\(\int \frac {d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx\) [509]
Optimal result
Integrand size = 21, antiderivative size = 246 \[
\int \frac {d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}
\]
[Out]
2*arctan((b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)*tan(1/2*x)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2))*(e+(-b*e+2*c*d)/(-4*a*c
+b^2)^(1/2))/(b-2*c-(-4*a*c+b^2)^(1/2))^(1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)+2*arctan((b-2*c+(-4*a*c+b^2)^(1
/2))^(1/2)*tan(1/2*x)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2))*(e+(b*e-2*c*d)/(-4*a*c+b^2)^(1/2))/(b-2*c+(-4*a*c+b^2)
^(1/2))^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)
Rubi [A] (verified)
Time = 0.88 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3374, 2738, 211}
\[
\int \frac {d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\frac {2 \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right ) \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \sqrt {-\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {-\sqrt {b^2-4 a c}+b-2 c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \sqrt {\sqrt {b^2-4 a c}+b-2 c}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b-2 c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}
\]
[In]
Int[(d + e*Cos[x])/(a + b*Cos[x] + c*Cos[x]^2),x]
[Out]
(2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sq
rt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(e - (2*c*d - b*
e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/
(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 2738
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*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 3374
Int[(cos[(d_.) + (e_.)*(x_)]*(B_.) + (A_))/((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + cos[(d_.) + (e_.)*(x_)]^2*
(c_.)), 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*Cos[d + e*x
]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Cos[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 \cos (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 \cos (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+2 c+\sqrt {b^2-4 a c}+\left (b-2 c+\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+2 c-\sqrt {b^2-4 a c}+\left (b-2 c-\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right ) \\ & = \frac {2 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {b-2 c-\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c-\sqrt {b^2-4 a c}} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}+\frac {2 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {b-2 c+\sqrt {b^2-4 a c}} \tan \left (\frac {x}{2}\right )}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-2 c+\sqrt {b^2-4 a c}} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.14 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98
\[
\int \frac {d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\frac {\sqrt {2} \left (-\frac {\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\left (b-2 c+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)-2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)-b \sqrt {b^2-4 a c}}}+\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \text {arctanh}\left (\frac {\left (-b+2 c+\sqrt {b^2-4 a c}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 c (a+c)+2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {-b^2+2 c (a+c)+b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c}}
\]
[In]
Integrate[(d + e*Cos[x])/(a + b*Cos[x] + c*Cos[x]^2),x]
[Out]
(Sqrt[2]*(-(((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2
+ 4*c*(a + c) - 2*b*Sqrt[b^2 - 4*a*c]]])/Sqrt[-b^2 + 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) + ((2*c*d + (-b + Sq
rt[b^2 - 4*a*c])*e)*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2
- 4*a*c]]])/Sqrt[-b^2 + 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c]
Maple [A] (verified)
Time = 5.29 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.03
| | |
method | result | size |
| | |
default |
\(2 \left (a -b +c \right ) \left (\frac {\left (\sqrt {-4 a c +b^{2}}\, d -e \sqrt {-4 a c +b^{2}}-2 a e +b d +b e -2 c d \right ) \arctan \left (\frac {\left (a -b +c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}+a -c \right ) \left (a -b +c \right )}}+\frac {\left (\sqrt {-4 a c +b^{2}}\, d -e \sqrt {-4 a c +b^{2}}+2 a e -b d -b e +2 c d \right ) \operatorname {arctanh}\left (\frac {\left (-a +b -c \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )}{2 \sqrt {-4 a c +b^{2}}\, \left (a -b +c \right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}-a +c \right ) \left (a -b +c \right )}}\right )\) |
\(254\) |
risch |
\(\text {Expression too large to display}\) |
\(8335\) |
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|
|
|
[In]
int((d+e*cos(x))/(a+b*cos(x)+c*cos(x)^2),x,method=_RETURNVERBOSE)
[Out]
2*(a-b+c)*(1/2*((-4*a*c+b^2)^(1/2)*d-e*(-4*a*c+b^2)^(1/2)-2*a*e+b*d+b*e-2*c*d)/(-4*a*c+b^2)^(1/2)/(a-b+c)/(((-
4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2)*arctan((a-b+c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)+a-c)*(a-b+c))^(1/2))+1/2*(
(-4*a*c+b^2)^(1/2)*d-e*(-4*a*c+b^2)^(1/2)+2*a*e-b*d-b*e+2*c*d)/(-4*a*c+b^2)^(1/2)/(a-b+c)/(((-4*a*c+b^2)^(1/2)
-a+c)*(a-b+c))^(1/2)*arctanh((-a+b-c)*tan(1/2*x)/(((-4*a*c+b^2)^(1/2)-a+c)*(a-b+c))^(1/2)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 6697 vs. \(2 (206) = 412\).
