\(\int \frac {\sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx\) [39]

Optimal result

Integrand size = 33, antiderivative size = 181 \[ \int \frac {\sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx=-\frac {2 a \left (a^2 c-2 b^2 c+a b d\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} (b c-a d)^2 f}+\frac {2 c^2 \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^2 \sqrt {c^2-d^2} f}+\frac {a^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3135, 3080, 2739, 632, 210} \[ \int \frac {\sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx=-\frac {2 a \left (a^2 c+a b d-2 b^2 c\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{3/2} (b c-a d)^2}+\frac {a^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}+\frac {2 c^2 \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \sqrt {c^2-d^2} (b c-a d)^2} \]

[In]

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

[Out]

(-2*a*(a^2*c - 2*b^2*c + a*b*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c - a*
d)^2*f) + (2*c^2*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^2*Sqrt[c^2 - d^2]*f) + (a^2*Co
s[e + f*x])/((a^2 - b^2)*(b*c - a*d)*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 3080

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

Rule 3135

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c
+ d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C
)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n +
3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && L
tQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rubi steps \begin{align*} \text {integral}& = \frac {a^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}-\frac {\int \frac {-a b c-\left (a^2 c-b^2 c+a b d\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx}{\left (a^2-b^2\right ) (b c-a d)} \\ & = \frac {a^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}+\frac {c^2 \int \frac {1}{c+d \sin (e+f x)} \, dx}{(b c-a d)^2}-\frac {\left (a \left (a^2 c-2 b^2 c+a b d\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) (b c-a d)^2} \\ & = \frac {a^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}+\frac {\left (2 c^2\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{(b c-a d)^2 f}-\frac {\left (2 a \left (a^2 c-2 b^2 c+a b 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 )}{\left (a^2-b^2\right ) (b c-a d)^2 f} \\ & = \frac {a^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}-\frac {\left (4 c^2\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{(b c-a d)^2 f}+\frac {\left (4 a \left (a^2 c-2 b^2 c+a b 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 )}{\left (a^2-b^2\right ) (b c-a d)^2 f} \\ & = -\frac {2 a \left (a^2 c-2 b^2 c+a b d\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} (b c-a d)^2 f}+\frac {2 c^2 \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^2 \sqrt {c^2-d^2} f}+\frac {a^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx=\frac {-\frac {2 a \left (a^2 c-2 b^2 c+a b d\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} (b c-a d)^2}+\frac {2 c^2 \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^2 \sqrt {c^2-d^2}}-\frac {a^2 \cos (e+f x)}{(a-b) (a+b) (-b c+a d) (a+b \sin (e+f x))}}{f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.03 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.35

method	result	size
derivativedivides	\(\frac {-\frac {2 a \left (\frac {\frac {b \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a^{2}-b^{2}}+\frac {a \left (a d -b c \right )}{a^{2}-b^{2}}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (a^{2} c +a b d -2 b^{2} c \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {8 c^{2} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (4 a^{2} d^{2}-8 a b c d +4 b^{2} c^{2}\right ) \sqrt {c^{2}-d^{2}}}}{f}\)	\(244\)
default	\(\frac {-\frac {2 a \left (\frac {\frac {b \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a^{2}-b^{2}}+\frac {a \left (a d -b c \right )}{a^{2}-b^{2}}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a +2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (a^{2} c +a b d -2 b^{2} c \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {8 c^{2} \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (4 a^{2} d^{2}-8 a b c d +4 b^{2} c^{2}\right ) \sqrt {c^{2}-d^{2}}}}{f}\)	\(244\)
risch	\(\frac {2 i a^{2} \left (i b +a \,{\mathrm e}^{i \left (f x +e \right )}\right )}{b \left (a^{2}-b^{2}\right ) \left (a d -b c \right ) f \left (-i {\mathrm e}^{2 i \left (f x +e \right )} b +2 a \,{\mathrm e}^{i \left (f x +e \right )}+i b \right )}+\frac {c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right )}{\sqrt {-c^{2}+d^{2}}\, \left (a d -b c \right )^{2} f}-\frac {c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right )}{\sqrt {-c^{2}+d^{2}}\, \left (a d -b c \right )^{2} f}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) c}{\sqrt {-a^{2}+b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b d}{\sqrt {-a^{2}+b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}-\frac {2 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2} c}{\sqrt {-a^{2}+b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) c}{\sqrt {-a^{2}+b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b d}{\sqrt {-a^{2}+b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}+\frac {2 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2} c}{\sqrt {-a^{2}+b^{2}}\, \left (a d -b c \right )^{2} \left (a +b \right ) \left (a -b \right ) f}\)	\(794\)

[In]

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

[Out]

1/f*(-2*a/(a^2*d^2-2*a*b*c*d+b^2*c^2)*((b*(a*d-b*c)/(a^2-b^2)*tan(1/2*f*x+1/2*e)+a*(a*d-b*c)/(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)+(a^2*c+a*b*d-2*b^2*c)/(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)))+8*c^2/(4*a^2*d^2-8*a*b*c*d+4*b^2*c^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*
x+1/2*e)+2*d)/(c^2-d^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (171) = 342\).

