d^6 + 288*a*b^13*c^4*d^4 - 64*a*b^13*c^6*d^2 - 64*a^2*b^12*c*d^7 + 32*a^2*b^12*c
^7*d - 224*a^4*b^10*c*d^7 + 1216*a^6*b^8*c*d^7 - 1280*a^8*b^6*c*d^7 + 384*a^10*b^4*c*d^7 - 768*a^2*b^12*c^3*d^
5 + 384*a^2*b^12*c^5*d^3 + 680*a^3*b^11*c^2*d^6 - 1680*a^3*b^11*c^4*d^4 - 96*a^3*b^11*c^6*d^2 + 3200*a^4*b^10*
c^3*d^5 - 96*a^4*b^10*c^5*d^3 - 2864*a^5*b^9*c^2*d^6 + 1376*a^5*b^9*c^4*d^4 + 48*a^5*b^9*c^6*d^2 - 2976*a^6*b^
8*c^3*d^5 - 64*a^6*b^8*c^5*d^3 + 2824*a^7*b^7*c^2*d^6 - 264*a^7*b^7*c^4*d^4 + 768*a^8*b^6*c^3*d^5 - 800*a^9*b^
5*c^2*d^6))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/2 - a*b*c*d^3*8i))/b^4 -
(((8*(a^2*b^11*d^8 + 10*a^4*b^9*d^8 + 13*a^6*b^7*d^8 - 60*a^8*b^5*d^8 + 36*a^10*b^3*d^8 - 32*a^3*b^10*c*d^7 -
128*a^5*b^8*c*d^7 + 352*a^7*b^6*c*d^7 - 192*a^9*b^4*c*d^7 + 24*a^2*b^11*c^2*d^6 + 144*a^2*b^11*c^4*d^4 - 384*a
^3*b^10*c^3*d^5 + 352*a^4*b^9*c^2*d^6 - 288*a^4*b^9*c^4*d^4 + 768*a^5*b^8*c^3*d^5 - 776*a^6*b^7*c^2*d^6 + 144*
a^6*b^7*c^4*d^4 - 384*a^7*b^6*c^3*d^5 + 400*a^8*b^5*c^2*d^6))/(b^12 - 2*a^2*b^10 + a^4*b^8) - (((8*(2*a*b^14*d
^4 + 4*a^3*b^12*c^4 - 4*a^5*b^10*c^4 + 6*a^3*b^12*d^4 - 14*a^5*b^10*d^4 + 6*a^7*b^8*d^4 + 24*a*b^14*c^2*d^2 -
32*a^2*b^13*c*d^3 - 16*a^2*b^13*c^3*d + 48*a^4*b^11*c*d^3 + 16*a^4*b^11*c^3*d - 16*a^6*b^9*c*d^3 - 24*a^3*b^12
*c^2*d^2))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(8*a^2*b^14*c^4 - 8*a^4*b^12*c^4 + 32*a^4*b^1
2*d^4 - 56*a^6*b^10*d^4 + 24*a^8*b^8*d^4 - 96*a^3*b^13*c*d^3 + 32*a^3*b^13*c^3*d + 160*a^5*b^11*c*d^3 - 64*a^7
*b^9*c*d^3 + 96*a^2*b^14*c^2*d^2 - 144*a^4*b^12*c^2*d^2 + 48*a^6*b^10*c^2*d^2 - 32*a*b^15*c^3*d))/(b^13 - 2*a^
2*b^11 + a^4*b^9) - (((8*(4*a^2*b^15 - 8*a^4*b^13 + 4*a^6*b^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 +
(f*x)/2)*(12*a*b^17 - 32*a^3*b^15 + 28*a^5*b^13 - 8*a^7*b^11))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a^2*d^4*3i + (b
^2*d^2*(12*c^2 + d^2)*1i)/2 - a*b*c*d^3*8i))/b^4)*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/2 - a*b*c*d^3*8i))
/b^4 + (8*tan(e/2 + (f*x)/2)*(2*a*b^13*d^8 - 4*a^3*b^11*c^8 + 19*a^3*b^11*d^8 + 16*a^5*b^9*d^8 - 197*a^7*b^7*d
