3.31 \(\int \frac{d-e x^2}{d^2+f x^2+e^2 x^4} \, dx\)
Optimal. Leaf size=78 \[ \frac{\log \left (x \sqrt{2 d e-f}+d+e x^2\right )}{2 \sqrt{2 d e-f}}-\frac{\log \left (-x \sqrt{2 d e-f}+d+e x^2\right )}{2 \sqrt{2 d e-f}} \]
[Out]
-Log[d - Sqrt[2*d*e - f]*x + e*x^2]/(2*Sqrt[2*d*e - f]) + Log[d + Sqrt[2*d*e - f]*x + e*x^2]/(2*Sqrt[2*d*e - f
])
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Rubi [A] time = 0.0480425, antiderivative size = 78, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used =
{1164, 628} \[ \frac{\log \left (x \sqrt{2 d e-f}+d+e x^2\right )}{2 \sqrt{2 d e-f}}-\frac{\log \left (-x \sqrt{2 d e-f}+d+e x^2\right )}{2 \sqrt{2 d e-f}} \]
Antiderivative was successfully verified.
[In]
Int[(d - e*x^2)/(d^2 + f*x^2 + e^2*x^4),x]
[Out]
-Log[d - Sqrt[2*d*e - f]*x + e*x^2]/(2*Sqrt[2*d*e - f]) + Log[d + Sqrt[2*d*e - f]*x + e*x^2]/(2*Sqrt[2*d*e - f
])
Rule 1164
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && !GtQ[b^2
- 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rubi steps
\begin{align*} \int \frac{d-e x^2}{d^2+f x^2+e^2 x^4} \, dx &=-\frac{\int \frac{\frac{\sqrt{2 d e-f}}{e}+2 x}{-\frac{d}{e}-\frac{\sqrt{2 d e-f} x}{e}-x^2} \, dx}{2 \sqrt{2 d e-f}}-\frac{\int \frac{\frac{\sqrt{2 d e-f}}{e}-2 x}{-\frac{d}{e}+\frac{\sqrt{2 d e-f} x}{e}-x^2} \, dx}{2 \sqrt{2 d e-f}}\\ &=-\frac{\log \left (d-\sqrt{2 d e-f} x+e x^2\right )}{2 \sqrt{2 d e-f}}+\frac{\log \left (d+\sqrt{2 d e-f} x+e x^2\right )}{2 \sqrt{2 d e-f}}\\ \end{align*}
Mathematica [B] time = 0.129162, size = 182, normalized size = 2.33 \[ \frac{\frac{\left (-\sqrt{f^2-4 d^2 e^2}+2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}\right )}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}-\frac{\left (\sqrt{f^2-4 d^2 e^2}+2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}\right )}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d - e*x^2)/(d^2 + f*x^2 + e^2*x^4),x]
[Out]
(((2*d*e + f - Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[f - Sqrt[-4*d^2*e^2 + f^2]]])/Sqrt[f - Sqrt[-
4*d^2*e^2 + f^2]] - ((2*d*e + f + Sqrt[-4*d^2*e^2 + f^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[f + Sqrt[-4*d^2*e^2 + f^2]
]])/Sqrt[f + Sqrt[-4*d^2*e^2 + f^2]])/(Sqrt[2]*Sqrt[-4*d^2*e^2 + f^2])
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Maple [A] time = 0.173, size = 69, normalized size = 0.9 \begin{align*}{\frac{1}{2}\ln \left ( d+e{x}^{2}+x\sqrt{2\,de-f} \right ){\frac{1}{\sqrt{2\,de-f}}}}-{\frac{1}{2}\ln \left ( -e{x}^{2}+x\sqrt{2\,de-f}-d \right ){\frac{1}{\sqrt{2\,de-f}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-e*x^2+d)/(e^2*x^4+f*x^2+d^2),x)
[Out]
1/2*ln(d+e*x^2+x*(2*d*e-f)^(1/2))/(2*d*e-f)^(1/2)-1/2/(2*d*e-f)^(1/2)*ln(-e*x^2+x*(2*d*e-f)^(1/2)-d)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{e x^{2} - d}{e^{2} x^{4} + f x^{2} + d^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e*x^2+d)/(e^2*x^4+f*x^2+d^2),x, algorithm="maxima")
[Out]
-integrate((e*x^2 - d)/(e^2*x^4 + f*x^2 + d^2), x)
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Fricas [A] time = 1.28469, size = 379, normalized size = 4.86 \begin{align*} \left [\frac{\log \left (\frac{e^{2} x^{4} +{\left (4 \, d e - f\right )} x^{2} + d^{2} + 2 \,{\left (e x^{3} + d x\right )} \sqrt{2 \, d e - f}}{e^{2} x^{4} + f x^{2} + d^{2}}\right )}{2 \, \sqrt{2 \, d e - f}}, \frac{\sqrt{-2 \, d e + f} \arctan \left (-\frac{\sqrt{-2 \, d e + f} e x}{2 \, d e - f}\right ) - \sqrt{-2 \, d e + f} \arctan \left (-\frac{{\left (e^{2} x^{3} -{\left (d e - f\right )} x\right )} \sqrt{-2 \, d e + f}}{2 \, d^{2} e - d f}\right )}{2 \, d e - f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e*x^2+d)/(e^2*x^4+f*x^2+d^2),x, algorithm="fricas")
[Out]
[1/2*log((e^2*x^4 + (4*d*e - f)*x^2 + d^2 + 2*(e*x^3 + d*x)*sqrt(2*d*e - f))/(e^2*x^4 + f*x^2 + d^2))/sqrt(2*d
*e - f), (sqrt(-2*d*e + f)*arctan(-sqrt(-2*d*e + f)*e*x/(2*d*e - f)) - sqrt(-2*d*e + f)*arctan(-(e^2*x^3 - (d*
e - f)*x)*sqrt(-2*d*e + f)/(2*d^2*e - d*f)))/(2*d*e - f)]
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Sympy [A] time = 0.599116, size = 110, normalized size = 1.41 \begin{align*} - \frac{\sqrt{\frac{1}{2 d e - f}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (- 2 d e \sqrt{\frac{1}{2 d e - f}} + f \sqrt{\frac{1}{2 d e - f}}\right )}{e} \right )}}{2} + \frac{\sqrt{\frac{1}{2 d e - f}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (2 d e \sqrt{\frac{1}{2 d e - f}} - f \sqrt{\frac{1}{2 d e - f}}\right )}{e} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e*x**2+d)/(e**2*x**4+f*x**2+d**2),x)
