\(\int \frac {d+e \sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx\) [503]

Optimal result

Integrand size = 21, antiderivative size = 242 \[ \int \frac {d+e \sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx=\frac {\sqrt {2} \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3373, 2739, 632, 210} \[ \int \frac {d+e \sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx=\frac {\sqrt {2} \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \left (b-\sqrt {b^2-4 a c}\right )+2 c}{\sqrt {2} \sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}\right )}{\sqrt {-b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}+\frac {\sqrt {2} \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \left (\sqrt {b^2-4 a c}+b\right )+2 c}{\sqrt {2} \sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}}\right )}{\sqrt {b \sqrt {b^2-4 a c}-2 c (a+c)+b^2}} \]

[In]

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

[Out]

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

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 3373

Int[((A_) + (B_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + (b_.)*sin[(d_.) + (e_.)*(x_)] + (c_.)*sin[(d_.) + (e_.)*(x
_)]^2), x_Symbol] :> Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/(b + q + 2*c*Sin[d + e*x
]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Sin[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B
}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{b+\sqrt {b^2-4 a c}+2 c \sin (x)} \, dx+\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{b-\sqrt {b^2-4 a c}+2 c \sin (x)} \, dx \\ & = \left (2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}+4 c x+\left (b+\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\left (2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}+4 c x+\left (b-\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right ) \\ & = -\left (\left (4 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (4 c^2-\left (b+\sqrt {b^2-4 a c}\right )^2\right )-x^2} \, dx,x,4 c+2 \left (b+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )\right )\right )-\left (4 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-8 \left (b^2-2 c (a+c)-b \sqrt {b^2-4 a c}\right )-x^2} \, dx,x,4 c+2 \left (b-\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )\right ) \\ & = \frac {\sqrt {2} \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-2 c (a+c)-b \sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-2 c (a+c)+b \sqrt {b^2-4 a c}}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.39 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.18 \[ \int \frac {d+e \sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx=\frac {\frac {\left (-2 i c d+\left (i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {2 c+\left (b-i \sqrt {-b^2+4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)-i b \sqrt {-b^2+4 a c}}}\right )}{\sqrt {b^2-2 c (a+c)-i b \sqrt {-b^2+4 a c}}}+\frac {\left (2 i c d+\left (-i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {2 c+\left (b+i \sqrt {-b^2+4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 c (a+c)+i b \sqrt {-b^2+4 a c}}}\right )}{\sqrt {b^2-2 c (a+c)+i b \sqrt {-b^2+4 a c}}}}{\sqrt {-\frac {b^2}{2}+2 a c}} \]

[In]

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

[Out]

((((-2*I)*c*d + (I*b + Sqrt[-b^2 + 4*a*c])*e)*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt
[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]] + (((2*I)*c*d
+ ((-I)*b + Sqrt[-b^2 + 4*a*c])*e)*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c
*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])/Sqrt[-1/2*b^2 + 2*a*c]

Maple [A] (verified)

Time = 5.27 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.09

method	result	size
default	\(2 a \left (-\frac {\sqrt {-4 a c +b^{2}}\, \left (\sqrt {-4 a c +b^{2}}\, d -2 a e +b d \right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+b +\sqrt {-4 a c +b^{2}}}{\sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{\left (4 a c -b^{2}\right ) a \sqrt {4 a c -2 b^{2}-2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}+\frac {\left (\sqrt {-4 a c +b^{2}}\, d +2 a e -b d \right ) \sqrt {-4 a c +b^{2}}\, \arctan \left (\frac {-2 a \tan \left (\frac {x}{2}\right )+\sqrt {-4 a c +b^{2}}-b}{\sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )}{\left (4 a c -b^{2}\right ) a \sqrt {4 a c -2 b^{2}+2 b \sqrt {-4 a c +b^{2}}+4 a^{2}}}\right )\)	\(263\)
risch	\(\text {Expression too large to display}\)	\(8331\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6695 vs. \(2 (208) = 416\).