Time = 6.40 (sec) , antiderivative size = 6697, normalized size of antiderivative = 27.22
\[
\int \frac {d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Too large to display}
\]
[In]
integrate((d+e*cos(x))/(a+b*cos(x)+c*cos(x)^2),x, algorithm="fricas")
[Out]
Too large to include
Sympy [F(-1)]
Timed out. \[
\int \frac {d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Timed out}
\]
[In]
integrate((d+e*cos(x))/(a+b*cos(x)+c*cos(x)**2),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\int { \frac {e \cos \left (x\right ) + d}{c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a} \,d x }
\]
[In]
integrate((d+e*cos(x))/(a+b*cos(x)+c*cos(x)^2),x, algorithm="maxima")
[Out]
integrate((e*cos(x) + d)/(c*cos(x)^2 + b*cos(x) + a), x)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 5300 vs. \(2 (206) = 412\).
Time = 2.04 (sec) , antiderivative size = 5300, normalized size of antiderivative = 21.54
\[
\int \frac {d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Too large to display}
\]
[In]
integrate((d+e*cos(x))/(a+b*cos(x)+c*cos(x)^2),x, algorithm="giac")
[Out]
((2*a^2*b^3 - 2*b^5 - 8*a^3*b*c - 12*a^2*b^2*c + 20*a*b^3*c + 4*b^4*c + 48*a^3*c^2 - 48*a^2*b*c^2 - 24*a*b^2*c
^2 - 6*b^3*c^2 + 32*a^2*c^3 + 24*a*b*c^3 + 4*b^2*c^3 - 16*a*c^4 + 3*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*
a*c)*(a - b + c))*a^2*b^2 - 2*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^3 - 5*sqrt(a^2 -
a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*b^4 - 12*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a -
b + c))*a^3*c + 8*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b*c + 34*sqrt(a^2 - a*b + b
*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^2*c + 6*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b +
c))*b^3*c - 56*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*c^2 - 24*sqrt(a^2 - a*b + b*c -
c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*b*c^2 - 5*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*
b^2*c^2 + 20*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*c^3 + 3*sqrt(a^2 - a*b + b*c - c^2
+ sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^2*b - 2*(b^2 - 4*a*c)*a^2*b - 2*sqrt(a^2 - a*b + b*c - c^
2 + sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b^2 - 5*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*
(a - b + c))*sqrt(b^2 - 4*a*c)*b^3 + 2*(b^2 - 4*a*c)*b^3 + 6*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a
- b + c))*sqrt(b^2 - 4*a*c)*a^2*c + 12*(b^2 - 4*a*c)*a^2*c + 10*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c
)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b*c - 12*(b^2 - 4*a*c)*a*b*c - 4*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*