Time = 65.07 (sec) , antiderivative size = 2837, normalized size of antiderivative = 15.67 \[ \int \frac {\sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/2*((a^3*b*c^2*d - a^3*b*d^3 + (a^4 - 2*a^2*b^2)*c^3 - (a^4 - 2*a^2*b^2)*c*d^2 + (a^2*b^2*c^2*d - a^2*b^2*d^
3 + (a^3*b - 2*a*b^3)*c^3 - (a^3*b - 2*a*b^3)*c*d^2)*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)) - ((a^4*b - 2*a^2*b^3 + b^5)*c^2*sin(f*x + e) + (a^5 -
2*a^3*b^2 + a*b^4)*c^2)*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 +
2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) -
c^2 - d^2)) + 2*((a^4*b - a^2*b^3)*c^3 - (a^5 - a^3*b^2)*c^2*d - (a^4*b - a^2*b^3)*c*d^2 + (a^5 - a^3*b^2)*d^3
)*cos(f*x + e))/(((a^4*b^3 - 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a^4*b^3
+ 3*a^2*b^5 - b^7)*c^2*d^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f*sin
(f*x + e) + ((a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2 + 3*
a^3*b^4 - a*b^6)*c^2*d^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f), -1/2*(
2*((a^4*b - 2*a^2*b^3 + b^5)*c^2*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*c^2)*sqrt(c^2 - d^2)*arctan(-(c*sin(
f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - (a^3*b*c^2*d - a^3*b*d^3 + (a^4 - 2*a^2*b^2)*c^3 - (a^4 - 2*a^
2*b^2)*c*d^2 + (a^2*b^2*c^2*d - a^2*b^2*d^3 + (a^3*b - 2*a*b^3)*c^3 - (a^3*b - 2*a*b^3)*c*d^2)*sin(f*x + e))*s
qrt(-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^4*b -
a^2*b^3)*c^3 - (a^5 - a^3*b^2)*c^2*d - (a^4*b - a^2*b^3)*c*d^2 + (a^5 - a^3*b^2)*d^3)*cos(f*x + e))/(((a^4*b^
3 - 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c^2*d
^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f*sin(f*x + e) + ((a^5*b^2 - 2
*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c^2*d^2
+ 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f), 1/2*(2*(a^3*b*c^2*d - a^3*b*d^3
+ (a^4 - 2*a^2*b^2)*c^3 - (a^4 - 2*a^2*b^2)*c*d^2 + (a^2*b^2*c^2*d - a^2*b^2*d^3 + (a^3*b - 2*a*b^3)*c^3 - (a
^3*b - 2*a*b^3)*c*d^2)*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
))) - ((a^4*b - 2*a^2*b^3 + b^5)*c^2*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*c^2)*sqrt(-c^2 + d^2)*log(((2*c^
2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sq
rt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*((a^4*b - a^2*b^3)*c^3 - (a^5 - a^3
*b^2)*c^2*d - (a^4*b - a^2*b^3)*c*d^2 + (a^5 - a^3*b^2)*d^3)*cos(f*x + e))/(((a^4*b^3 - 2*a^2*b^5 + b^7)*c^4 -
2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c^2*d^2 + 2*(a^5*b^2 - 2*a^3*b^
4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f*sin(f*x + e) + ((a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^4 - 2*
(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c^2*d^2 + 2*(a^6*b - 2*a^4*b^3 + a
^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f), ((a^3*b*c^2*d - a^3*b*d^3 + (a^4 - 2*a^2*b^2)*c^3 - (a^4
- 2*a^2*b^2)*c*d^2 + (a^2*b^2*c^2*d - a^2*b^2*d^3 + (a^3*b - 2*a*b^3)*c^3 - (a^3*b - 2*a*b^3)*c*d^2)*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))) - ((a^4*b - 2*a^2*b^3 + b^5)
*c^2*sin(f*x + e) + (a^5 - 2*a^3*b^2 + a*b^4)*c^2)*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^
2)*cos(f*x + e))) + ((a^4*b - a^2*b^3)*c^3 - (a^5 - a^3*b^2)*c^2*d - (a^4*b - a^2*b^3)*c*d^2 + (a^5 - a^3*b^2)
*d^3)*cos(f*x + e))/(((a^4*b^3 - 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b - 3*a^4
*b^3 + 3*a^2*b^5 - b^7)*c^2*d^2 + 2*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*c*d^3 - (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^4)*f
*sin(f*x + e) + ((a^5*b^2 - 2*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c^3*d + (a^7 - 3*a^5*b^2
+ 3*a^3*b^4 - a*b^6)*c^2*d^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*c*d^3 - (a^7 - 2*a^5*b^2 + a^3*b^4)*d^4)*f)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(sin(f*x+e)^2/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e)),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*d^2-4*c^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.61 \[ \int \frac {\sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx=\frac {2 \, {\left (\frac {{\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )} c^{2}}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c^{2} - d^{2}}} - \frac {{\left (a^{3} c - 2 \, a b^{2} c + a^{2} b d\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{2} c^{2} - b^{4} c^{2} - 2 \, a^{3} b c d + 2 \, a b^{3} c d + a^{4} d^{2} - a^{2} b^{2} d^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{2}}{{\left (a^{2} b c - b^{3} c - a^{3} d + a b^{2} d\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a\right )}}\right )}}{f} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 27.41 (sec) , antiderivative size = 23933, normalized size of antiderivative = 132.23 \[ \int \frac {\sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- ((2*a^2)/((a^2 - b^2)*(a*d - b*c)) + (2*a*b*tan(e/2 + (f*x)/2))/((a^2 - b^2)*(a*d - b*c)))/(f*(a + 2*b*tan(e