^8 + 228*a^9*b^5*d^8 - 72*a^11*b^3*d^8 + 48*a*b^13*c^2*d^6 + 288*a*b^13*c^4*d^4 - 64*a*b^13*c^6*d^2 - 64*a^2*b
^12*c*d^7 + 32*a^2*b^12*c^7*d - 224*a^4*b^10*c*d^7 + 1216*a^6*b^8*c*d^7 - 1280*a^8*b^6*c*d^7 + 384*a^10*b^4*c*
d^7 - 768*a^2*b^12*c^3*d^5 + 384*a^2*b^12*c^5*d^3 + 680*a^3*b^11*c^2*d^6 - 1680*a^3*b^11*c^4*d^4 - 96*a^3*b^11
*c^6*d^2 + 3200*a^4*b^10*c^3*d^5 - 96*a^4*b^10*c^5*d^3 - 2864*a^5*b^9*c^2*d^6 + 1376*a^5*b^9*c^4*d^4 + 48*a^5*
b^9*c^6*d^2 - 2976*a^6*b^8*c^3*d^5 - 64*a^6*b^8*c^5*d^3 + 2824*a^7*b^7*c^2*d^6 - 264*a^7*b^7*c^4*d^4 + 768*a^8
*b^6*c^3*d^5 - 800*a^9*b^5*c^2*d^6))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/
2 - a*b*c*d^3*8i))/b^4 + (16*tan(e/2 + (f*x)/2)*(216*a^12*d^12 + 8*a^4*b^8*d^12 + 82*a^6*b^6*d^12 + 126*a^8*b^
4*d^12 - 432*a^10*b^2*d^12 - 8*a*b^11*c^3*d^9 - 192*a*b^11*c^5*d^7 - 1152*a*b^11*c^7*d^5 - 24*a^3*b^9*c*d^11 -
504*a^5*b^7*c*d^11 - 1488*a^7*b^5*c*d^11 + 3744*a^9*b^3*c*d^11 + 24*a^2*b^10*c^2*d^10 + 834*a^2*b^10*c^4*d^8
+ 6576*a^2*b^10*c^6*d^6 + 288*a^2*b^10*c^8*d^4 - 1432*a^3*b^9*c^3*d^9 - 15744*a^3*b^9*c^5*d^7 + 384*a^3*b^9*c^
7*d^5 + 1212*a^4*b^8*c^2*d^10 + 20406*a^4*b^8*c^4*d^8 - 7504*a^4*b^8*c^6*d^6 - 288*a^4*b^8*c^8*d^4 - 15360*a^5
*b^7*c^3*d^9 + 22464*a^5*b^7*c^5*d^7 + 768*a^5*b^7*c^7*d^5 + 6636*a^6*b^6*c^2*d^10 - 32976*a^6*b^6*c^4*d^8 + 9
28*a^6*b^6*c^6*d^6 + 27808*a^7*b^5*c^3*d^9 - 6528*a^7*b^5*c^5*d^7 - 13776*a^8*b^4*c^2*d^10 + 11736*a^8*b^4*c^4
*d^8 - 11008*a^9*b^3*c^3*d^9 + 5904*a^10*b^2*c^2*d^10 - 1728*a^11*b*c*d^11))/(b^13 - 2*a^2*b^11 + a^4*b^9)))*(
a^2*d^4*3i + (b^2*d^2*(12*c^2 + d^2)*1i)/2 - a*b*c*d^3*8i)*2i)/(b^4*f) + (atan((((a*d - b*c)^3*(-(a + b)^3*(a
- b)^3)^(1/2)*((8*(a^2*b^11*d^8 + 10*a^4*b^9*d^8 + 13*a^6*b^7*d^8 - 60*a^8*b^5*d^8 + 36*a^10*b^3*d^8 - 32*a^3*
b^10*c*d^7 - 128*a^5*b^8*c*d^7 + 352*a^7*b^6*c*d^7 - 192*a^9*b^4*c*d^7 + 24*a^2*b^11*c^2*d^6 + 144*a^2*b^11*c^
4*d^4 - 384*a^3*b^10*c^3*d^5 + 352*a^4*b^9*c^2*d^6 - 288*a^4*b^9*c^4*d^4 + 768*a^5*b^8*c^3*d^5 - 776*a^6*b^7*c
^2*d^6 + 144*a^6*b^7*c^4*d^4 - 384*a^7*b^6*c^3*d^5 + 400*a^8*b^5*c^2*d^6))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*
tan(e/2 + (f*x)/2)*(2*a*b^13*d^8 - 4*a^3*b^11*c^8 + 19*a^3*b^11*d^8 + 16*a^5*b^9*d^8 - 197*a^7*b^7*d^8 + 228*a
