3.28 \(\int \frac{d+e x^2}{d^2-b x^2+e^2 x^4} \, dx\)

Optimal. Leaf size=78 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b+2 d e}-2 e x}{\sqrt{b-2 d e}}\right )}{\sqrt{b-2 d e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b+2 d e}+2 e x}{\sqrt{b-2 d e}}\right )}{\sqrt{b-2 d e}} \]

[Out]

ArcTanh[(Sqrt[b + 2*d*e] - 2*e*x)/Sqrt[b - 2*d*e]]/Sqrt[b - 2*d*e] - ArcTanh[(Sqrt[b + 2*d*e] + 2*e*x)/Sqrt[b
- 2*d*e]]/Sqrt[b - 2*d*e]

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Rubi [A]  time = 0.0980389, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1161, 618, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b+2 d e}-2 e x}{\sqrt{b-2 d e}}\right )}{\sqrt{b-2 d e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b+2 d e}+2 e x}{\sqrt{b-2 d e}}\right )}{\sqrt{b-2 d e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[(Sqrt[b + 2*d*e] - 2*e*x)/Sqrt[b - 2*d*e]]/Sqrt[b - 2*d*e] - ArcTanh[(Sqrt[b + 2*d*e] + 2*e*x)/Sqrt[b
- 2*d*e]]/Sqrt[b - 2*d*e]

Rule 1161

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), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/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[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

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

Mathematica [B]  time = 0.109107, size = 189, normalized size = 2.42 \[ \frac{\frac{\left (\sqrt{b^2-4 d^2 e^2}+b+2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{-\sqrt{b^2-4 d^2 e^2}-b}}\right )}{\sqrt{-\sqrt{b^2-4 d^2 e^2}-b}}+\frac{\left (\sqrt{b^2-4 d^2 e^2}-b-2 d e\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{b^2-4 d^2 e^2}-b}}\right )}{\sqrt{\sqrt{b^2-4 d^2 e^2}-b}}}{\sqrt{2} \sqrt{b^2-4 d^2 e^2}} \]

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]]
])/Sqrt[-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 = 0.179, size = 75, normalized size = 1. \begin{align*}{\arctan \left ({ \left ( 2\,ex+\sqrt{2\,de+b} \right ){\frac{1}{\sqrt{2\,de-b}}}} \right ){\frac{1}{\sqrt{2\,de-b}}}}-{\arctan \left ({ \left ( -2\,ex+\sqrt{2\,de+b} \right ){\frac{1}{\sqrt{2\,de-b}}}} \right ){\frac{1}{\sqrt{2\,de-b}}}} \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/(2*d*e-b)^(1/2)*arctan((2*e*x+(2*d*e+b)^(1/2))/(2*d*e-b)^(1/2))-1/(2*d*e-b)^(1/2)*arctan((-2*e*x+(2*d*e+b)^(
1/2))/(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 = 1.34758, size = 375, normalized size = 4.81 \begin{align*} \left [-\frac{\sqrt{-2 \, d e + b} \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 \,{\left (2 \, d e - b\right )}}, \frac{\sqrt{2 \, d e - b} \arctan \left (\frac{e x}{\sqrt{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*sqrt(-2*d*e + b)*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))/(2*d*e - b), (sqrt(2*d*e - b)*arctan(e*x/sqrt(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 = 0.45524, size = 110, normalized size = 1.41 \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 = 1.62429, size = 5387, normalized size = 69.06 \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))*c
os(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))^3*e*sin(1/2*
real_part(arccos(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(arccos(1/2*b*e^(-1)/abs(d))))^3*e*sin(1/2*real_part(arcco
s(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*b*e^(-1
)/abs(d))))^2*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(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))*c
osh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))^2*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^3*sinh(1/2
*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcc
os(1/2*b*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(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(arccos(1/2*b*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d)))
)^3*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*e*sin(1/2
*real_part(arccos(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(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(1/2*real_part(arcc
os(1/2*b*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arccos(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(arccos(1/2*b
*e^(-1)/abs(d))))*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d)
)))*sinh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d)))))*arctan(((d^2)^(1/4)*cos(1/2*arccos(1/2*b*e^(-1)/abs(d)))
*e^(-1/2) + x)*e^(1/2)/((d^2)^(1/4)*sin(1/2*arccos(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(1/2*
real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))^3*e*sin(1/2*real_pa
rt(arccos(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(arccos(1/2*b*e^(-1)/abs(d))))^3*e*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d
))))^2*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(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(arccos(1/2*b*e^(-1)/abs(d))))^2*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^3*sinh(1/2*imag_p
art(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*
b*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(1/2*b*e^(-1)/ab
s(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(arccos(1/2*b*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^3*sin
h(1/2*imag_part(arccos(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) + sq
rt(-4*d^2*e^2 + b^2)*b*(d^2)^(3/4)*e^(9/2))*cos(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*e*sin(1/2*real_p
art(arccos(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))^3 + (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))*e*sin(1/2*real_part(arccos(1/2*
b*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arccos(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(arccos(1/2*b*e^(-1)
/abs(d))))*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*sin
h(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d)))))*arctan(-((d^2)^(1/4)*cos(1/2*arccos(1/2*b*e^(-1)/abs(d)))*e^(-1
/2) - x)*e^(1/2)/((d^2)^(1/4)*sin(1/2*arccos(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(1/2*real_par
t(arccos(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/
2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))^3*e*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))^2
*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_
part(arccos(1/2*b*e^(-1)/abs(d))))^2*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(ar
ccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arccos(1/2*b*e^(-
1)/abs(d))))*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*
cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*sinh(1/2*
imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^3*e*sinh(1/2*imag_part(arc
cos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*b*e^(-
1)/abs(d))))^2*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d)
)))*cosh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*sinh(1/2*
imag_part(arccos(1/2*b*e^(-1)/abs(d)))))*log(2*(d^2)^(1/4)*x*cos(1/2*arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*ima
g_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos
(1/2*b*e^(-1)/abs(d))))^3*e*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1
)/abs(d))))^3*cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))^2*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))^2*e*sin(1/2
*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arccos(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(1/2*real_part(arcc
os(1/2*b*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))))*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d))
))*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arccos(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(1/2*re
al_part(arccos(1/2*b*e^(-1)/abs(d))))^3*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arc
cos(1/2*b*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*b*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(1/2*b*e^(-1)/abs(d)))))*log(-
2*(d^2)^(1/4)*x*cos(1/2*arccos(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)