/2 + (f*x)/2) + a*tan(e/2 + (f*x)/2)^2)) - (c^2*atan(((c^2*(d^2 - c^2)^(1/2)*((32*(2*a^4*b^4*c^6 - a^2*b^6*c^6
- a^6*b^2*c^6 + a^8*c^4*d^2 - 3*a^3*b^5*c^5*d + 2*a^5*b^3*c^5*d + 2*a^7*b*c^3*d^3 + 8*a^4*b^4*c^4*d^2 - 5*a^5
*b^3*c^3*d^3 + a^6*b^2*c^2*d^4 - 6*a^6*b^2*c^4*d^2))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^
4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d
- 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^7*b*c^6 - 2*a*b^7*c^6 - 2*a^8*c^5*d + 9*a^3*b^5*c^6 - 8*a^5*b^3
*c^6 + 2*a^8*c^3*d^3 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c^5*d + 2*a^6*b^2*c*d^5 + 10*a^6*b^2*c^5*d + 4*a^7*b*c^2*d
^4 - 4*a^7*b*c^4*d^2 - 8*a^3*b^5*c^4*d^2 + 16*a^4*b^4*c^3*d^3 - 10*a^5*b^3*c^2*d^4 + 13*a^5*b^3*c^4*d^2 - 13*a
^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*
d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (c^2*(d^2 - c^2)^
(1/2)*((32*(a*b^9*c^7 - a^3*b^7*c^7 + a^10*c^2*d^5 - 3*a^2*b^8*c^6*d + 2*a^4*b^6*c^6*d + a^6*b^4*c^6*d - a^7*b
^3*c*d^6 - 4*a^9*b*c^3*d^4 + a^3*b^7*c^5*d^2 + 6*a^4*b^6*c^4*d^3 - 9*a^5*b^5*c^3*d^4 + 3*a^5*b^5*c^5*d^2 + 5*a
^6*b^4*c^2*d^5 - 12*a^6*b^4*c^4*d^3 + 13*a^7*b^3*c^3*d^4 - 4*a^7*b^3*c^5*d^2 - 6*a^8*b^2*c^2*d^5 + 6*a^8*b^2*c
^4*d^3 + a^9*b*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*
b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2
+ (f*x)/2)*(4*a^2*b^8*c^7 - 6*a^4*b^6*c^7 + 2*a^6*b^4*c^7 + 2*a^10*c^3*d^4 - 20*a^3*b^7*c^6*d + 26*a^5*b^5*c^
6*d - 2*a^6*b^4*c*d^6 - 8*a^7*b^3*c^6*d + 2*a^8*b^2*c*d^6 + 2*a^9*b*c^2*d^5 - 8*a^9*b*c^4*d^3 - 10*a^2*b^8*c^5
*d^2 + 20*a^3*b^7*c^4*d^3 - 20*a^4*b^6*c^3*d^4 + 42*a^4*b^6*c^5*d^2 + 10*a^5*b^5*c^2*d^5 - 48*a^5*b^5*c^4*d^3
+ 32*a^6*b^4*c^3*d^4 - 44*a^6*b^4*c^5*d^2 - 12*a^7*b^3*c^2*d^5 + 36*a^7*b^3*c^4*d^3 - 14*a^8*b^2*c^3*d^4 + 12*
a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d
^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) +
(c^2*(d^2 - c^2)^(1/2)*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a
^5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a
^3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^
4*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5
*a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))
/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*
c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*
c^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^
2*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 1
0*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b
^9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c
^6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6
*d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2
- 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3
- 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^
6*b*c*d^2)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4
- a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d))*1i)/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 -
2*a*b*c*d^3 + 2*a*b*c^3*d) + (c^2*(d^2 - c^2)^(1/2)*((32*(2*a^4*b^4*c^6 - a^2*b^6*c^6 - a^6*b^2*c^6 + a^8*c^4
*d^2 - 3*a^3*b^5*c^5*d + 2*a^5*b^3*c^5*d + 2*a^7*b*c^3*d^3 + 8*a^4*b^4*c^4*d^2 - 5*a^5*b^3*c^3*d^3 + a^6*b^2*c
^2*d^4 - 6*a^6*b^2*c^4*d^2))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 -
3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*t
an(e/2 + (f*x)/2)*(2*a^7*b*c^6 - 2*a*b^7*c^6 - 2*a^8*c^5*d + 9*a^3*b^5*c^6 - 8*a^5*b^3*c^6 + 2*a^8*c^3*d^3 + 2
*a^2*b^6*c^5*d - 13*a^4*b^4*c^5*d + 2*a^6*b^2*c*d^5 + 10*a^6*b^2*c^5*d + 4*a^7*b*c^2*d^4 - 4*a^7*b*c^4*d^2 - 8
*a^3*b^5*c^4*d^2 + 16*a^4*b^4*c^3*d^3 - 10*a^5*b^3*c^2*d^4 + 13*a^5*b^3*c^4*d^2 - 13*a^6*b^2*c^3*d^3))/(a^7*d^
3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d +
6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (c^2*(d^2 - c^2)^(1/2)*((32*(a*b^9*c^7 -
a^3*b^7*c^7 + a^10*c^2*d^5 - 3*a^2*b^8*c^6*d + 2*a^4*b^6*c^6*d + a^6*b^4*c^6*d - a^7*b^3*c*d^6 - 4*a^9*b*c^3*d
^4 + a^3*b^7*c^5*d^2 + 6*a^4*b^6*c^4*d^3 - 9*a^5*b^5*c^3*d^4 + 3*a^5*b^5*c^5*d^2 + 5*a^6*b^4*c^2*d^5 - 12*a^6*
b^4*c^4*d^3 + 13*a^7*b^3*c^3*d^4 - 4*a^7*b^3*c^5*d^2 - 6*a^8*b^2*c^2*d^5 + 6*a^8*b^2*c^4*d^3 + a^9*b*c*d^6))/(
a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^
