3.337 \(\int \frac{a+b x+c x^2}{d+e x^2+f x^4} \, dx\)

Optimal. Leaf size=209 \[ \frac{\left (c-\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e-\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e-\sqrt{e^2-4 d f}}}+\frac{\left (\frac{c e-2 a f}{\sqrt{e^2-4 d f}}+c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{\sqrt{e^2-4 d f}+e}}\right )}{\sqrt{2} \sqrt{f} \sqrt{\sqrt{e^2-4 d f}+e}}-\frac{b \tanh ^{-1}\left (\frac{e+2 f x^2}{\sqrt{e^2-4 d f}}\right )}{\sqrt{e^2-4 d f}} \]

[Out]

((c - (c*e - 2*a*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e - Sqrt[e^2 - 4*d*f]]])/(Sqrt[2]*Sqrt[
f]*Sqrt[e - Sqrt[e^2 - 4*d*f]]) + ((c + (c*e - 2*a*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e + S
qrt[e^2 - 4*d*f]]])/(Sqrt[2]*Sqrt[f]*Sqrt[e + Sqrt[e^2 - 4*d*f]]) - (b*ArcTanh[(e + 2*f*x^2)/Sqrt[e^2 - 4*d*f]
])/Sqrt[e^2 - 4*d*f]

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Rubi [A]  time = 0.370806, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {1673, 1166, 205, 12, 1107, 618, 206} \[ \frac{\left (c-\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e-\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e-\sqrt{e^2-4 d f}}}+\frac{\left (\frac{c e-2 a f}{\sqrt{e^2-4 d f}}+c\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{\sqrt{e^2-4 d f}+e}}\right )}{\sqrt{2} \sqrt{f} \sqrt{\sqrt{e^2-4 d f}+e}}-\frac{b \tanh ^{-1}\left (\frac{e+2 f x^2}{\sqrt{e^2-4 d f}}\right )}{\sqrt{e^2-4 d f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c - (c*e - 2*a*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e - Sqrt[e^2 - 4*d*f]]])/(Sqrt[2]*Sqrt[
f]*Sqrt[e - Sqrt[e^2 - 4*d*f]]) + ((c + (c*e - 2*a*f)/Sqrt[e^2 - 4*d*f])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e + S
qrt[e^2 - 4*d*f]]])/(Sqrt[2]*Sqrt[f]*Sqrt[e + Sqrt[e^2 - 4*d*f]]) - (b*ArcTanh[(e + 2*f*x^2)/Sqrt[e^2 - 4*d*f]
])/Sqrt[e^2 - 4*d*f]

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 618

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{a+b x+c x^2}{d+e x^2+f x^4} \, dx &=\int \frac{b x}{d+e x^2+f x^4} \, dx+\int \frac{a+c x^2}{d+e x^2+f x^4} \, dx\\ &=b \int \frac{x}{d+e x^2+f x^4} \, dx+\frac{1}{2} \left (c-\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \int \frac{1}{\frac{e}{2}-\frac{1}{2} \sqrt{e^2-4 d f}+f x^2} \, dx+\frac{1}{2} \left (c+\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \int \frac{1}{\frac{e}{2}+\frac{1}{2} \sqrt{e^2-4 d f}+f x^2} \, dx\\ &=\frac{\left (c-\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e-\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e-\sqrt{e^2-4 d f}}}+\frac{\left (c+\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e+\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e+\sqrt{e^2-4 d f}}}+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{d+e x+f x^2} \, dx,x,x^2\right )\\ &=\frac{\left (c-\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e-\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e-\sqrt{e^2-4 d f}}}+\frac{\left (c+\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e+\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e+\sqrt{e^2-4 d f}}}-b \operatorname{Subst}\left (\int \frac{1}{e^2-4 d f-x^2} \, dx,x,e+2 f x^2\right )\\ &=\frac{\left (c-\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e-\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e-\sqrt{e^2-4 d f}}}+\frac{\left (c+\frac{c e-2 a f}{\sqrt{e^2-4 d f}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e+\sqrt{e^2-4 d f}}}\right )}{\sqrt{2} \sqrt{f} \sqrt{e+\sqrt{e^2-4 d f}}}-\frac{b \tanh ^{-1}\left (\frac{e+2 f x^2}{\sqrt{e^2-4 d f}}\right )}{\sqrt{e^2-4 d f}}\\ \end{align*}