Time = 6.01 (sec) , antiderivative size = 6695, normalized size of antiderivative = 27.67 \[ \int \frac {d+e \sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 42.93 (sec) , antiderivative size = 10465, normalized size of antiderivative = 43.24 \[ \int \frac {d+e \sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan(((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2
)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c -
b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e
+ 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2
*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2)
- 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^
2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*
c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^
2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3
*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c
^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*
e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6
+ 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^
(1/2)*(tan(x/2)*(96*a*b^4 + 256*a^4*c - 64*a^3*b^2 + 512*a^2*c^3 + 768*a^3*c^2 - 128*a*b^2*c^2 - 576*a^2*b^2*c
) + 32*a^2*b^3 + 128*a^2*b*c^2 - 32*a*b^3*c - 128*a^3*b*c) + tan(x/2)*(64*a^2*b^2*e - 256*a^2*c^2*e - 64*a*b^3
*d - 256*a^3*c*e + 256*a^2*b*c*d + 64*a*b^2*c*e) - 32*a^2*b^2*d + 128*a^2*c^2*d + 32*a*b^3*e + 128*a^3*c*d - 3
2*a*b^2*c*d - 128*a^2*b*c*e) - tan(x/2)*(64*a^3*e^2 + 32*a*b^2*d^2 - 64*a*b^2*e^2 - 128*a*c^2*d^2 - 64*a^2*c*d
^2 + 128*a^2*c*e^2 - 64*a^2*b*d*e + 128*a*b*c*d*e) + 32*a^2*b*e^2 + 32*a*b*c*d^2 - 128*a^2*c*d*e)*1i + (-(b^4*
d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) +
2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2)
+ 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*
d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b
^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^
2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d
*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b
^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 -
8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*
(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^
2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c -
b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4
+ 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*(tan(x/
2)*(96*a*b^4 + 256*a^4*c - 64*a^3*b^2 + 512*a^2*c^3 + 768*a^3*c^2 - 128*a*b^2*c^2 - 576*a^2*b^2*c) + 32*a^2*b^
3 + 128*a^2*b*c^2 - 32*a*b^3*c - 128*a^3*b*c) - tan(x/2)*(64*a^2*b^2*e - 256*a^2*c^2*e - 64*a*b^3*d - 256*a^3*
c*e + 256*a^2*b*c*d + 64*a*b^2*c*e) + 32*a^2*b^2*d - 128*a^2*c^2*d - 32*a*b^3*e - 128*a^3*c*d + 32*a*b^2*c*d +
128*a^2*b*c*e) - tan(x/2)*(64*a^3*e^2 + 32*a*b^2*d^2 - 64*a*b^2*e^2 - 128*a*c^2*d^2 - 64*a^2*c*d^2 + 128*a^2*
c*e^2 - 64*a^2*b*d*e + 128*a*b*c*d*e) + 32*a^2*b*e^2 + 32*a*b*c*d^2 - 128*a^2*c*d*e)*1i)/(2*tan(x/2)*(64*a^2*e
^3 - 64*a*b*d*e^2 + 64*a*c*d^2*e) + (-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^
3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a
*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 +
6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4
*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 +
b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2
- 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*
a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^
2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*(
(-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(
1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3
)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a
^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 3
2*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*(tan(x/2)*(96*a*b^4 + 256*a^4*c - 64*a^3*b^2 + 512*a^2*c^3 + 768*a^3*c^2 -
128*a*b^2*c^2 - 576*a^2*b^2*c) + 32*a^2*b^3 + 128*a^2*b*c^2 - 32*a*b^3*c - 128*a^3*b*c) + tan(x/2)*(64*a^2*b^
2*e - 256*a^2*c^2*e - 64*a*b^3*d - 256*a^3*c*e + 256*a^2*b*c*d + 64*a*b^2*c*e) - 32*a^2*b^2*d + 128*a^2*c^2*d
+ 32*a*b^3*e + 128*a^3*c*d - 32*a*b^2*c*d - 128*a^2*b*c*e) - tan(x/2)*(64*a^3*e^2 + 32*a*b^2*d^2 - 64*a*b^2*e^