2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(4*a^2*b^8*c
^7 - 6*a^4*b^6*c^7 + 2*a^6*b^4*c^7 + 2*a^10*c^3*d^4 - 20*a^3*b^7*c^6*d + 26*a^5*b^5*c^6*d - 2*a^6*b^4*c*d^6 -
8*a^7*b^3*c^6*d + 2*a^8*b^2*c*d^6 + 2*a^9*b*c^2*d^5 - 8*a^9*b*c^4*d^3 - 10*a^2*b^8*c^5*d^2 + 20*a^3*b^7*c^4*d^
3 - 20*a^4*b^6*c^3*d^4 + 42*a^4*b^6*c^5*d^2 + 10*a^5*b^5*c^2*d^5 - 48*a^5*b^5*c^4*d^3 + 32*a^6*b^4*c^3*d^4 - 4
4*a^6*b^4*c^5*d^2 - 12*a^7*b^3*c^2*d^5 + 36*a^7*b^3*c^4*d^3 - 14*a^8*b^2*c^3*d^4 + 12*a^8*b^2*c^5*d^2 + 2*a*b^
9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6
*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (c^2*(d^2 - c^2)^(1/2)*(
(32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^7*b^5*c
*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 + 3*a^4*b
^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6*c^6*d^
2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2 + 3*a^9
*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c^3 + 2*
a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2
+ 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^7 - 8*a
^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40*a^4*b^
8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7 + 15*a^
11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a^4*b^8*
c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*b^6*c^3
*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^4*c^3*d
^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^3*d^5 +
20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*
b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2)))/(a^2*d^4 -
b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*
d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d))*1i)/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3
*d))/((64*(a^5*b*c^5 - 2*a^3*b^3*c^5 + a^4*b^2*c^4*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*
b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*
d - 3*a^6*b*c*d^2) + (64*tan(e/2 + (f*x)/2)*(2*a^6*c^5 + 4*a^2*b^4*c^5 - 6*a^4*b^2*c^5 - 6*a^3*b^3*c^4*d + 2*a
^4*b^2*c^3*d^2 + 4*a^5*b*c^4*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^
3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (
c^2*(d^2 - c^2)^(1/2)*((32*(2*a^4*b^4*c^6 - a^2*b^6*c^6 - a^6*b^2*c^6 + a^8*c^4*d^2 - 3*a^3*b^5*c^5*d + 2*a^5*
b^3*c^5*d + 2*a^7*b*c^3*d^3 + 8*a^4*b^4*c^4*d^2 - 5*a^5*b^3*c^3*d^3 + a^6*b^2*c^2*d^4 - 6*a^6*b^2*c^4*d^2))/(a
^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2
*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^7*b*c^6
- 2*a*b^7*c^6 - 2*a^8*c^5*d + 9*a^3*b^5*c^6 - 8*a^5*b^3*c^6 + 2*a^8*c^3*d^3 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c^5
*d + 2*a^6*b^2*c*d^5 + 10*a^6*b^2*c^5*d + 4*a^7*b*c^2*d^4 - 4*a^7*b*c^4*d^2 - 8*a^3*b^5*c^4*d^2 + 16*a^4*b^4*c
^3*d^3 - 10*a^5*b^3*c^2*d^4 + 13*a^5*b^3*c^4*d^2 - 13*a^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a
^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2
*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (c^2*(d^2 - c^2)^(1/2)*((32*(a*b^9*c^7 - a^3*b^7*c^7 + a^10*c^2*d^5 - 3*
a^2*b^8*c^6*d + 2*a^4*b^6*c^6*d + a^6*b^4*c^6*d - a^7*b^3*c*d^6 - 4*a^9*b*c^3*d^4 + a^3*b^7*c^5*d^2 + 6*a^4*b^
6*c^4*d^3 - 9*a^5*b^5*c^3*d^4 + 3*a^5*b^5*c^5*d^2 + 5*a^6*b^4*c^2*d^5 - 12*a^6*b^4*c^4*d^3 + 13*a^7*b^3*c^3*d^
4 - 4*a^7*b^3*c^5*d^2 - 6*a^8*b^2*c^2*d^5 + 6*a^8*b^2*c^4*d^3 + a^9*b*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c
^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b
^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(4*a^2*b^8*c^7 - 6*a^4*b^6*c^7 + 2*a^6*b^4*
c^7 + 2*a^10*c^3*d^4 - 20*a^3*b^7*c^6*d + 26*a^5*b^5*c^6*d - 2*a^6*b^4*c*d^6 - 8*a^7*b^3*c^6*d + 2*a^8*b^2*c*d
^6 + 2*a^9*b*c^2*d^5 - 8*a^9*b*c^4*d^3 - 10*a^2*b^8*c^5*d^2 + 20*a^3*b^7*c^4*d^3 - 20*a^4*b^6*c^3*d^4 + 42*a^4
*b^6*c^5*d^2 + 10*a^5*b^5*c^2*d^5 - 48*a^5*b^5*c^4*d^3 + 32*a^6*b^4*c^3*d^4 - 44*a^6*b^4*c^5*d^2 - 12*a^7*b^3*
c^2*d^5 + 36*a^7*b^3*c^4*d^3 - 14*a^8*b^2*c^3*d^4 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 +
2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^
2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (c^2*(d^2 - c^2)^(1/2)*((32*(2*a^4*b^8*c^8 - a^2*b^10*c
^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*
b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5
- 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b
^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d
^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3
*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2
*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*