Mathematica [A]  time = 0.254728, size = 234, normalized size = 1.12 \[ \frac{\frac{\sqrt{2} \left (2 a f+c \left (\sqrt{e^2-4 d f}-e\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{e-\sqrt{e^2-4 d f}}}\right )}{\sqrt{f} \sqrt{e-\sqrt{e^2-4 d f}}}+\frac{\sqrt{2} \left (c \left (\sqrt{e^2-4 d f}+e\right )-2 a f\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{\sqrt{e^2-4 d f}+e}}\right )}{\sqrt{f} \sqrt{\sqrt{e^2-4 d f}+e}}+b \log \left (\sqrt{e^2-4 d f}-e-2 f x^2\right )-b \log \left (\sqrt{e^2-4 d f}+e+2 f x^2\right )}{2 \sqrt{e^2-4 d f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[2]*(2*a*f + c*(-e + Sqrt[e^2 - 4*d*f]))*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[e - Sqrt[e^2 - 4*d*f]]])/(Sqrt[
f]*Sqrt[e - Sqrt[e^2 - 4*d*f]]) + (Sqrt[2]*(-2*a*f + c*(e + Sqrt[e^2 - 4*d*f]))*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqr
t[e + Sqrt[e^2 - 4*d*f]]])/(Sqrt[f]*Sqrt[e + Sqrt[e^2 - 4*d*f]]) + b*Log[-e + Sqrt[e^2 - 4*d*f] - 2*f*x^2] - b
*Log[e + Sqrt[e^2 - 4*d*f] + 2*f*x^2])/(2*Sqrt[e^2 - 4*d*f])