^9*b^5*d^8 - 72*a^11*b^3*d^8 + 48*a*b^13*c^2*d^6 + 288*a*b^13*c^4*d^4 - 64*a*b^13*c^6*d^2 - 64*a^2*b^12*c*d^7
+ 32*a^2*b^12*c^7*d - 224*a^4*b^10*c*d^7 + 1216*a^6*b^8*c*d^7 - 1280*a^8*b^6*c*d^7 + 384*a^10*b^4*c*d^7 - 768*
a^2*b^12*c^3*d^5 + 384*a^2*b^12*c^5*d^3 + 680*a^3*b^11*c^2*d^6 - 1680*a^3*b^11*c^4*d^4 - 96*a^3*b^11*c^6*d^2 +
3200*a^4*b^10*c^3*d^5 - 96*a^4*b^10*c^5*d^3 - 2864*a^5*b^9*c^2*d^6 + 1376*a^5*b^9*c^4*d^4 + 48*a^5*b^9*c^6*d^
2 - 2976*a^6*b^8*c^3*d^5 - 64*a^6*b^8*c^5*d^3 + 2824*a^7*b^7*c^2*d^6 - 264*a^7*b^7*c^4*d^4 + 768*a^8*b^6*c^3*d
^5 - 800*a^9*b^5*c^2*d^6))/(b^13 - 2*a^2*b^11 + a^4*b^9) + ((a*d - b*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(2*
a*b^14*d^4 + 4*a^3*b^12*c^4 - 4*a^5*b^10*c^4 + 6*a^3*b^12*d^4 - 14*a^5*b^10*d^4 + 6*a^7*b^8*d^4 + 24*a*b^14*c^
2*d^2 - 32*a^2*b^13*c*d^3 - 16*a^2*b^13*c^3*d + 48*a^4*b^11*c*d^3 + 16*a^4*b^11*c^3*d - 16*a^6*b^9*c*d^3 - 24*
a^3*b^12*c^2*d^2))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(8*a^2*b^14*c^4 - 8*a^4*b^12*c^4 + 32
*a^4*b^12*d^4 - 56*a^6*b^10*d^4 + 24*a^8*b^8*d^4 - 96*a^3*b^13*c*d^3 + 32*a^3*b^13*c^3*d + 160*a^5*b^11*c*d^3
- 64*a^7*b^9*c*d^3 + 96*a^2*b^14*c^2*d^2 - 144*a^4*b^12*c^2*d^2 + 48*a^6*b^10*c^2*d^2 - 32*a*b^15*c^3*d))/(b^1
3 - 2*a^2*b^11 + a^4*b^9) + (((8*(4*a^2*b^15 - 8*a^4*b^13 + 4*a^6*b^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*ta
n(e/2 + (f*x)/2)*(12*a*b^17 - 32*a^3*b^15 + 28*a^5*b^13 - 8*a^7*b^11))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a*d - b
*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^2*d - 4*b^2*d + a*b*c))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(3*a
^2*d - 4*b^2*d + a*b*c))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(3*a^2*d - 4*b^2*d + a*b*c)*1i)/(b^10 - 3*a
^2*b^8 + 3*a^4*b^6 - a^6*b^4) + ((a*d - b*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(a^2*b^11*d^8 + 10*a^4*b^9*d^8
+ 13*a^6*b^7*d^8 - 60*a^8*b^5*d^8 + 36*a^10*b^3*d^8 - 32*a^3*b^10*c*d^7 - 128*a^5*b^8*c*d^7 + 352*a^7*b^6*c*d
^7 - 192*a^9*b^4*c*d^7 + 24*a^2*b^11*c^2*d^6 + 144*a^2*b^11*c^4*d^4 - 384*a^3*b^10*c^3*d^5 + 352*a^4*b^9*c^2*d