a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*
a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^
4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 -
20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55
*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^
9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3
- b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6
*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2
*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d
)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d) - (c^2*(d^2 - c^2)^(1/2)*((32*
(2*a^4*b^4*c^6 - a^2*b^6*c^6 - a^6*b^2*c^6 + a^8*c^4*d^2 - 3*a^3*b^5*c^5*d + 2*a^5*b^3*c^5*d + 2*a^7*b*c^3*d^3
+ 8*a^4*b^4*c^4*d^2 - 5*a^5*b^3*c^3*d^3 + a^6*b^2*c^2*d^4 - 6*a^6*b^2*c^4*d^2))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^
5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^
5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^7*b*c^6 - 2*a*b^7*c^6 - 2*a^8*c^5*d
+ 9*a^3*b^5*c^6 - 8*a^5*b^3*c^6 + 2*a^8*c^3*d^3 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c^5*d + 2*a^6*b^2*c*d^5 + 10*a
^6*b^2*c^5*d + 4*a^7*b*c^2*d^4 - 4*a^7*b*c^4*d^2 - 8*a^3*b^5*c^4*d^2 + 16*a^4*b^4*c^3*d^3 - 10*a^5*b^3*c^2*d^4
+ 13*a^5*b^3*c^4*d^2 - 13*a^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 -
2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*
b*c*d^2) - (c^2*(d^2 - c^2)^(1/2)*((32*(a*b^9*c^7 - a^3*b^7*c^7 + a^10*c^2*d^5 - 3*a^2*b^8*c^6*d + 2*a^4*b^6*c
^6*d + a^6*b^4*c^6*d - a^7*b^3*c*d^6 - 4*a^9*b*c^3*d^4 + a^3*b^7*c^5*d^2 + 6*a^4*b^6*c^4*d^3 - 9*a^5*b^5*c^3*d
^4 + 3*a^5*b^5*c^5*d^2 + 5*a^6*b^4*c^2*d^5 - 12*a^6*b^4*c^4*d^3 + 13*a^7*b^3*c^3*d^4 - 4*a^7*b^3*c^5*d^2 - 6*a
^8*b^2*c^2*d^5 + 6*a^8*b^2*c^4*d^3 + a^9*b*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*
d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d -
3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(4*a^2*b^8*c^7 - 6*a^4*b^6*c^7 + 2*a^6*b^4*c^7 + 2*a^10*c^3*d^4 - 20*a
^3*b^7*c^6*d + 26*a^5*b^5*c^6*d - 2*a^6*b^4*c*d^6 - 8*a^7*b^3*c^6*d + 2*a^8*b^2*c*d^6 + 2*a^9*b*c^2*d^5 - 8*a^
9*b*c^4*d^3 - 10*a^2*b^8*c^5*d^2 + 20*a^3*b^7*c^4*d^3 - 20*a^4*b^6*c^3*d^4 + 42*a^4*b^6*c^5*d^2 + 10*a^5*b^5*c
^2*d^5 - 48*a^5*b^5*c^4*d^3 + 32*a^6*b^4*c^3*d^4 - 44*a^6*b^4*c^5*d^2 - 12*a^7*b^3*c^2*d^5 + 36*a^7*b^3*c^4*d^
3 - 14*a^8*b^2*c^3*d^4 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3
+ a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b
^6*c^2*d - 3*a^6*b*c*d^2) - (c^2*(d^2 - c^2)^(1/2)*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2
*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^
5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^
6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*
d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a
*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 -
3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*t
an(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^
5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*
c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 4
0*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^
5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*
b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*
c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3
- a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*
c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*
c^3*d)))/(a^2*d^4 - b^2*c^4 - a^2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))/(a^2*d^4 - b^2*c^4 - a^
2*c^2*d^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)))*(d^2 - c^2)^(1/2)*2i)/(f*(a^2*d^4 - b^2*c^4 - a^2*c^2*d
^2 + b^2*c^2*d^2 - 2*a*b*c*d^3 + 2*a*b*c^3*d)) - (a*atan(((a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(2*a^4*b^4*c^6
- a^2*b^6*c^6 - a^6*b^2*c^6 + a^8*c^4*d^2 - 3*a^3*b^5*c^5*d + 2*a^5*b^3*c^5*d + 2*a^7*b*c^3*d^3 + 8*a^4*b^4*c^
4*d^2 - 5*a^5*b^3*c^3*d^3 + a^6*b^2*c^2*d^4 - 6*a^6*b^2*c^4*d^2))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3
*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3
*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^7*b*c^6 - 2*a*b^7*c^6 - 2*a^8*c^5*d + 9*a^3*b^5*c^
6 - 8*a^5*b^3*c^6 + 2*a^8*c^3*d^3 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c^5*d + 2*a^6*b^2*c*d^5 + 10*a^6*b^2*c^5*d +
4*a^7*b*c^2*d^4 - 4*a^7*b*c^4*d^2 - 8*a^3*b^5*c^4*d^2 + 16*a^4*b^4*c^3*d^3 - 10*a^5*b^3*c^2*d^4 + 13*a^5*b^3*c
^4*d^2 - 13*a^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 -
3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (a*(