2 - 128*a*c^2*d^2 - 64*a^2*c*d^2 + 128*a^2*c*e^2 - 64*a^2*b*d*e + 128*a*b*c*d*e) + 32*a^2*b*e^2 + 32*a*b*c*d^2
- 128*a^2*c*d*e) - (-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 + b*e^2*
(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e - 2*a*d
*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 -
8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c
^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c -
b^2)^3)^(1/2) - 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2
- 2*b^2*c^2*d^2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1
/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c
^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4
*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2
*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*
c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(
a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 +
10*a*b^4*c)))^(1/2)*(tan(x/2)*(96*a*b^4 + 256*a^4*c - 64*a^3*b^2 + 512*a^2*c^3 + 768*a^3*c^2 - 128*a*b^2*c^2 -
576*a^2*b^2*c) + 32*a^2*b^3 + 128*a^2*b*c^2 - 32*a*b^3*c - 128*a^3*b*c) - tan(x/2)*(64*a^2*b^2*e - 256*a^2*c^
2*e - 64*a*b^3*d - 256*a^3*c*e + 256*a^2*b*c*d + 64*a*b^2*c*e) + 32*a^2*b^2*d - 128*a^2*c^2*d - 32*a*b^3*e - 1
28*a^3*c*d + 32*a*b^2*c*d + 128*a^2*b*c*e) - tan(x/2)*(64*a^3*e^2 + 32*a*b^2*d^2 - 64*a*b^2*e^2 - 128*a*c^2*d^
2 - 64*a^2*c*d^2 + 128*a^2*c*e^2 - 64*a^2*b*d*e + 128*a*b*c*d*e) + 32*a^2*b*e^2 + 32*a*b*c*d^2 - 128*a^2*c*d*e
) + 64*a^2*d*e^2 + 64*a*c*d^3 - 64*a*b*d^2*e))*(-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 + b*d^2*(-(4*a*c - b^2)^3)^(
1/2) - 8*a^3*c*e^2 + b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^
2*d^2 - 2*a*b^3*d*e - 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e - 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*
b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^
4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*2i + atan(((-(b^4*d^2 - b^4
*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 - b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2
*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*
c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(
a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 +
10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 - b*e^2
*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e + 2*a*
d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2
- 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*
c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c
- b^2)^3)^(1/2) - 8*a^3*c*e^2 - b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2
- 2*b^2*c^2*d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(
1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*
c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*(tan(x/2)*(96*a*
b^4 + 256*a^4*c - 64*a^3*b^2 + 512*a^2*c^3 + 768*a^3*c^2 - 128*a*b^2*c^2 - 576*a^2*b^2*c) + 32*a^2*b^3 + 128*a
^2*b*c^2 - 32*a*b^3*c - 128*a^3*b*c) + tan(x/2)*(64*a^2*b^2*e - 256*a^2*c^2*e - 64*a*b^3*d - 256*a^3*c*e + 256
*a^2*b*c*d + 64*a*b^2*c*e) - 32*a^2*b^2*d + 128*a^2*c^2*d + 32*a*b^3*e + 128*a^3*c*d - 32*a*b^2*c*d - 128*a^2*
b*c*e) - tan(x/2)*(64*a^3*e^2 + 32*a*b^2*d^2 - 64*a*b^2*e^2 - 128*a*c^2*d^2 - 64*a^2*c*d^2 + 128*a^2*c*e^2 - 6
4*a^2*b*d*e + 128*a*b*c*d*e) + 32*a^2*b*e^2 + 32*a*b*c*d^2 - 128*a^2*c*d*e)*1i + (-(b^4*d^2 - b^4*e^2 + 8*a*c^
3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 - b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*
c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d
*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6
+ 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))
^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 - b*e^2*(-(4*a*c - b
^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(4*a*c
- b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d
*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b
^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/
2) - 8*a^3*c*e^2 - b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*
d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^