^6 - 288*a^4*b^9*c^4*d^4 + 768*a^5*b^8*c^3*d^5 - 776*a^6*b^7*c^2*d^6 + 144*a^6*b^7*c^4*d^4 - 384*a^7*b^6*c^3*d
^5 + 400*a^8*b^5*c^2*d^6))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(2*a*b^13*d^8 - 4*a^3*b^11*c^
8 + 19*a^3*b^11*d^8 + 16*a^5*b^9*d^8 - 197*a^7*b^7*d^8 + 228*a^9*b^5*d^8 - 72*a^11*b^3*d^8 + 48*a*b^13*c^2*d^6
+ 288*a*b^13*c^4*d^4 - 64*a*b^13*c^6*d^2 - 64*a^2*b^12*c*d^7 + 32*a^2*b^12*c^7*d - 224*a^4*b^10*c*d^7 + 1216*
a^6*b^8*c*d^7 - 1280*a^8*b^6*c*d^7 + 384*a^10*b^4*c*d^7 - 768*a^2*b^12*c^3*d^5 + 384*a^2*b^12*c^5*d^3 + 680*a^
3*b^11*c^2*d^6 - 1680*a^3*b^11*c^4*d^4 - 96*a^3*b^11*c^6*d^2 + 3200*a^4*b^10*c^3*d^5 - 96*a^4*b^10*c^5*d^3 - 2
864*a^5*b^9*c^2*d^6 + 1376*a^5*b^9*c^4*d^4 + 48*a^5*b^9*c^6*d^2 - 2976*a^6*b^8*c^3*d^5 - 64*a^6*b^8*c^5*d^3 +
2824*a^7*b^7*c^2*d^6 - 264*a^7*b^7*c^4*d^4 + 768*a^8*b^6*c^3*d^5 - 800*a^9*b^5*c^2*d^6))/(b^13 - 2*a^2*b^11 +
a^4*b^9) - ((a*d - b*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(2*a*b^14*d^4 + 4*a^3*b^12*c^4 - 4*a^5*b^10*c^4 + 6
*a^3*b^12*d^4 - 14*a^5*b^10*d^4 + 6*a^7*b^8*d^4 + 24*a*b^14*c^2*d^2 - 32*a^2*b^13*c*d^3 - 16*a^2*b^13*c^3*d +
48*a^4*b^11*c*d^3 + 16*a^4*b^11*c^3*d - 16*a^6*b^9*c*d^3 - 24*a^3*b^12*c^2*d^2))/(b^12 - 2*a^2*b^10 + a^4*b^8)
+ (8*tan(e/2 + (f*x)/2)*(8*a^2*b^14*c^4 - 8*a^4*b^12*c^4 + 32*a^4*b^12*d^4 - 56*a^6*b^10*d^4 + 24*a^8*b^8*d^4
- 96*a^3*b^13*c*d^3 + 32*a^3*b^13*c^3*d + 160*a^5*b^11*c*d^3 - 64*a^7*b^9*c*d^3 + 96*a^2*b^14*c^2*d^2 - 144*a
^4*b^12*c^2*d^2 + 48*a^6*b^10*c^2*d^2 - 32*a*b^15*c^3*d))/(b^13 - 2*a^2*b^11 + a^4*b^9) - (((8*(4*a^2*b^15 - 8
*a^4*b^13 + 4*a^6*b^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^17 - 32*a^3*b^15 + 28*a
^5*b^13 - 8*a^7*b^11))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a*d - b*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^2*d - 4*
b^2*d + a*b*c))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(3*a^2*d - 4*b^2*d + a*b*c))/(b^10 - 3*a^2*b^8 + 3*a
^4*b^6 - a^6*b^4))*(3*a^2*d - 4*b^2*d + a*b*c)*1i)/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))/((16*(54*a^11*d^1
2 + 4*a^5*b^6*d^12 + 9*a^7*b^4*d^12 - 81*a^9*b^2*d^12 - 32*a*b^10*c^6*d^6 - 384*a*b^10*c^8*d^4 - 12*a^4*b^7*c*