a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b^2*c - 4*(b^2 - 4*a*c)*b^2*c + 28*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 -
4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*c^2 + 8*(b^2 - 4*a*c)*a*c^2 + 7*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2
- 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b*c^2 + 6*(b^2 - 4*a*c)*b*c^2 - 10*sqrt(a^2 - a*b + b*c - c^2 + sqrt(
b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*c^3 - 4*(b^2 - 4*a*c)*c^3)*d*abs(a - b + c) - (4*a^3*b^2 - 6*a^2*b
^3 - 4*a*b^4 + 6*b^5 - 16*a^4*c + 24*a^3*b*c + 40*a^2*b^2*c - 44*a*b^3*c - 8*b^4*c - 96*a^3*c^2 + 80*a^2*b*c^2
+ 52*a*b^2*c^2 + 2*b^3*c^2 - 80*a^2*c^3 - 8*a*b*c^3 - 3*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b
+ c))*a^2*b^2 + 2*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^3 + 5*sqrt(a^2 - a*b + b*c
- c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*b^4 + 12*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^
3*c - 8*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b*c - 34*sqrt(a^2 - a*b + b*c - c^2 +
sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^2*c - 6*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*b^3*c +
56*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*c^2 + 24*sqrt(a^2 - a*b + b*c - c^2 + sqrt
(b^2 - 4*a*c)*(a - b + c))*a*b*c^2 + 5*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*b^2*c^2 - 2
0*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*a*c)*(a - b + c))*a*c^3 + 6*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2
- 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^3 - 4*(b^2 - 4*a*c)*a^3 - sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 - 4*
a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^2*b + 6*(b^2 - 4*a*c)*a^2*b - 12*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2 -
4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b^2 + 4*(b^2 - 4*a*c)*a*b^2 - 5*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2
- 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b^3 - 6*(b^2 - 4*a*c)*b^3 + 28*sqrt(a^2 - a*b + b*c - c^2 + sqrt(b^2
- 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^2*c - 24*(b^2 - 4*a*c)*a^2*c + 26*sqrt(a^2 - a*b + b*c - c^2 + sqrt(
b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b*c + 20*(b^2 - 4*a*c)*a*b*c + 6*sqrt(a^2 - a*b + b*c - c^2 + sq
rt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b^2*c + 8*(b^2 - 4*a*c)*b^2*c - 10*sqrt(a^2 - a*b + b*c - c^2 +
sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*c^2 - 20*(b^2 - 4*a*c)*a*c^2 - 5*sqrt(a^2 - a*b + b*c - c^
2 + sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b*c^2 - 2*(b^2 - 4*a*c)*b*c^2)*e*abs(a - b + c))*(pi*floo
