3.30 \(\int \frac{d-e x^2}{d^2+b x^2+e^2 x^4} \, dx\)
Optimal. Leaf size=78 \[ \frac{\log \left (x \sqrt{2 d e-b}+d+e x^2\right )}{2 \sqrt{2 d e-b}}-\frac{\log \left (-x \sqrt{2 d e-b}+d+e x^2\right )}{2 \sqrt{2 d e-b}} \]
[Out]
-Log[d - Sqrt[-b + 2*d*e]*x + e*x^2]/(2*Sqrt[-b + 2*d*e]) + Log[d + Sqrt[-b + 2*d*e]*x + e*x^2]/(2*Sqrt[-b + 2
*d*e])
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Rubi [A] time = 0.0515393, 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} \[ \frac{\log \left (x \sqrt{2 d e-b}+d+e x^2\right )}{2 \sqrt{2 d e-b}}-\frac{\log \left (-x \sqrt{2 d e-b}+d+e x^2\right )}{2 \sqrt{2 d e-b}} \]
Antiderivative was successfully verified.
[In]
Int[(d - e*x^2)/(d^2 + b*x^2 + e^2*x^4),x]
[Out]
-Log[d - Sqrt[-b + 2*d*e]*x + e*x^2]/(2*Sqrt[-b + 2*d*e]) + Log[d + Sqrt[-b + 2*d*e]*x + e*x^2]/(2*Sqrt[-b + 2
*d*e])
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+b x^2+e^2 x^4} \, dx &=-\frac{\int \frac{\frac{\sqrt{-b+2 d e}}{e}+2 x}{-\frac{d}{e}-\frac{\sqrt{-b+2 d e} x}{e}-x^2} \, dx}{2 \sqrt{-b+2 d e}}-\frac{\int \frac{\frac{\sqrt{-b+2 d e}}{e}-2 x}{-\frac{d}{e}+\frac{\sqrt{-b+2 d e} x}{e}-x^2} \, dx}{2 \sqrt{-b+2 d e}}\\ &=-\frac{\log \left (d-\sqrt{-b+2 d e} x+e x^2\right )}{2 \sqrt{-b+2 d e}}+\frac{\log \left (d+\sqrt{-b+2 d e} x+e x^2\right )}{2 \sqrt{-b+2 d e}}\\ \end{align*}
Mathematica [B] time = 0.120087, size = 182, normalized size = 2.33 \[ \frac{\frac{\left (-\sqrt{b^2-4 d^2 e^2}+b+2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}\right )}{\sqrt{b-\sqrt{b^2-4 d^2 e^2}}}-\frac{\left (\sqrt{b^2-4 d^2 e^2}+b+2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}\right )}{\sqrt{\sqrt{b^2-4 d^2 e^2}+b}}}{\sqrt{2} \sqrt{b^2-4 d^2 e^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d - e*x^2)/(d^2 + b*x^2 + e^2*x^4),x]
[Out]
(((b + 2*d*e - Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b - Sqrt[b^2 - 4*d^2*e^2]]])/Sqrt[b - Sqrt[b^2
- 4*d^2*e^2]] - ((b + 2*d*e + Sqrt[b^2 - 4*d^2*e^2])*ArcTan[(Sqrt[2]*e*x)/Sqrt[b + Sqrt[b^2 - 4*d^2*e^2]]])/S
qrt[b + Sqrt[b^2 - 4*d^2*e^2]])/(Sqrt[2]*Sqrt[b^2 - 4*d^2*e^2])
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Maple [A] time = 0.173, size = 88, normalized size = 1.1 \begin{align*}{\frac{1}{-4\,de+2\,b}\sqrt{2\,de-b}\ln \left ( -e{x}^{2}+x\sqrt{2\,de-b}-d \right ) }-{\frac{1}{-4\,de+2\,b}\sqrt{2\,de-b}\ln \left ( d+e{x}^{2}+x\sqrt{2\,de-b} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-e*x^2+d)/(e^2*x^4+b*x^2+d^2),x)
[Out]
1/(-4*d*e+2*b)*(2*d*e-b)^(1/2)*ln(-e*x^2+x*(2*d*e-b)^(1/2)-d)-1/(-4*d*e+2*b)*(2*d*e-b)^(1/2)*ln(d+e*x^2+x*(2*d
*e-b)^(1/2))
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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} + b 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+b*x^2+d^2),x, algorithm="maxima")
[Out]
-integrate((e*x^2 - d)/(e^2*x^4 + b*x^2 + d^2), x)
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Fricas [A] time = 1.32831, size = 378, normalized size = 4.85 \begin{align*} \left [\frac{\log \left (\frac{e^{2} x^{4} +{\left (4 \, d e - b\right )} x^{2} + d^{2} + 2 \,{\left (e x^{3} + d x\right )} \sqrt{2 \, d e - b}}{e^{2} x^{4} + b x^{2} + d^{2}}\right )}{2 \, \sqrt{2 \, d e - b}}, -\frac{\sqrt{-2 \, d e + b} \arctan \left (\frac{\sqrt{-2 \, d e + b} e x}{2 \, d e - b}\right ) - \sqrt{-2 \, d e + b} \arctan \left (\frac{{\left (e^{2} x^{3} -{\left (d e - b\right )} x\right )} \sqrt{-2 \, d e + b}}{2 \, d^{2} e - b d}\right )}{2 \, d e - b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e*x^2+d)/(e^2*x^4+b*x^2+d^2),x, algorithm="fricas")
[Out]
[1/2*log((e^2*x^4 + (4*d*e - b)*x^2 + d^2 + 2*(e*x^3 + d*x)*sqrt(2*d*e - b))/(e^2*x^4 + b*x^2 + d^2))/sqrt(2*d
*e - b), -(sqrt(-2*d*e + b)*arctan(sqrt(-2*d*e + b)*e*x/(2*d*e - b)) - sqrt(-2*d*e + b)*arctan((e^2*x^3 - (d*e
- b)*x)*sqrt(-2*d*e + b)/(2*d^2*e - b*d)))/(2*d*e - b)]
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Sympy [A] time = 0.574788, size = 121, normalized size = 1.55 \begin{align*} \frac{\sqrt{- \frac{1}{b - 2 d e}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (- b \sqrt{- \frac{1}{b - 2 d e}} + 2 d e \sqrt{- \frac{1}{b - 2 d e}}\right )}{e} \right )}}{2} - \frac{\sqrt{- \frac{1}{b - 2 d e}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (b \sqrt{- \frac{1}{b - 2 d e}} - 2 d e \sqrt{- \frac{1}{b - 2 d e}}\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+b*x**2+d**2),x)
[Out]