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Maple [B]  time = 0.044, size = 616, normalized size = 3. \begin{align*}{\frac{b}{8\,df-2\,{e}^{2}}\sqrt{-4\,df+{e}^{2}}\ln \left ( 2\,f{x}^{2}+\sqrt{-4\,df+{e}^{2}}+e \right ) }+2\,{\frac{f\sqrt{2}cd}{ \left ( 4\,df-{e}^{2} \right ) \sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}\arctan \left ({\frac{fx\sqrt{2}}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}} \right ) }-{\frac{c\sqrt{2}{e}^{2}}{8\,df-2\,{e}^{2}}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}}+{\frac{f\sqrt{2}a}{4\,df-{e}^{2}}\sqrt{-4\,df+{e}^{2}}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}}-{\frac{c\sqrt{2}e}{8\,df-2\,{e}^{2}}\sqrt{-4\,df+{e}^{2}}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( e+\sqrt{-4\,df+{e}^{2}} \right ) f}}}}-{\frac{b}{8\,df-2\,{e}^{2}}\sqrt{-4\,df+{e}^{2}}\ln \left ( -2\,f{x}^{2}+\sqrt{-4\,df+{e}^{2}}-e \right ) }-2\,{\frac{f\sqrt{2}cd}{ \left ( 4\,df-{e}^{2} \right ) \sqrt{ \left ( \sqrt{-4\,df+{e}^{2}}-e \right ) f}}{\it Artanh} \left ({\frac{fx\sqrt{2}}{\sqrt{ \left ( \sqrt{-4\,df+{e}^{2}}-e \right ) f}}} \right ) }+{\frac{c\sqrt{2}{e}^{2}}{8\,df-2\,{e}^{2}}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( \sqrt{-4\,df+{e}^{2}}-e \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{-4\,df+{e}^{2}}-e \right ) f}}}}+{\frac{f\sqrt{2}a}{4\,df-{e}^{2}}\sqrt{-4\,df+{e}^{2}}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( \sqrt{-4\,df+{e}^{2}}-e \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{-4\,df+{e}^{2}}-e \right ) f}}}}-{\frac{c\sqrt{2}e}{8\,df-2\,{e}^{2}}\sqrt{-4\,df+{e}^{2}}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{ \left ( \sqrt{-4\,df+{e}^{2}}-e \right ) f}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{-4\,df+{e}^{2}}-e \right ) f}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{c x^{2} + b x + a}{f x^{4} + e x^{2} + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [C]  time = 3.09984, size = 9341, normalized size = 44.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_
part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*sin(5/4*
pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) - (4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e
^2)*e - (d*f^3)^(3/4)*e^2)*c*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*sin(5/4*pi + 1/2*real_p
art(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 - 9*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*
f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*cosh(1/2*imag_part(arcsin(
1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_
part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) + 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f
^3)^(3/4)*e^2)*c*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sin(5/4*pi + 1/2*real_part(arcsin(1
/2*sqrt(d*f)*e/(d*abs(f)))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) + 9*(4*(d*f^3)^(3/4)*d*
f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*
e/(d*abs(f)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(5/4*pi + 1/2*real_part(arcsin(1/
2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - 3*(4*(d*f^3)^(3/4)*d*f
- (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(
f)))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*
f)*e/(d*abs(f)))))^2 - 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(
5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*
e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 + (4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/
4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))
^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 - 2*(4*sqrt(d*f)*d*f^3 + sqrt(d*f)*sqrt(-4*d*f +
e^2)*f^2*e + sqrt(d*f)*f^2*e^2)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1/2*ima
g_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))
+ 4*(4*sqrt(d*f)*d*f^3 - sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e + sqrt(d*f)*f^2*e^2)*b*cos(5/4*pi + 1/2*real_part(
arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(5/4*pi + 1/2*
real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) - 2*(4*
sqrt(d*f)*d*f^3 - sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e - sqrt(d*f)*f^2*e^2)*b*cos(5/4*pi + 1/2*real_part(arcsin(
1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_pa
rt(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 + (4*(d*f^3)^(1/4)*d*f^3 - (d*f^3)^(1/4)*sqrt(-4*d*f + e^2)*f^2*e -
(d*f^3)^(1/4)*f^2*e^2)*a*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(5/4*pi + 1/2*real_part(ar
csin(1/2*sqrt(d*f)*e/(d*abs(f))))) - (4*(d*f^3)^(1/4)*d*f^3 - (d*f^3)^(1/4)*sqrt(-4*d*f + e^2)*f^2*e - (d*f^3)
^(1/4)*f^2*e^2)*a*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/
2*sqrt(d*f)*e/(d*abs(f))))))*arctan(-((d/f)^(1/4)*cos(5/4*pi + 1/2*arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))) - x)/((
d/f)^(1/4)*sin(5/4*pi + 1/2*arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))))/(4*d^2*f^4 - d*f^3*e^2) + 1/2*(3*(4*(d*f^3)^
(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sq
rt(d*f)*e/(d*abs(f)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*sin(1/4*pi + 1/2*real_part
(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) - (4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3
/4)*e^2)*c*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqr
t(d*f)*e/(d*abs(f)))))^3 - 9*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*
cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d
*abs(f)))))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sq
rt(d*f)*e/(d*abs(f))))) + 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*c
osh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*
abs(f)))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) + 9*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*
sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*
cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*a
bs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*s
qrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(1/4*pi
+ 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))
^2 - 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(1/4*pi + 1/2*real_
part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*si
nh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 + (4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e
^2)*e - (d*f^3)^(3/4)*e^2)*c*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*sinh(1/2*imag_p
art(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 + 2*(4*sqrt(d*f)*d*f^3 + sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e - sqrt(
d*f)*f^2*e^2)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1/2*imag_part(arcsin(1/2*
sqrt(d*f)*e/(d*abs(f)))))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) + 4*(4*sqrt(d*f)*d
*f^3 + sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e - sqrt(d*f)*f^2*e^2)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*
f)*e/(d*abs(f)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(1/4*pi + 1/2*real_part(arcsin(1
/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) - 2*(4*sqrt(d*f)*d*f^3 +
sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e + sqrt(d*f)*f^2*e^2)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d