r(1/2*x/pi + 1/2) + arctan(2*sqrt(1/2)*tan(1/2*x)/sqrt((2*a - 2*c + sqrt(-4*(a + b + c)*(a - b + c) + 4*(a - c
)^2))/(a - b + c))))/(3*a^5*b^2 - 5*a^4*b^3 - 6*a^3*b^4 + 10*a^2*b^5 + 3*a*b^6 - 5*b^7 - 12*a^6*c + 20*a^5*b*c
+ 47*a^4*b^2*c - 60*a^3*b^3*c - 46*a^2*b^4*c + 40*a*b^5*c + 11*b^6*c - 92*a^5*c^2 + 80*a^4*b*c^2 + 182*a^3*b^
2*c^2 - 94*a^2*b^3*c^2 - 78*a*b^4*c^2 - 6*b^5*c^2 - 184*a^4*c^3 + 56*a^3*b*c^3 + 166*a^2*b^2*c^3 + 36*a*b^3*c^
3 - 6*b^4*c^3 - 120*a^3*c^4 - 48*a^2*b*c^4 + 23*a*b^2*c^4 + 11*b^3*c^4 + 4*a^2*c^5 - 44*a*b*c^5 - 5*b^2*c^5 +
20*a*c^6) - ((2*a^2*b^3 - 2*b^5 - 8*a^3*b*c - 12*a^2*b^2*c + 20*a*b^3*c + 4*b^4*c + 48*a^3*c^2 - 48*a^2*b*c^2
- 24*a*b^2*c^2 - 6*b^3*c^2 + 32*a^2*c^3 + 24*a*b*c^3 + 4*b^2*c^3 - 16*a*c^4 - 3*sqrt(a^2 - a*b + b*c - c^2 - s
qrt(b^2 - 4*a*c)*(a - b + c))*a^2*b^2 + 2*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^3 +
5*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*b^4 + 12*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 -
4*a*c)*(a - b + c))*a^3*c - 8*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b*c - 34*sqrt(a
^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^2*c - 6*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*
c)*(a - b + c))*b^3*c + 56*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*c^2 + 24*sqrt(a^2 -
a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b*c^2 + 5*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(
a - b + c))*b^2*c^2 - 20*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*c^3 + 3*sqrt(a^2 - a*b
+ b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^2*b - 2*(b^2 - 4*a*c)*a^2*b - 2*sqrt(a^2 - a*
b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b^2 - 5*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b
^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b^3 + 2*(b^2 - 4*a*c)*b^3 + 6*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2
- 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^2*c + 12*(b^2 - 4*a*c)*a^2*c + 10*sqrt(a^2 - a*b + b*c - c^2 - sqrt
(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b*c - 12*(b^2 - 4*a*c)*a*b*c - 4*sqrt(a^2 - a*b + b*c - c^2 - s
qrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b^2*c - 4*(b^2 - 4*a*c)*b^2*c + 28*sqrt(a^2 - a*b + b*c - c^2
- sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*c^2 + 8*(b^2 - 4*a*c)*a*c^2 + 7*sqrt(a^2 - a*b + b*c - c^
2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b*c^2 + 6*(b^2 - 4*a*c)*b*c^2 - 10*sqrt(a^2 - a*b + b*c -
c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*c^3 - 4*(b^2 - 4*a*c)*c^3)*d*abs(a - b + c) - (4*a^3*b
^2 - 6*a^2*b^3 - 4*a*b^4 + 6*b^5 - 16*a^4*c + 24*a^3*b*c + 40*a^2*b^2*c - 44*a*b^3*c - 8*b^4*c - 96*a^3*c^2 +