2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*
c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*(tan(x/2)*(96*a*b^4 + 256*a^4
*c - 64*a^3*b^2 + 512*a^2*c^3 + 768*a^3*c^2 - 128*a*b^2*c^2 - 576*a^2*b^2*c) + 32*a^2*b^3 + 128*a^2*b*c^2 - 32
*a*b^3*c - 128*a^3*b*c) - tan(x/2)*(64*a^2*b^2*e - 256*a^2*c^2*e - 64*a*b^3*d - 256*a^3*c*e + 256*a^2*b*c*d +
64*a*b^2*c*e) + 32*a^2*b^2*d - 128*a^2*c^2*d - 32*a*b^3*e - 128*a^3*c*d + 32*a*b^2*c*d + 128*a^2*b*c*e) - tan(
x/2)*(64*a^3*e^2 + 32*a*b^2*d^2 - 64*a*b^2*e^2 - 128*a*c^2*d^2 - 64*a^2*c*d^2 + 128*a^2*c*e^2 - 64*a^2*b*d*e +
128*a*b*c*d*e) + 32*a^2*b*e^2 + 32*a*b*c*d^2 - 128*a^2*c*d*e)*1i)/(2*tan(x/2)*(64*a^2*e^3 - 64*a*b*d*e^2 + 64
*a*c*d^2*e) + (-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 - b*e^2*(-(4*a
*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(
4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b
*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8
*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^
3)^(1/2) - 8*a^3*c*e^2 - b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^
2*c^2*d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) -
6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 1
6*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 +
8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 - b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 +
8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e
+ 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^
4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b
^4*c)))^(1/2)*(tan(x/2)*(96*a*b^4 + 256*a^4*c - 64*a^3*b^2 + 512*a^2*c^3 + 768*a^3*c^2 - 128*a*b^2*c^2 - 576*a
^2*b^2*c) + 32*a^2*b^3 + 128*a^2*b*c^2 - 32*a*b^3*c - 128*a^3*b*c) + tan(x/2)*(64*a^2*b^2*e - 256*a^2*c^2*e -
64*a*b^3*d - 256*a^3*c*e + 256*a^2*b*c*d + 64*a*b^2*c*e) - 32*a^2*b^2*d + 128*a^2*c^2*d + 32*a*b^3*e + 128*a^3
*c*d - 32*a*b^2*c*d - 128*a^2*b*c*e) - tan(x/2)*(64*a^3*e^2 + 32*a*b^2*d^2 - 64*a*b^2*e^2 - 128*a*c^2*d^2 - 64
*a^2*c*d^2 + 128*a^2*c*e^2 - 64*a^2*b*d*e + 128*a*b*c*d*e) + 32*a^2*b*e^2 + 32*a*b*c*d^2 - 128*a^2*c*d*e) - (-
(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 - b*e^2*(-(4*a*c - b^2)^3)^(1/
2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(4*a*c - b^2)^3)^
(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2
*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*
a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3
*c*e^2 - b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*
b^3*d*e + 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 +
6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*
c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*((-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b
*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 - b*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 -
8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e + 2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a
*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2
*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*(t
an(x/2)*(96*a*b^4 + 256*a^4*c - 64*a^3*b^2 + 512*a^2*c^3 + 768*a^3*c^2 - 128*a*b^2*c^2 - 576*a^2*b^2*c) + 32*a
^2*b^3 + 128*a^2*b*c^2 - 32*a*b^3*c - 128*a^3*b*c) - tan(x/2)*(64*a^2*b^2*e - 256*a^2*c^2*e - 64*a*b^3*d - 256
*a^3*c*e + 256*a^2*b*c*d + 64*a*b^2*c*e) + 32*a^2*b^2*d - 128*a^2*c^2*d - 32*a*b^3*e - 128*a^3*c*d + 32*a*b^2*
c*d + 128*a^2*b*c*e) - tan(x/2)*(64*a^3*e^2 + 32*a*b^2*d^2 - 64*a*b^2*e^2 - 128*a*c^2*d^2 - 64*a^2*c*d^2 + 128
*a^2*c*e^2 - 64*a^2*b*d*e + 128*a*b*c*d*e) + 32*a^2*b*e^2 + 32*a*b*c*d^2 - 128*a^2*c*d*e) + 64*a^2*d*e^2 + 64*
a*c*d^3 - 64*a*b*d^2*e))*(-(b^4*d^2 - b^4*e^2 + 8*a*c^3*d^2 - b*d^2*(-(4*a*c - b^2)^3)^(1/2) - 8*a^3*c*e^2 - b
*e^2*(-(4*a*c - b^2)^3)^(1/2) + 2*a^2*b^2*e^2 + 8*a^2*c^2*d^2 - 8*a^2*c^2*e^2 - 2*b^2*c^2*d^2 - 2*a*b^3*d*e +
2*a*d*e*(-(4*a*c - b^2)^3)^(1/2) + 2*b^3*c*d*e + 2*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d^2 + 6*a*b^2*c*
e^2 - 8*a*b*c^2*d*e + 8*a^2*b*c*d*e)/(2*(a^2*b^4 - b^6 + 16*a^2*c^4 + 32*a^3*c^3 + 16*a^4*c^2 + b^4*c^2 - 8*a*
b^2*c^3 - 8*a^3*b^2*c - 32*a^2*b^2*c^2 + 10*a*b^4*c)))^(1/2)*2i