sqrt(-1/(b - 2*d*e))*log(d/e + x**2 + x*(-b*sqrt(-1/(b - 2*d*e)) + 2*d*e*sqrt(-1/(b - 2*d*e)))/e)/2 - sqrt(-1/
(b - 2*d*e))*log(d/e + x**2 + x*(b*sqrt(-1/(b - 2*d*e)) - 2*d*e*sqrt(-1/(b - 2*d*e)))/e)/2
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Giac [C] time = 1.63989, 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+b*x^2+d^2),x, algorithm="giac")
[Out]
-1/2*(3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*
cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*
e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d)))) - (4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(
9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*e*sin(
5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3 - 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/
2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*
cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*
sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d)))) + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - s
qrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*e*sin(5/4*pi +
1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d)))) + 9*(4*(d^2)^(
3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*r
eal_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))*e*sin(5/4*pi + 1/2*r
eal_part(arcsin(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2 - 3*(4*(d^2)^(3/4)*d
^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cosh(1/2*imag_part(arcsi
n(1/2*b*e^(-1)/abs(d))))*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsi
n(1/2*b*e^(-1)/abs(d))))^2 - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*
b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*e*sin(5/4*pi + 1/2*real_part
(arcsin(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3 + (4*(d^2)^(3/4)*d^2*e^(13/2
) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*e*sin(5/4*pi + 1/2*real_part(arcsi
n(1/2*b*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3 - (4*(d^2)^(1/4)*d^3*e^(15/2) -
b^2*(d^2)^(1/4)*d*e^(11/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(1/4)*d*e^(11/2))*cosh(1/2*imag_part(arcsin(1/2*b*
e^(-1)/abs(d))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d)))) + (4*(d^2)^(1/4)*d^3*e^(15/2) - b^2*
(d^2)^(1/4)*d*e^(11/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(1/4)*d*e^(11/2))*sin(5/4*pi + 1/2*real_part(arcsin(1/
2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d)))))*arctan(-((d^2)^(1/4)*cos(5/4*pi + 1/2*a
rcsin(1/2*b*e^(-1)/abs(d)))*e^(-1/2) - x)*e^(1/2)/((d^2)^(1/4)*sin(5/4*pi + 1/2*arcsin(1/2*b*e^(-1)/abs(d)))))
/(4*d^4*e^8 - b^2*d^2*e^6) - 1/2*(3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 +
b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcs
in(1/2*b*e^(-1)/abs(d))))^3*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d)))) - (4*(d^2)^(3/4)*d^2*e^
(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2
*b*e^(-1)/abs(d))))^3*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3 - 9*(4*(d^2)^(3/4)*d^2*e^(1
3/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcs
in(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*e*sin(1/4*pi + 1/2*real_part(ar
csin(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d)))) + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) -
b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1
)/abs(d))))^2*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsin(1/2*b*e^(
-1)/abs(d)))) + 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)
*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/ab
s(d))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(
d))))^2 - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/
2))*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d)))
)^3*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2 - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/
2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*
e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3
+ (4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*e*sin(
1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3 - (4
*(d^2)^(1/4)*d^3*e^(15/2) - b^2*(d^2)^(1/4)*d*e^(11/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(1/4)*d*e^(11/2))*cosh
(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d)))) + (4*(d^
2)^(1/4)*d^3*e^(15/2) - b^2*(d^2)^(1/4)*d*e^(11/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(1/4)*d*e^(11/2))*sin(1/4*
pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d)))))*arctan(-((d
^2)^(1/4)*cos(1/4*pi + 1/2*arcsin(1/2*b*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*b*e^(-1)/abs(d)))))/(4*d^4*e^8 - b^2*d^2*e^6) + 1/4*((4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^
(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))
^3*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*e - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9
/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*c
osh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2
- 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos
(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*e*s
inh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d)))) + 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sq
rt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*cosh(1/2*
imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*sinh(1/
2*imag_part(arcsin(1/2*b*e^(-1)/abs(d)))) + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*
d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*imag
_part(arcsin(1/2*b*e^(-1)/abs(d))))*e*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2 - 9*(4*(d^2)^(3/4)*d^
2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_par
t(arcsin(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))*e*sin(5/4*pi + 1/2*real_part(
arcsin(1/2*b*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2 - (4*(d^2)^(3/4)*d^2*e^(13/
2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin
(1/2*b*e^(-1)/abs(d))))^3*e*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3 + 3*(4*(d^2)^(3/4)*d^2*e^(13/2)
- b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1
/2*b*e^(-1)/abs(d))))*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arcsin(1
/2*b*e^(-1)/abs(d))))^3 - (4*(d^2)^(1/4)*d^3*e^(15/2) - b^2*(d^2)^(1/4)*d*e^(11/2) - sqrt(-4*d^2*e^2 + b^2)*b*
(d^2)^(1/4)*d*e^(11/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2
*b*e^(-1)/abs(d)))) + (4*(d^2)^(1/4)*d^3*e^(15/2) - b^2*(d^2)^(1/4)*d*e^(11/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2
)^(1/4)*d*e^(11/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*b*e
^(-1)/abs(d)))))*log(-2*(d^2)^(1/4)*x*cos(5/4*pi + 1/2*arcsin(1/2*b*e^(-1)/abs(d)))*e^(-1/2) + x^2 + sqrt(d^2)
*e^(-1))/(4*d^4*e^8 - b^2*d^2*e^6) + 1/4*((4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*
e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*imag_par
t(arcsin(1/2*b*e^(-1)/abs(d))))^3*e - 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^
2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(ar
csin(1/2*b*e^(-1)/abs(d))))^3*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2 - 3*(4*(d^2)^(3/4)*
d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_p
art(arcsin(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*e*sinh(1/2*imag_part(ar
csin(1/2*b*e^(-1)/abs(d)))) + 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)
*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2
*b*e^(-1)/abs(d))))^2*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arcsin(1
/2*b*e^(-1)/abs(d)))) + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^
2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arcsin(1/2*b*e
^(-1)/abs(d))))*e*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2 - 9*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^
2)^(3/4)*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1
)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/
abs(d))))^2*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^2 - (4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)
*e^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d)
)))^3*e*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))^3 + 3*(4*(d^2)^(3/4)*d^2*e^(13/2) - b^2*(d^2)^(3/4)*e
^(9/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d)))
)*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d)))
)^3 - (4*(d^2)^(1/4)*d^3*e^(15/2) - b^2*(d^2)^(1/4)*d*e^(11/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(1/4)*d*e^(11/
2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d))))
+ (4*(d^2)^(1/4)*d^3*e^(15/2) - b^2*(d^2)^(1/4)*d*e^(11/2) - sqrt(-4*d^2*e^2 + b^2)*b*(d^2)^(1/4)*d*e^(11/2))*
cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arcsin(1/2*b*e^(-1)/abs(d)))))*log
(-2*(d^2)^(1/4)*x*cos(1/4*pi + 1/2*arcsin(1/2*b*e^(-1)/abs(d)))*e^(-1/2) + x^2 + sqrt(d^2)*e^(-1))/(4*d^4*e^8
- b^2*d^2*e^6)