80*a^2*b*c^2 + 52*a*b^2*c^2 + 2*b^3*c^2 - 80*a^2*c^3 - 8*a*b*c^3 + 3*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4
*a*c)*(a - b + c))*a^2*b^2 - 2*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^3 - 5*sqrt(a^2
- a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*b^4 - 12*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a
- b + c))*a^3*c + 8*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*b*c + 34*sqrt(a^2 - a*b +
b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b^2*c + 6*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b +
c))*b^3*c - 56*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a^2*c^2 - 24*sqrt(a^2 - a*b + b*c
- c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*b*c^2 - 5*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))
*b^2*c^2 + 20*sqrt(a^2 - a*b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*a*c^3 + 6*sqrt(a^2 - a*b + b*c - c^2
- sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^3 - 4*(b^2 - 4*a*c)*a^3 - sqrt(a^2 - a*b + b*c - c^2 - s
qrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^2*b + 6*(b^2 - 4*a*c)*a^2*b - 12*sqrt(a^2 - a*b + b*c - c^2
- sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b^2 + 4*(b^2 - 4*a*c)*a*b^2 - 5*sqrt(a^2 - a*b + b*c - c^
2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b^3 - 6*(b^2 - 4*a*c)*b^3 + 28*sqrt(a^2 - a*b + b*c - c^2
- sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a^2*c - 24*(b^2 - 4*a*c)*a^2*c + 26*sqrt(a^2 - a*b + b*c -
c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*b*c + 20*(b^2 - 4*a*c)*a*b*c + 6*sqrt(a^2 - a*b + b*
c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b^2*c + 8*(b^2 - 4*a*c)*b^2*c - 10*sqrt(a^2 - a*b +
b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*a*c^2 - 20*(b^2 - 4*a*c)*a*c^2 - 5*sqrt(a^2 - a*
b + b*c - c^2 - sqrt(b^2 - 4*a*c)*(a - b + c))*sqrt(b^2 - 4*a*c)*b*c^2 - 2*(b^2 - 4*a*c)*b*c^2)*e*abs(a - b +
c))*(pi*floor(1/2*x/pi + 1/2) + arctan(2*sqrt(1/2)*tan(1/2*x)/sqrt((2*a - 2*c - sqrt(-4*(a + b + c)*(a - b + c
) + 4*(a - c)^2))/(a - b + c))))/(3*a^5*b^2 - 5*a^4*b^3 - 6*a^3*b^4 + 10*a^2*b^5 + 3*a*b^6 - 5*b^7 - 12*a^6*c
+ 20*a^5*b*c + 47*a^4*b^2*c - 60*a^3*b^3*c - 46*a^2*b^4*c + 40*a*b^5*c + 11*b^6*c - 92*a^5*c^2 + 80*a^4*b*c^2
+ 182*a^3*b^2*c^2 - 94*a^2*b^3*c^2 - 78*a*b^4*c^2 - 6*b^5*c^2 - 184*a^4*c^3 + 56*a^3*b*c^3 + 166*a^2*b^2*c^3 +
36*a*b^3*c^3 - 6*b^4*c^3 - 120*a^3*c^4 - 48*a^2*b*c^4 + 23*a*b^2*c^4 + 11*b^3*c^4 + 4*a^2*c^5 - 44*a*b*c^5 -
5*b^2*c^5 + 20*a*c^6)
Mupad [B] (verification not implemented)
Time = 44.54 (sec) , antiderivative size = 11781, normalized size of antiderivative = 47.89
\[
\int \frac {d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx=\text {Too large to display}
\]
[In]