-(a + b)^3*(a - b)^3)^(1/2)*((32*(a*b^9*c^7 - a^3*b^7*c^7 + a^10*c^2*d^5 - 3*a^2*b^8*c^6*d + 2*a^4*b^6*c^6*d +
a^6*b^4*c^6*d - a^7*b^3*c*d^6 - 4*a^9*b*c^3*d^4 + a^3*b^7*c^5*d^2 + 6*a^4*b^6*c^4*d^3 - 9*a^5*b^5*c^3*d^4 + 3
*a^5*b^5*c^5*d^2 + 5*a^6*b^4*c^2*d^5 - 12*a^6*b^4*c^4*d^3 + 13*a^7*b^3*c^3*d^4 - 4*a^7*b^3*c^5*d^2 - 6*a^8*b^2
*c^2*d^5 + 6*a^8*b^2*c^4*d^3 + a^9*b*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 -
2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*
b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(4*a^2*b^8*c^7 - 6*a^4*b^6*c^7 + 2*a^6*b^4*c^7 + 2*a^10*c^3*d^4 - 20*a^3*b^7
*c^6*d + 26*a^5*b^5*c^6*d - 2*a^6*b^4*c*d^6 - 8*a^7*b^3*c^6*d + 2*a^8*b^2*c*d^6 + 2*a^9*b*c^2*d^5 - 8*a^9*b*c^
4*d^3 - 10*a^2*b^8*c^5*d^2 + 20*a^3*b^7*c^4*d^3 - 20*a^4*b^6*c^3*d^4 + 42*a^4*b^6*c^5*d^2 + 10*a^5*b^5*c^2*d^5
- 48*a^5*b^5*c^4*d^3 + 32*a^6*b^4*c^3*d^4 - 44*a^6*b^4*c^5*d^2 - 12*a^7*b^3*c^2*d^5 + 36*a^7*b^3*c^4*d^3 - 14
*a^8*b^2*c^3*d^4 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3
*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2
*d - 3*a^6*b*c*d^2) + (a*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7
*a^5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5
*a^3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*
c^4*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 -
5*a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7
))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^
4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^1
1*c^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*
a^2*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d +
10*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3
*b^9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7
*c^6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c
^6*d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d
^2 - 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^
3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*
a^6*b*c*d^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(a^2*c - 2*b^2*c + a*b*d))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a
^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5
*c*d + 6*a^5*b^3*c*d))*(a^2*c - 2*b^2*c + a*b*d))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2
*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d
))*(a^2*c - 2*b^2*c + a*b*d)*1i)/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^
2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d) + (a*(-(a + b)^
3*(a - b)^3)^(1/2)*((32*(2*a^4*b^4*c^6 - a^2*b^6*c^6 - a^6*b^2*c^6 + a^8*c^4*d^2 - 3*a^3*b^5*c^5*d + 2*a^5*b^3
*c^5*d + 2*a^7*b*c^3*d^3 + 8*a^4*b^4*c^4*d^2 - 5*a^5*b^3*c^3*d^3 + a^6*b^2*c^2*d^4 - 6*a^6*b^2*c^4*d^2))/(a^7*
d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d
+ 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^7*b*c^6 - 2
*a*b^7*c^6 - 2*a^8*c^5*d + 9*a^3*b^5*c^6 - 8*a^5*b^3*c^6 + 2*a^8*c^3*d^3 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c^5*d
+ 2*a^6*b^2*c*d^5 + 10*a^6*b^2*c^5*d + 4*a^7*b*c^2*d^4 - 4*a^7*b*c^4*d^2 - 8*a^3*b^5*c^4*d^2 + 16*a^4*b^4*c^3*
d^3 - 10*a^5*b^3*c^2*d^4 + 13*a^5*b^3*c^4*d^2 - 13*a^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*
b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d
+ 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(a*b^9*c^7 - a^3*b^7*c^7 + a^10*c^2*d^
5 - 3*a^2*b^8*c^6*d + 2*a^4*b^6*c^6*d + a^6*b^4*c^6*d - a^7*b^3*c*d^6 - 4*a^9*b*c^3*d^4 + a^3*b^7*c^5*d^2 + 6*
a^4*b^6*c^4*d^3 - 9*a^5*b^5*c^3*d^4 + 3*a^5*b^5*c^5*d^2 + 5*a^6*b^4*c^2*d^5 - 12*a^6*b^4*c^4*d^3 + 13*a^7*b^3*
c^3*d^4 - 4*a^7*b^3*c^5*d^2 - 6*a^8*b^2*c^2*d^5 + 6*a^8*b^2*c^4*d^3 + a^9*b*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2
*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3
*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(4*a^2*b^8*c^7 - 6*a^4*b^6*c^7 + 2*a^
6*b^4*c^7 + 2*a^10*c^3*d^4 - 20*a^3*b^7*c^6*d + 26*a^5*b^5*c^6*d - 2*a^6*b^4*c*d^6 - 8*a^7*b^3*c^6*d + 2*a^8*b
^2*c*d^6 + 2*a^9*b*c^2*d^5 - 8*a^9*b*c^4*d^3 - 10*a^2*b^8*c^5*d^2 + 20*a^3*b^7*c^4*d^3 - 20*a^4*b^6*c^3*d^4 +
42*a^4*b^6*c^5*d^2 + 10*a^5*b^5*c^2*d^5 - 48*a^5*b^5*c^4*d^3 + 32*a^6*b^4*c^3*d^4 - 44*a^6*b^4*c^5*d^2 - 12*a^
7*b^3*c^2*d^5 + 36*a^7*b^3*c^4*d^3 - 14*a^8*b^2*c^3*d^4 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*
c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^
3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (a*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c
^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*
a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c
^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7