d^11 - 60*a^6*b^5*c*d^11 + 648*a^8*b^3*c*d^11 - 4*a^2*b^9*c^3*d^9 + 96*a^2*b^9*c^5*d^7 + 2256*a^2*b^9*c^7*d^5
+ 192*a^2*b^9*c^9*d^3 + 12*a^3*b^8*c^2*d^10 - 63*a^3*b^8*c^4*d^8 - 5784*a^3*b^8*c^6*d^6 - 690*a^3*b^8*c^8*d^4
- 24*a^3*b^8*c^10*d^2 - 76*a^4*b^7*c^3*d^9 + 8592*a^4*b^7*c^5*d^7 + 480*a^4*b^7*c^7*d^5 + 32*a^4*b^7*c^9*d^3 +
126*a^5*b^6*c^2*d^10 - 8277*a^5*b^6*c^4*d^8 + 1552*a^5*b^6*c^6*d^6 + 132*a^5*b^6*c^8*d^4 + 5424*a^6*b^5*c^3*d
^9 - 4128*a^6*b^5*c^5*d^7 - 384*a^6*b^5*c^7*d^5 - 2394*a^7*b^4*c^2*d^10 + 4860*a^7*b^4*c^4*d^8 + 400*a^7*b^4*c
^6*d^6 - 3472*a^8*b^3*c^3*d^9 - 192*a^8*b^3*c^5*d^7 + 1584*a^9*b^2*c^2*d^10 + 36*a^9*b^2*c^4*d^8 - 432*a^10*b*
c*d^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (16*tan(e/2 + (f*x)/2)*(216*a^12*d^12 + 8*a^4*b^8*d^12 + 82*a^6*b^6*d
^12 + 126*a^8*b^4*d^12 - 432*a^10*b^2*d^12 - 8*a*b^11*c^3*d^9 - 192*a*b^11*c^5*d^7 - 1152*a*b^11*c^7*d^5 - 24*
a^3*b^9*c*d^11 - 504*a^5*b^7*c*d^11 - 1488*a^7*b^5*c*d^11 + 3744*a^9*b^3*c*d^11 + 24*a^2*b^10*c^2*d^10 + 834*a
^2*b^10*c^4*d^8 + 6576*a^2*b^10*c^6*d^6 + 288*a^2*b^10*c^8*d^4 - 1432*a^3*b^9*c^3*d^9 - 15744*a^3*b^9*c^5*d^7
+ 384*a^3*b^9*c^7*d^5 + 1212*a^4*b^8*c^2*d^10 + 20406*a^4*b^8*c^4*d^8 - 7504*a^4*b^8*c^6*d^6 - 288*a^4*b^8*c^8
*d^4 - 15360*a^5*b^7*c^3*d^9 + 22464*a^5*b^7*c^5*d^7 + 768*a^5*b^7*c^7*d^5 + 6636*a^6*b^6*c^2*d^10 - 32976*a^6
*b^6*c^4*d^8 + 928*a^6*b^6*c^6*d^6 + 27808*a^7*b^5*c^3*d^9 - 6528*a^7*b^5*c^5*d^7 - 13776*a^8*b^4*c^2*d^10 + 1
1736*a^8*b^4*c^4*d^8 - 11008*a^9*b^3*c^3*d^9 + 5904*a^10*b^2*c^2*d^10 - 1728*a^11*b*c*d^11))/(b^13 - 2*a^2*b^1
1 + a^4*b^9) + ((a*d - b*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(a^2*b^11*d^8 + 10*a^4*b^9*d^8 + 13*a^6*b^7*d^8
- 60*a^8*b^5*d^8 + 36*a^10*b^3*d^8 - 32*a^3*b^10*c*d^7 - 128*a^5*b^8*c*d^7 + 352*a^7*b^6*c*d^7 - 192*a^9*b^4*
c*d^7 + 24*a^2*b^11*c^2*d^6 + 144*a^2*b^11*c^4*d^4 - 384*a^3*b^10*c^3*d^5 + 352*a^4*b^9*c^2*d^6 - 288*a^4*b^9*
c^4*d^4 + 768*a^5*b^8*c^3*d^5 - 776*a^6*b^7*c^2*d^6 + 144*a^6*b^7*c^4*d^4 - 384*a^7*b^6*c^3*d^5 + 400*a^8*b^5*
c^2*d^6))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(2*a*b^13*d^8 - 4*a^3*b^11*c^8 + 19*a^3*b^11*d