int((d + e*cos(x))/(a + b*cos(x) + c*cos(x)^2),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)*(256*a^2*c^2*d - 32*b^4*e - 32*a^2*b^2*d - 32*a^2*b^2*e - 32*b^4*
d + 256*a^2*c^2*e - 32*b^2*c^2*d - 32*b^2*c^2*e + tan(x/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)*(64*a*b^4 + 256
*a*c^4 - 256*a^4*c - 64*b^4*c - 128*a^2*b^3 + 64*a^3*b^2 + 256*a^2*c^3 - 256*a^3*c^2 - 64*b^2*c^3 + 128*b^3*c^
2 + 192*a*b^2*c^2 - 192*a^2*b^2*c - 512*a*b*c^3 + 512*a^3*b*c) + 64*a*b^3*d + 64*a*b^3*e + 128*a*c^3*d + 128*a
^3*c*d + 128*a*c^3*e + 128*a^3*c*e + 64*b^3*c*d + 64*b^3*c*e - 256*a*b*c^2*d + 64*a*b^2*c*d - 256*a^2*b*c*d -
256*a*b*c^2*e + 64*a*b^2*c*e - 256*a^2*b*c*e) + tan(x/2)*(64*a^3*e^2 - 32*b^3*d^2 - 32*b^3*e^2 + 64*c^3*d^2 +
32*a*b^2*d^2 + 96*a*b^2*e^2 - 128*a^2*b*e^2 - 64*a^2*c*d^2 - 64*a*c^2*e^2 - 128*b*c^2*d^2 + 96*b^2*c*d^2 + 32*
b^2*c*e^2 + 64*a*b^2*d*e - 64*a^2*b*d*e + 256*a*c^2*d*e + 256*a^2*c*d*e - 64*b*c^2*d*e + 64*b^2*c*d*e - 384*a*
b*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)*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)*(256*a^2*c^2*d - 32*b^
4*e - 32*a^2*b^2*d - 32*a^2*b^2*e - 32*b^4*d + 256*a^2*c^2*e - 32*b^2*c^2*d - 32*b^2*c^2*e - tan(x/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)*(64*a*b^4 + 256*a*c^4 - 256*a^4*c - 64*b^4*c - 128*a^2*b^3 + 64*a^3*b^2 + 256*a^2*
c^3 - 256*a^3*c^2 - 64*b^2*c^3 + 128*b^3*c^2 + 192*a*b^2*c^2 - 192*a^2*b^2*c - 512*a*b*c^3 + 512*a^3*b*c) + 64
*a*b^3*d + 64*a*b^3*e + 128*a*c^3*d + 128*a^3*c*d + 128*a*c^3*e + 128*a^3*c*e + 64*b^3*c*d + 64*b^3*c*e - 256*
a*b*c^2*d + 64*a*b^2*c*d - 256*a^2*b*c*d - 256*a*b*c^2*e + 64*a*b^2*c*e - 256*a^2*b*c*e) - tan(x/2)*(64*a^3*e^
2 - 32*b^3*d^2 - 32*b^3*e^2 + 64*c^3*d^2 + 32*a*b^2*d^2 + 96*a*b^2*e^2 - 128*a^2*b*e^2 - 64*a^2*c*d^2 - 64*a*c
^2*e^2 - 128*b*c^2*d^2 + 96*b^2*c*d^2 + 32*b^2*c*e^2 + 64*a*b^2*d*e - 64*a^2*b*d*e + 256*a*c^2*d*e + 256*a^2*c
*d*e - 64*b*c^2*d*e + 64*b^2*c*d*e - 384*a*b*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)*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)*(256*a^2*c^2*d - 32*b^4*e - 32*a^2*b^2*d - 32*a^2*b^2*e - 32*b^4*d + 256*a^2*c^2*e - 32*b
^2*c^2*d - 32*b^2*c^2*e + tan(x/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)*(64*a*b^4 + 256*a*c^4 - 256*a^4*c - 64*
b^4*c - 128*a^2*b^3 + 64*a^3*b^2 + 256*a^2*c^3 - 256*a^3*c^2 - 64*b^2*c^3 + 128*b^3*c^2 + 192*a*b^2*c^2 - 192*
a^2*b^2*c - 512*a*b*c^3 + 512*a^3*b*c) + 64*a*b^3*d + 64*a*b^3*e + 128*a*c^3*d + 128*a^3*c*d + 128*a*c^3*e + 1
28*a^3*c*e + 64*b^3*c*d + 64*b^3*c*e - 256*a*b*c^2*d + 64*a*b^2*c*d - 256*a^2*b*c*d - 256*a*b*c^2*e + 64*a*b^2
*c*e - 256*a^2*b*c*e) + tan(x/2)*(64*a^3*e^2 - 32*b^3*d^2 - 32*b^3*e^2 + 64*c^3*d^2 + 32*a*b^2*d^2 + 96*a*b^2*
e^2 - 128*a^2*b*e^2 - 64*a^2*c*d^2 - 64*a*c^2*e^2 - 128*b*c^2*d^2 + 96*b^2*c*d^2 + 32*b^2*c*e^2 + 64*a*b^2*d*e