*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^
2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a
^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c
*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 +
2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d
- 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^1
0*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2
*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d
^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6
+ 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2
*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2
+ 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(a^2*c - 2*b^2*c + a*b*d))/(
a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2
+ 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(a^2*c - 2*b^2*c + a*b*d))/(a^8*d^2 - b^8*c^2 +
3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^
7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(a^2*c - 2*b^2*c + a*b*d)*1i)/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3
*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b
^5*c*d + 6*a^5*b^3*c*d))/((64*(a^5*b*c^5 - 2*a^3*b^3*c^5 + a^4*b^2*c^4*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3
- a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*
c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (64*tan(e/2 + (f*x)/2)*(2*a^6*c^5 + 4*a^2*b^4*c^5 - 6*a^4*b^2*c^5 - 6
*a^3*b^3*c^4*d + 2*a^4*b^2*c^3*d^2 + 4*a^5*b*c^4*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^
4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d
- 3*a^6*b*c*d^2) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(2*a^4*b^4*c^6 - a^2*b^6*c^6 - a^6*b^2*c^6 + a^8*c^4*d
^2 - 3*a^3*b^5*c^5*d + 2*a^5*b^3*c^5*d + 2*a^7*b*c^3*d^3 + 8*a^4*b^4*c^4*d^2 - 5*a^5*b^3*c^3*d^3 + a^6*b^2*c^2
*d^4 - 6*a^6*b^2*c^4*d^2))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*
a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan
(e/2 + (f*x)/2)*(2*a^7*b*c^6 - 2*a*b^7*c^6 - 2*a^8*c^5*d + 9*a^3*b^5*c^6 - 8*a^5*b^3*c^6 + 2*a^8*c^3*d^3 + 2*a
^2*b^6*c^5*d - 13*a^4*b^4*c^5*d + 2*a^6*b^2*c*d^5 + 10*a^6*b^2*c^5*d + 4*a^7*b*c^2*d^4 - 4*a^7*b*c^4*d^2 - 8*a
^3*b^5*c^4*d^2 + 16*a^4*b^4*c^3*d^3 - 10*a^5*b^3*c^2*d^4 + 13*a^5*b^3*c^4*d^2 - 13*a^6*b^2*c^3*d^3))/(a^7*d^3
- b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*
a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(a*b^9
*c^7 - a^3*b^7*c^7 + a^10*c^2*d^5 - 3*a^2*b^8*c^6*d + 2*a^4*b^6*c^6*d + a^6*b^4*c^6*d - a^7*b^3*c*d^6 - 4*a^9*
b*c^3*d^4 + a^3*b^7*c^5*d^2 + 6*a^4*b^6*c^4*d^3 - 9*a^5*b^5*c^3*d^4 + 3*a^5*b^5*c^5*d^2 + 5*a^6*b^4*c^2*d^5 -
12*a^6*b^4*c^4*d^3 + 13*a^7*b^3*c^3*d^4 - 4*a^7*b^3*c^5*d^2 - 6*a^8*b^2*c^2*d^5 + 6*a^8*b^2*c^4*d^3 + a^9*b*c*
d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3
*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(4*a^
2*b^8*c^7 - 6*a^4*b^6*c^7 + 2*a^6*b^4*c^7 + 2*a^10*c^3*d^4 - 20*a^3*b^7*c^6*d + 26*a^5*b^5*c^6*d - 2*a^6*b^4*c
*d^6 - 8*a^7*b^3*c^6*d + 2*a^8*b^2*c*d^6 + 2*a^9*b*c^2*d^5 - 8*a^9*b*c^4*d^3 - 10*a^2*b^8*c^5*d^2 + 20*a^3*b^7
*c^4*d^3 - 20*a^4*b^6*c^3*d^4 + 42*a^4*b^6*c^5*d^2 + 10*a^5*b^5*c^2*d^5 - 48*a^5*b^5*c^4*d^3 + 32*a^6*b^4*c^3*
d^4 - 44*a^6*b^4*c^5*d^2 - 12*a^7*b^3*c^2*d^5 + 36*a^7*b^3*c^4*d^3 - 14*a^8*b^2*c^3*d^4 + 12*a^8*b^2*c^5*d^2 +
2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*
d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) + (a*((32*(2*a^4*b^
8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*
b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 -
5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5
*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5
+ 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 -
a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c
^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 +
7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a
^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6
- 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*
a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^
6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b
^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2 - 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*
c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6
*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2))*(-(a + b)^3*(a - b)^3)^(1
/2)*(a^2*c - 2*b^2*c + a*b*d))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2
+ 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(a^2*c - 2*b^2*c