*abs(f)))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt
(d*f)*e/(d*abs(f)))))^2 + (4*(d*f^3)^(1/4)*d*f^3 - (d*f^3)^(1/4)*sqrt(-4*d*f + e^2)*f^2*e - (d*f^3)^(1/4)*f^2*
e^2)*a*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)
*e/(d*abs(f))))) - (4*(d*f^3)^(1/4)*d*f^3 - (d*f^3)^(1/4)*sqrt(-4*d*f + e^2)*f^2*e - (d*f^3)^(1/4)*f^2*e^2)*a*
sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*a
bs(f))))))*arctan(-((d/f)^(1/4)*cos(1/4*pi + 1/2*arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))) - x)/((d/f)^(1/4)*sin(1/4
*pi + 1/2*arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))))/(4*d^2*f^4 - d*f^3*e^2) - 1/4*((4*(d*f^3)^(3/4)*d*f - (d*f^3)^
(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))
)))^3*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 - 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(
-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1/
2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)
))))^2 - 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*r
eal_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh
(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) + 9*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2
)*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1/2*imag_part(
arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(
1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) + 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)
*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*cosh(1/2*imag_part
(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - 9*(4*(d*f^3)
^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*s
qrt(d*f)*e/(d*abs(f)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(5/4*pi + 1/2*real_part(ar
csin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - (4*(d*f^3)^(3
/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt
(d*f)*e/(d*abs(f)))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 + 3*(4*(d*f^3)^(3/4)*d*f - (
d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*
abs(f)))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(1/2*imag_part(arcsin(1/2*sqr
t(d*f)*e/(d*abs(f)))))^3 + (4*sqrt(d*f)*d*f^3 + sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e + sqrt(d*f)*f^2*e^2)*b*cos(
5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs
(f)))))^2 - (4*sqrt(d*f)*d*f^3 + sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e - sqrt(d*f)*f^2*e^2)*b*cosh(1/2*imag_part(
arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - 2*(
4*sqrt(d*f)*d*f^3 + sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e - sqrt(d*f)*f^2*e^2)*b*cos(5/4*pi + 1/2*real_part(arcsi
n(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(a
rcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) + 2*(4*sqrt(d*f)*d*f^3 + sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e - sqrt(d*f)*f^
2*e^2)*b*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*
f)*e/(d*abs(f)))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) + (4*sqrt(d*f)*d*f^3 - sqrt(d*f)*
sqrt(-4*d*f + e^2)*f^2*e - sqrt(d*f)*f^2*e^2)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))
)^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - (4*sqrt(d*f)*d*f^3 - sqrt(d*f)*sqrt(-4*d*f + e
^2)*f^2*e + sqrt(d*f)*f^2*e^2)*b*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(1/2*im
ag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 + (4*(d*f^3)^(1/4)*d*f^3 - (d*f^3)^(1/4)*sqrt(-4*d*f + e^2)*f^2
*e - (d*f^3)^(1/4)*f^2*e^2)*a*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1/2*imag_pa
rt(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) - (4*(d*f^3)^(1/4)*d*f^3 - (d*f^3)^(1/4)*sqrt(-4*d*f + e^2)*f^2*e - (d
*f^3)^(1/4)*f^2*e^2)*a*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcs
in(1/2*sqrt(d*f)*e/(d*abs(f))))))*log(-2*x*(d/f)^(1/4)*cos(5/4*pi + 1/2*arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))) +
x^2 + sqrt(d/f))/(4*d^2*f^4 - d*f^3*e^2) - 1/4*((4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d
*f^3)^(3/4)*e^2)*c*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*cosh(1/2*imag_part(arcsin
(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 - 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)
*e^2)*c*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f
)*e/(d*abs(f)))))^3*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - 3*(4*(d*f^3)^(3/4)*d*f
- (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e
/(d*abs(f)))))^3*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(
d*f)*e/(d*abs(f))))) + 9*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(
1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f
)))))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(d
*f)*e/(d*abs(f))))) + 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(1
/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(
f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - 9*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt
(-4*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1
/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))
)))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - (4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4
*d*f + e^2)*e - (d*f^3)^(3/4)*e^2)*c*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3*sinh(1/
2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 + 3*(4*(d*f^3)^(3/4)*d*f - (d*f^3)^(3/4)*sqrt(-4*d*f + e^2)
*e - (d*f^3)^(3/4)*e^2)*c*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(1/4*pi + 1/2*rea
l_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^3 - (4*s
qrt(d*f)*d*f^3 + sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e - sqrt(d*f)*f^2*e^2)*b*cos(1/4*pi + 1/2*real_part(arcsin(1
/2*sqrt(d*f)*e/(d*abs(f)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - (4*sqrt(d*f)*d*f^3
+ sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e - sqrt(d*f)*f^2*e^2)*b*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f
)))))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2 - 2*(4*sqrt(d*f)*d*f^3 + sqrt(d*f)*s
qrt(-4*d*f + e^2)*f^2*e - sqrt(d*f)*f^2*e^2)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))
^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)
)))) + 2*(4*sqrt(d*f)*d*f^3 - sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e + sqrt(d*f)*f^2*e^2)*b*cosh(1/2*imag_part(arc
sin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(1/2*i
mag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))) - (4*sqrt(d*f)*d*f^3 - sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e - sqrt
(d*f)*f^2*e^2)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(1/2*imag_part(arcsin(1
/2*sqrt(d*f)*e/(d*abs(f)))))^2 + (4*sqrt(d*f)*d*f^3 + sqrt(d*f)*sqrt(-4*d*f + e^2)*f^2*e + sqrt(d*f)*f^2*e^2)*
b*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/
(d*abs(f)))))^2 + (4*(d*f^3)^(1/4)*d*f^3 - (d*f^3)^(1/4)*sqrt(-4*d*f + e^2)*f^2*e - (d*f^3)^(1/4)*f^2*e^2)*a*c
os(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*ab
s(f))))) - (4*(d*f^3)^(1/4)*d*f^3 - (d*f^3)^(1/4)*sqrt(-4*d*f + e^2)*f^2*e - (d*f^3)^(1/4)*f^2*e^2)*a*cos(1/4*
pi + 1/2*real_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(d*f)*e/(d*abs(f))))
))*log(-2*x*(d/f)^(1/4)*cos(1/4*pi + 1/2*arcsin(1/2*sqrt(d*f)*e/(d*abs(f)))) + x^2 + sqrt(d/f))/(4*d^2*f^4 - d
*f^3*e^2)