- 64*a^2*b*d*e + 256*a*c^2*d*e + 256*a^2*c*d*e - 64*b*c^2*d*e + 64*b^2*c*d*e - 384*a*b*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)*(256*a^2*c^2*d - 32*b^4*e - 32*a^2*b^2*d - 32*a^2
*b^2*e - 32*b^4*d + 256*a^2*c^2*e - 32*b^2*c^2*d - 32*b^2*c^2*e - tan(x/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)
*(64*a*b^4 + 256*a*c^4 - 256*a^4*c - 64*b^4*c - 128*a^2*b^3 + 64*a^3*b^2 + 256*a^2*c^3 - 256*a^3*c^2 - 64*b^2*
c^3 + 128*b^3*c^2 + 192*a*b^2*c^2 - 192*a^2*b^2*c - 512*a*b*c^3 + 512*a^3*b*c) + 64*a*b^3*d + 64*a*b^3*e + 128
*a*c^3*d + 128*a^3*c*d + 128*a*c^3*e + 128*a^3*c*e + 64*b^3*c*d + 64*b^3*c*e - 256*a*b*c^2*d + 64*a*b^2*c*d -
256*a^2*b*c*d - 256*a*b*c^2*e + 64*a*b^2*c*e - 256*a^2*b*c*e) - tan(x/2)*(64*a^3*e^2 - 32*b^3*d^2 - 32*b^3*e^2
+ 64*c^3*d^2 + 32*a*b^2*d^2 + 96*a*b^2*e^2 - 128*a^2*b*e^2 - 64*a^2*c*d^2 - 64*a*c^2*e^2 - 128*b*c^2*d^2 + 96
*b^2*c*d^2 + 32*b^2*c*e^2 + 64*a*b^2*d*e - 64*a^2*b*d*e + 256*a*c^2*d*e + 256*a^2*c*d*e - 64*b*c^2*d*e + 64*b^
2*c*d*e - 384*a*b*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) - 64*a^2*e^3 + 64*c^2*d^3 + 64*a^2*d*e^2 - 64*b^
2*d*e^2 + 64*b^2*d^2*e - 64*c^2*d^2*e + 64*a*b*e^3 + 64*a*c*d^3 - 64*a*c*e^3 - 64*b*c*d^3 - 64*a*b*d^2*e + 64*
a*c*d*e^2 - 64*a*c*d^2*e + 64*b*c*d*e^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)*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)*(256*a^2*c^2*d - 32*b^4*e - 32*a^2*b^2*d - 32*a^2*b^2*e - 32*b^4*d + 256*a^2*c^2*e - 32*b^2*c^
2*d - 32*b^2*c^2*e + tan(x/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)*(64*a*b^4 + 256*a*c^4 - 256*a^4*c - 64*b^4*c
- 128*a^2*b^3 + 64*a^3*b^2 + 256*a^2*c^3 - 256*a^3*c^2 - 64*b^2*c^3 + 128*b^3*c^2 + 192*a*b^2*c^2 - 192*a^2*b
^2*c - 512*a*b*c^3 + 512*a^3*b*c) + 64*a*b^3*d + 64*a*b^3*e + 128*a*c^3*d + 128*a^3*c*d + 128*a*c^3*e + 128*a^
3*c*e + 64*b^3*c*d + 64*b^3*c*e - 256*a*b*c^2*d + 64*a*b^2*c*d - 256*a^2*b*c*d - 256*a*b*c^2*e + 64*a*b^2*c*e
- 256*a^2*b*c*e) + tan(x/2)*(64*a^3*e^2 - 32*b^3*d^2 - 32*b^3*e^2 + 64*c^3*d^2 + 32*a*b^2*d^2 + 96*a*b^2*e^2 -
128*a^2*b*e^2 - 64*a^2*c*d^2 - 64*a*c^2*e^2 - 128*b*c^2*d^2 + 96*b^2*c*d^2 + 32*b^2*c*e^2 + 64*a*b^2*d*e - 64
*a^2*b*d*e + 256*a*c^2*d*e + 256*a^2*c*d*e - 64*b*c^2*d*e + 64*b^2*c*d*e - 384*a*b*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)*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)*(256*a^2*c^2*d - 32*b^4*e - 32*a^2*b^2*d - 32*a^2*b
^2*e - 32*b^4*d + 256*a^2*c^2*e - 32*b^2*c^2*d - 32*b^2*c^2*e - tan(x/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)*(
64*a*b^4 + 256*a*c^4 - 256*a^4*c - 64*b^4*c - 128*a^2*b^3 + 64*a^3*b^2 + 256*a^2*c^3 - 256*a^3*c^2 - 64*b^2*c^