+ a*b*d))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*
a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*(a^2*c - 2*b^2*c + a*b*d))/(a^8*d^2
- b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^
7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d) - (a*(-(a + b)^3*(a - b)^3)^(1/2)*((32*(2*a^4*b^4*c^6 - a
^2*b^6*c^6 - a^6*b^2*c^6 + a^8*c^4*d^2 - 3*a^3*b^5*c^5*d + 2*a^5*b^3*c^5*d + 2*a^7*b*c^3*d^3 + 8*a^4*b^4*c^4*d
^2 - 5*a^5*b^3*c^3*d^3 + a^6*b^2*c^2*d^4 - 6*a^6*b^2*c^4*d^2))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^
3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*
b^6*c^2*d - 3*a^6*b*c*d^2) + (32*tan(e/2 + (f*x)/2)*(2*a^7*b*c^6 - 2*a*b^7*c^6 - 2*a^8*c^5*d + 9*a^3*b^5*c^6 -
8*a^5*b^3*c^6 + 2*a^8*c^3*d^3 + 2*a^2*b^6*c^5*d - 13*a^4*b^4*c^5*d + 2*a^6*b^2*c*d^5 + 10*a^6*b^2*c^5*d + 4*a
^7*b*c^2*d^4 - 4*a^7*b*c^4*d^2 - 8*a^3*b^5*c^4*d^2 + 16*a^4*b^4*c^3*d^3 - 10*a^5*b^3*c^2*d^4 + 13*a^5*b^3*c^4*
d^2 - 13*a^6*b^2*c^3*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*
a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (a*(-(a
+ b)^3*(a - b)^3)^(1/2)*((32*(a*b^9*c^7 - a^3*b^7*c^7 + a^10*c^2*d^5 - 3*a^2*b^8*c^6*d + 2*a^4*b^6*c^6*d + a^
6*b^4*c^6*d - a^7*b^3*c*d^6 - 4*a^9*b*c^3*d^4 + a^3*b^7*c^5*d^2 + 6*a^4*b^6*c^4*d^3 - 9*a^5*b^5*c^3*d^4 + 3*a^
5*b^5*c^5*d^2 + 5*a^6*b^4*c^2*d^5 - 12*a^6*b^4*c^4*d^3 + 13*a^7*b^3*c^3*d^4 - 4*a^7*b^3*c^5*d^2 - 6*a^8*b^2*c^
2*d^5 + 6*a^8*b^2*c^4*d^3 + a^9*b*c*d^6))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a
^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c
*d^2) + (32*tan(e/2 + (f*x)/2)*(4*a^2*b^8*c^7 - 6*a^4*b^6*c^7 + 2*a^6*b^4*c^7 + 2*a^10*c^3*d^4 - 20*a^3*b^7*c^
6*d + 26*a^5*b^5*c^6*d - 2*a^6*b^4*c*d^6 - 8*a^7*b^3*c^6*d + 2*a^8*b^2*c*d^6 + 2*a^9*b*c^2*d^5 - 8*a^9*b*c^4*d
^3 - 10*a^2*b^8*c^5*d^2 + 20*a^3*b^7*c^4*d^3 - 20*a^4*b^6*c^3*d^4 + 42*a^4*b^6*c^5*d^2 + 10*a^5*b^5*c^2*d^5 -
48*a^5*b^5*c^4*d^3 + 32*a^6*b^4*c^3*d^4 - 44*a^6*b^4*c^5*d^2 - 12*a^7*b^3*c^2*d^5 + 36*a^7*b^3*c^4*d^3 - 14*a^
8*b^2*c^3*d^4 + 12*a^8*b^2*c^5*d^2 + 2*a*b^9*c^6*d))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^
4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d
- 3*a^6*b*c*d^2) - (a*((32*(2*a^4*b^8*c^8 - a^2*b^10*c^8 - a^6*b^6*c^8 + a^12*c^2*d^6 + 2*a^3*b^9*c^7*d - 7*a^
5*b^7*c^7*d - a^7*b^5*c*d^7 + 4*a^7*b^5*c^7*d + 2*a^9*b^3*c*d^7 - 4*a^11*b*c^3*d^5 - 4*a^2*b^10*c^6*d^2 + 5*a^
3*b^9*c^5*d^3 + 3*a^4*b^8*c^6*d^2 - 5*a^5*b^7*c^3*d^5 - 10*a^5*b^7*c^5*d^3 + 4*a^6*b^6*c^2*d^6 + 5*a^6*b^6*c^4
*d^4 + 6*a^6*b^6*c^6*d^2 + 6*a^7*b^5*c^3*d^5 + 5*a^7*b^5*c^5*d^3 - 7*a^8*b^4*c^2*d^6 - 10*a^8*b^4*c^4*d^4 - 5*
a^8*b^4*c^6*d^2 + 3*a^9*b^3*c^3*d^5 + 2*a^10*b^2*c^2*d^6 + 5*a^10*b^2*c^4*d^4 + a*b^11*c^7*d - a^11*b*c*d^7))/
(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 - 2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c
^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6*b*c*d^2) - (32*tan(e/2 + (f*x)/2)*(3*a*b^11*c
^8 - 3*a^12*c*d^7 - 8*a^3*b^9*c^8 + 7*a^5*b^7*c^8 - 2*a^7*b^5*c^8 + 2*a^12*c^3*d^5 - 4*a*b^11*c^6*d^2 - 15*a^2
*b^10*c^7*d + 40*a^4*b^8*c^7*d + 4*a^6*b^6*c*d^7 - 35*a^6*b^6*c^7*d - 11*a^8*b^4*c*d^7 + 10*a^8*b^4*c^7*d + 10
*a^10*b^2*c*d^7 + 15*a^11*b*c^2*d^6 - 10*a^11*b*c^4*d^4 + 20*a^2*b^10*c^5*d^3 - 40*a^3*b^9*c^4*d^4 + 41*a^3*b^
9*c^6*d^2 + 40*a^4*b^8*c^3*d^5 - 85*a^4*b^8*c^5*d^3 - 20*a^5*b^7*c^2*d^6 + 125*a^5*b^7*c^4*d^4 - 90*a^5*b^7*c^
6*d^2 - 113*a^6*b^6*c^3*d^5 + 130*a^6*b^6*c^5*d^3 + 55*a^7*b^5*c^2*d^6 - 140*a^7*b^5*c^4*d^4 + 73*a^7*b^5*c^6*
d^2 + 108*a^8*b^4*c^3*d^5 - 85*a^8*b^4*c^5*d^3 - 50*a^9*b^3*c^2*d^6 + 65*a^9*b^3*c^4*d^4 - 20*a^9*b^3*c^6*d^2
- 37*a^10*b^2*c^3*d^5 + 20*a^10*b^2*c^5*d^3))/(a^7*d^3 - b^7*c^3 + 2*a^2*b^5*c^3 - a^4*b^3*c^3 + a^3*b^4*d^3 -
2*a^5*b^2*d^3 - 3*a^2*b^5*c*d^2 - 6*a^3*b^4*c^2*d + 6*a^4*b^3*c*d^2 + 3*a^5*b^2*c^2*d + 3*a*b^6*c^2*d - 3*a^6
*b*c*d^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(a^2*c - 2*b^2*c + a*b*d))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*
b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*
d + 6*a^5*b^3*c*d))*(a^2*c - 2*b^2*c + a*b*d))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^
2 - a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))*
(a^2*c - 2*b^2*c + a*b*d))/(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 - a^2*b^6*d^2 + 3*
a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d)))*(-(a + b)^3*(a - b)
^3)^(1/2)*(a^2*c - 2*b^2*c + a*b*d)*2i)/(f*(a^8*d^2 - b^8*c^2 + 3*a^2*b^6*c^2 - 3*a^4*b^4*c^2 + a^6*b^2*c^2 -
a^2*b^6*d^2 + 3*a^4*b^4*d^2 - 3*a^6*b^2*d^2 + 2*a*b^7*c*d - 2*a^7*b*c*d - 6*a^3*b^5*c*d + 6*a^5*b^3*c*d))