^8 + 16*a^5*b^9*d^8 - 197*a^7*b^7*d^8 + 228*a^9*b^5*d^8 - 72*a^11*b^3*d^8 + 48*a*b^13*c^2*d^6 + 288*a*b^13*c^4
*d^4 - 64*a*b^13*c^6*d^2 - 64*a^2*b^12*c*d^7 + 32*a^2*b^12*c^7*d - 224*a^4*b^10*c*d^7 + 1216*a^6*b^8*c*d^7 - 1
280*a^8*b^6*c*d^7 + 384*a^10*b^4*c*d^7 - 768*a^2*b^12*c^3*d^5 + 384*a^2*b^12*c^5*d^3 + 680*a^3*b^11*c^2*d^6 -
1680*a^3*b^11*c^4*d^4 - 96*a^3*b^11*c^6*d^2 + 3200*a^4*b^10*c^3*d^5 - 96*a^4*b^10*c^5*d^3 - 2864*a^5*b^9*c^2*d
^6 + 1376*a^5*b^9*c^4*d^4 + 48*a^5*b^9*c^6*d^2 - 2976*a^6*b^8*c^3*d^5 - 64*a^6*b^8*c^5*d^3 + 2824*a^7*b^7*c^2*
d^6 - 264*a^7*b^7*c^4*d^4 + 768*a^8*b^6*c^3*d^5 - 800*a^9*b^5*c^2*d^6))/(b^13 - 2*a^2*b^11 + a^4*b^9) + ((a*d
- b*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(2*a*b^14*d^4 + 4*a^3*b^12*c^4 - 4*a^5*b^10*c^4 + 6*a^3*b^12*d^4 - 1
4*a^5*b^10*d^4 + 6*a^7*b^8*d^4 + 24*a*b^14*c^2*d^2 - 32*a^2*b^13*c*d^3 - 16*a^2*b^13*c^3*d + 48*a^4*b^11*c*d^3
+ 16*a^4*b^11*c^3*d - 16*a^6*b^9*c*d^3 - 24*a^3*b^12*c^2*d^2))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (
f*x)/2)*(8*a^2*b^14*c^4 - 8*a^4*b^12*c^4 + 32*a^4*b^12*d^4 - 56*a^6*b^10*d^4 + 24*a^8*b^8*d^4 - 96*a^3*b^13*c*
d^3 + 32*a^3*b^13*c^3*d + 160*a^5*b^11*c*d^3 - 64*a^7*b^9*c*d^3 + 96*a^2*b^14*c^2*d^2 - 144*a^4*b^12*c^2*d^2 +
48*a^6*b^10*c^2*d^2 - 32*a*b^15*c^3*d))/(b^13 - 2*a^2*b^11 + a^4*b^9) + (((8*(4*a^2*b^15 - 8*a^4*b^13 + 4*a^6
*b^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(12*a*b^17 - 32*a^3*b^15 + 28*a^5*b^13 - 8*a^7*b
^11))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a*d - b*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^2*d - 4*b^2*d + a*b*c))/(
b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(3*a^2*d - 4*b^2*d + a*b*c))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4)
)*(3*a^2*d - 4*b^2*d + a*b*c))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4) - ((a*d - b*c)^3*(-(a + b)^3*(a - b)^3
)^(1/2)*((8*(a^2*b^11*d^8 + 10*a^4*b^9*d^8 + 13*a^6*b^7*d^8 - 60*a^8*b^5*d^8 + 36*a^10*b^3*d^8 - 32*a^3*b^10*c
*d^7 - 128*a^5*b^8*c*d^7 + 352*a^7*b^6*c*d^7 - 192*a^9*b^4*c*d^7 + 24*a^2*b^11*c^2*d^6 + 144*a^2*b^11*c^4*d^4
- 384*a^3*b^10*c^3*d^5 + 352*a^4*b^9*c^2*d^6 - 288*a^4*b^9*c^4*d^4 + 768*a^5*b^8*c^3*d^5 - 776*a^6*b^7*c^2*d^6