3 + 128*b^3*c^2 + 192*a*b^2*c^2 - 192*a^2*b^2*c - 512*a*b*c^3 + 512*a^3*b*c) + 64*a*b^3*d + 64*a*b^3*e + 128*a
*c^3*d + 128*a^3*c*d + 128*a*c^3*e + 128*a^3*c*e + 64*b^3*c*d + 64*b^3*c*e - 256*a*b*c^2*d + 64*a*b^2*c*d - 25
6*a^2*b*c*d - 256*a*b*c^2*e + 64*a*b^2*c*e - 256*a^2*b*c*e) - tan(x/2)*(64*a^3*e^2 - 32*b^3*d^2 - 32*b^3*e^2 +
64*c^3*d^2 + 32*a*b^2*d^2 + 96*a*b^2*e^2 - 128*a^2*b*e^2 - 64*a^2*c*d^2 - 64*a*c^2*e^2 - 128*b*c^2*d^2 + 96*b
^2*c*d^2 + 32*b^2*c*e^2 + 64*a*b^2*d*e - 64*a^2*b*d*e + 256*a*c^2*d*e + 256*a^2*c*d*e - 64*b*c^2*d*e + 64*b^2*
c*d*e - 384*a*b*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)*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)*(256*a^2
*c^2*d - 32*b^4*e - 32*a^2*b^2*d - 32*a^2*b^2*e - 32*b^4*d + 256*a^2*c^2*e - 32*b^2*c^2*d - 32*b^2*c^2*e + tan
(x/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)*(64*a*b^4 + 256*a*c^4 - 256*a^4*c - 64*b^4*c - 128*a^2*b^3 + 64*a^3*
b^2 + 256*a^2*c^3 - 256*a^3*c^2 - 64*b^2*c^3 + 128*b^3*c^2 + 192*a*b^2*c^2 - 192*a^2*b^2*c - 512*a*b*c^3 + 512
*a^3*b*c) + 64*a*b^3*d + 64*a*b^3*e + 128*a*c^3*d + 128*a^3*c*d + 128*a*c^3*e + 128*a^3*c*e + 64*b^3*c*d + 64*
b^3*c*e - 256*a*b*c^2*d + 64*a*b^2*c*d - 256*a^2*b*c*d - 256*a*b*c^2*e + 64*a*b^2*c*e - 256*a^2*b*c*e) + tan(x
/2)*(64*a^3*e^2 - 32*b^3*d^2 - 32*b^3*e^2 + 64*c^3*d^2 + 32*a*b^2*d^2 + 96*a*b^2*e^2 - 128*a^2*b*e^2 - 64*a^2*
c*d^2 - 64*a*c^2*e^2 - 128*b*c^2*d^2 + 96*b^2*c*d^2 + 32*b^2*c*e^2 + 64*a*b^2*d*e - 64*a^2*b*d*e + 256*a*c^2*d
*e + 256*a^2*c*d*e - 64*b*c^2*d*e + 64*b^2*c*d*e - 384*a*b*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)*(256*a^2*c^2*d - 32*b^4*e - 32*a^2*b^2*d - 32*a^2*b^2*e - 32*b^4*d + 256*a^2*c
^2*e - 32*b^2*c^2*d - 32*b^2*c^2*e - tan(x/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)*(64*a*b^4 + 256*a*c^4 - 256*
a^4*c - 64*b^4*c - 128*a^2*b^3 + 64*a^3*b^2 + 256*a^2*c^3 - 256*a^3*c^2 - 64*b^2*c^3 + 128*b^3*c^2 + 192*a*b^2
*c^2 - 192*a^2*b^2*c - 512*a*b*c^3 + 512*a^3*b*c) + 64*a*b^3*d + 64*a*b^3*e + 128*a*c^3*d + 128*a^3*c*d + 128*
a*c^3*e + 128*a^3*c*e + 64*b^3*c*d + 64*b^3*c*e - 256*a*b*c^2*d + 64*a*b^2*c*d - 256*a^2*b*c*d - 256*a*b*c^2*e
+ 64*a*b^2*c*e - 256*a^2*b*c*e) - tan(x/2)*(64*a^3*e^2 - 32*b^3*d^2 - 32*b^3*e^2 + 64*c^3*d^2 + 32*a*b^2*d^2
+ 96*a*b^2*e^2 - 128*a^2*b*e^2 - 64*a^2*c*d^2 - 64*a*c^2*e^2 - 128*b*c^2*d^2 + 96*b^2*c*d^2 + 32*b^2*c*e^2 + 6
4*a*b^2*d*e - 64*a^2*b*d*e + 256*a*c^2*d*e + 256*a^2*c*d*e - 64*b*c^2*d*e + 64*b^2*c*d*e - 384*a*b*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) - 64*a^2*e^3 + 64*c^2*d^3 + 64*a^2*d*e^2 - 64*b^2*d*e^2 + 64*b^2*d^2*e - 64*c
^2*d^2*e + 64*a*b*e^3 + 64*a*c*d^3 - 64*a*c*e^3 - 64*b*c*d^3 - 64*a*b*d^2*e + 64*a*c*d*e^2 - 64*a*c*d^2*e + 64
*b*c*d*e^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)*2i