[Out]
-sqrt(1/(2*d*e - f))*log(d/e + x**2 + x*(-2*d*e*sqrt(1/(2*d*e - f)) + f*sqrt(1/(2*d*e - f)))/e)/2 + sqrt(1/(2*
d*e - f))*log(d/e + x**2 + x*(2*d*e*sqrt(1/(2*d*e - f)) - f*sqrt(1/(2*d*e - f)))/e)/2
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Giac [C] time = 1.63378, size = 5779, normalized size = 74.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e*x^2+d)/(e^2*x^4+f*x^2+d^2),x, algorithm="giac")
[Out]
-1/2*(3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*
cos(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*
e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d)))) - (4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(
9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*e*sin(
5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3 - 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/
2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*
cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*
sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d)))) + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - s
qrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*e*sin(5/4*pi +
1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d)))) + 9*(4*(d^2)^(
3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*r
eal_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))*e*sin(5/4*pi + 1/2*r
eal_part(arcsin(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2 - 3*(4*(d^2)^(3/4)*d
^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cosh(1/2*imag_part(arcsi
n(1/2*f*e^(-1)/abs(d))))*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsi
n(1/2*f*e^(-1)/abs(d))))^2 - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*
(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*e*sin(5/4*pi + 1/2*real_part
(arcsin(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3 + (4*(d^2)^(3/4)*d^2*e^(13/2
) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*e*sin(5/4*pi + 1/2*real_part(arcsi
n(1/2*f*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3 - (4*(d^2)^(1/4)*d^3*e^(15/2) -
(d^2)^(1/4)*d*f^2*e^(11/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(1/4)*d*f*e^(11/2))*cosh(1/2*imag_part(arcsin(1/2*f*
e^(-1)/abs(d))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d)))) + (4*(d^2)^(1/4)*d^3*e^(15/2) - (d^2
)^(1/4)*d*f^2*e^(11/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(1/4)*d*f*e^(11/2))*sin(5/4*pi + 1/2*real_part(arcsin(1/
2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d)))))*arctan(-((d^2)^(1/4)*cos(5/4*pi + 1/2*a
rcsin(1/2*f*e^(-1)/abs(d)))*e^(-1/2) - x)*e^(1/2)/((d^2)^(1/4)*sin(5/4*pi + 1/2*arcsin(1/2*f*e^(-1)/abs(d)))))
/(4*d^4*e^8 - d^2*f^2*e^6) - 1/2*(3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 +
f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcs
in(1/2*f*e^(-1)/abs(d))))^3*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d)))) - (4*(d^2)^(3/4)*d^2*e^
(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2
*f*e^(-1)/abs(d))))^3*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3 - 9*(4*(d^2)^(3/4)*d^2*e^(1
3/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcs
in(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*e*sin(1/4*pi + 1/2*real_part(ar
csin(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d)))) + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) -
(d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1
)/abs(d))))^2*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsin(1/2*f*e^(
-1)/abs(d)))) + 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f
*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/ab
s(d))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(
d))))^2 - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/
2))*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d)))
)^3*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2 - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/
2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*
e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3
+ (4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*e*sin(
1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3 - (4
*(d^2)^(1/4)*d^3*e^(15/2) - (d^2)^(1/4)*d*f^2*e^(11/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(1/4)*d*f*e^(11/2))*cosh
(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d)))) + (4*(d^
2)^(1/4)*d^3*e^(15/2) - (d^2)^(1/4)*d*f^2*e^(11/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(1/4)*d*f*e^(11/2))*sin(1/4*
pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d)))))*arctan(-((d
^2)^(1/4)*cos(1/4*pi + 1/2*arcsin(1/2*f*e^(-1)/abs(d)))*e^(-1/2) - x)*e^(1/2)/((d^2)^(1/4)*sin(1/4*pi + 1/2*ar
csin(1/2*f*e^(-1)/abs(d)))))/(4*d^4*e^8 - d^2*f^2*e^6) + 1/4*((4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^
(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))
^3*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*e - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9
/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*c
osh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2
- 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos
(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*e*s
inh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d)))) + 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sq
rt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*cosh(1/2*
imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*sinh(1/
2*imag_part(arcsin(1/2*f*e^(-1)/abs(d)))) + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*
d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*imag
_part(arcsin(1/2*f*e^(-1)/abs(d))))*e*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2 - 9*(4*(d^2)^(3/4)*d^
2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*real_par
t(arcsin(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))*e*sin(5/4*pi + 1/2*real_part(
arcsin(1/2*f*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2 - (4*(d^2)^(3/4)*d^2*e^(13/
2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin
(1/2*f*e^(-1)/abs(d))))^3*e*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3 + 3*(4*(d^2)^(3/4)*d^2*e^(13/2)
- (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1
/2*f*e^(-1)/abs(d))))*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arcsin(1
/2*f*e^(-1)/abs(d))))^3 - (4*(d^2)^(1/4)*d^3*e^(15/2) - (d^2)^(1/4)*d*f^2*e^(11/2) - sqrt(-4*d^2*e^2 + f^2)*(d
^2)^(1/4)*d*f*e^(11/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2
*f*e^(-1)/abs(d)))) + (4*(d^2)^(1/4)*d^3*e^(15/2) - (d^2)^(1/4)*d*f^2*e^(11/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^
(1/4)*d*f*e^(11/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*f*e
^(-1)/abs(d)))))*log(-2*(d^2)^(1/4)*x*cos(5/4*pi + 1/2*arcsin(1/2*f*e^(-1)/abs(d)))*e^(-1/2) + x^2 + sqrt(d^2)
*e^(-1))/(4*d^4*e^8 - d^2*f^2*e^6) + 1/4*((4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*
e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*imag_par
t(arcsin(1/2*f*e^(-1)/abs(d))))^3*e - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^
2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(ar
csin(1/2*f*e^(-1)/abs(d))))^3*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2 - 3*(4*(d^2)^(3/4)*
d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_p
art(arcsin(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*e*sinh(1/2*imag_part(ar
csin(1/2*f*e^(-1)/abs(d)))) + 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)
*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2
*f*e^(-1)/abs(d))))^2*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arcsin(1
/2*f*e^(-1)/abs(d)))) + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)
^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arcsin(1/2*f*e
^(-1)/abs(d))))*e*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2 - 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(
3/4)*f^2*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1
)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/
abs(d))))^2*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^2 - (4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2
*e^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d)
)))^3*e*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))^3 + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - (d^2)^(3/4)*f^2*e
^(9/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d)))
)*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d)))
)^3 - (4*(d^2)^(1/4)*d^3*e^(15/2) - (d^2)^(1/4)*d*f^2*e^(11/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(1/4)*d*f*e^(11/
2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d))))
+ (4*(d^2)^(1/4)*d^3*e^(15/2) - (d^2)^(1/4)*d*f^2*e^(11/2) - sqrt(-4*d^2*e^2 + f^2)*(d^2)^(1/4)*d*f*e^(11/2))*
cos(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*f*e^(-1)/abs(d)))))*log
(-2*(d^2)^(1/4)*x*cos(1/4*pi + 1/2*arcsin(1/2*f*e^(-1)/abs(d)))*e^(-1/2) + x^2 + sqrt(d^2)*e^(-1))/(4*d^4*e^8
- d^2*f^2*e^6)