+ 144*a^6*b^7*c^4*d^4 - 384*a^7*b^6*c^3*d^5 + 400*a^8*b^5*c^2*d^6))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/
2 + (f*x)/2)*(2*a*b^13*d^8 - 4*a^3*b^11*c^8 + 19*a^3*b^11*d^8 + 16*a^5*b^9*d^8 - 197*a^7*b^7*d^8 + 228*a^9*b^5
*d^8 - 72*a^11*b^3*d^8 + 48*a*b^13*c^2*d^6 + 288*a*b^13*c^4*d^4 - 64*a*b^13*c^6*d^2 - 64*a^2*b^12*c*d^7 + 32*a
^2*b^12*c^7*d - 224*a^4*b^10*c*d^7 + 1216*a^6*b^8*c*d^7 - 1280*a^8*b^6*c*d^7 + 384*a^10*b^4*c*d^7 - 768*a^2*b^
12*c^3*d^5 + 384*a^2*b^12*c^5*d^3 + 680*a^3*b^11*c^2*d^6 - 1680*a^3*b^11*c^4*d^4 - 96*a^3*b^11*c^6*d^2 + 3200*
a^4*b^10*c^3*d^5 - 96*a^4*b^10*c^5*d^3 - 2864*a^5*b^9*c^2*d^6 + 1376*a^5*b^9*c^4*d^4 + 48*a^5*b^9*c^6*d^2 - 29
76*a^6*b^8*c^3*d^5 - 64*a^6*b^8*c^5*d^3 + 2824*a^7*b^7*c^2*d^6 - 264*a^7*b^7*c^4*d^4 + 768*a^8*b^6*c^3*d^5 - 8
00*a^9*b^5*c^2*d^6))/(b^13 - 2*a^2*b^11 + a^4*b^9) - ((a*d - b*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(2*a*b^14
*d^4 + 4*a^3*b^12*c^4 - 4*a^5*b^10*c^4 + 6*a^3*b^12*d^4 - 14*a^5*b^10*d^4 + 6*a^7*b^8*d^4 + 24*a*b^14*c^2*d^2
- 32*a^2*b^13*c*d^3 - 16*a^2*b^13*c^3*d + 48*a^4*b^11*c*d^3 + 16*a^4*b^11*c^3*d - 16*a^6*b^9*c*d^3 - 24*a^3*b^
12*c^2*d^2))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2 + (f*x)/2)*(8*a^2*b^14*c^4 - 8*a^4*b^12*c^4 + 32*a^4*b
^12*d^4 - 56*a^6*b^10*d^4 + 24*a^8*b^8*d^4 - 96*a^3*b^13*c*d^3 + 32*a^3*b^13*c^3*d + 160*a^5*b^11*c*d^3 - 64*a
^7*b^9*c*d^3 + 96*a^2*b^14*c^2*d^2 - 144*a^4*b^12*c^2*d^2 + 48*a^6*b^10*c^2*d^2 - 32*a*b^15*c^3*d))/(b^13 - 2*
a^2*b^11 + a^4*b^9) - (((8*(4*a^2*b^15 - 8*a^4*b^13 + 4*a^6*b^11))/(b^12 - 2*a^2*b^10 + a^4*b^8) + (8*tan(e/2
+ (f*x)/2)*(12*a*b^17 - 32*a^3*b^15 + 28*a^5*b^13 - 8*a^7*b^11))/(b^13 - 2*a^2*b^11 + a^4*b^9))*(a*d - b*c)^3*
(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^2*d - 4*b^2*d + a*b*c))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(3*a^2*d -
4*b^2*d + a*b*c))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*(3*a^2*d - 4*b^2*d + a*b*c))/(b^10 - 3*a^2*b^8 +
3*a^4*b^6 - a^6*b^4)))*(a*d - b*c)^3*(-(a + b)^3*(a - b)^3)^(1/2)*(3*a^2*d - 4*b^2*d + a*b*c)*2i)/(f*(b^10 - 3
*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))