3.29 \(\int \frac{d+e x^2}{d^2-f x^2+e^2 x^4} \, dx\)
Optimal. Leaf size=86 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2 d e+f}+2 e x}{\sqrt{2 d e-f}}\right )}{\sqrt{2 d e-f}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 d e+f}-2 e x}{\sqrt{2 d e-f}}\right )}{\sqrt{2 d e-f}} \]
[Out]
-(ArcTan[(Sqrt[2*d*e + f] - 2*e*x)/Sqrt[2*d*e - f]]/Sqrt[2*d*e - f]) + ArcTan[(Sqrt[2*d*e + f] + 2*e*x)/Sqrt[2
*d*e - f]]/Sqrt[2*d*e - f]
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Rubi [A] time = 0.103915, antiderivative size = 86, 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, 204} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2 d e+f}+2 e x}{\sqrt{2 d e-f}}\right )}{\sqrt{2 d e-f}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 d e+f}-2 e x}{\sqrt{2 d e-f}}\right )}{\sqrt{2 d e-f}} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x^2)/(d^2 - f*x^2 + e^2*x^4),x]
[Out]
-(ArcTan[(Sqrt[2*d*e + f] - 2*e*x)/Sqrt[2*d*e - f]]/Sqrt[2*d*e - f]) + ArcTan[(Sqrt[2*d*e + f] + 2*e*x)/Sqrt[2
*d*e - f]]/Sqrt[2*d*e - f]
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 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{d+e x^2}{d^2-f x^2+e^2 x^4} \, dx &=\frac{\int \frac{1}{\frac{d}{e}-\frac{\sqrt{2 d e+f} x}{e}+x^2} \, dx}{2 e}+\frac{\int \frac{1}{\frac{d}{e}+\frac{\sqrt{2 d e+f} x}{e}+x^2} \, dx}{2 e}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{2 d e-f}{e^2}-x^2} \, dx,x,-\frac{\sqrt{2 d e+f}}{e}+2 x\right )}{e}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{2 d e-f}{e^2}-x^2} \, dx,x,\frac{\sqrt{2 d e+f}}{e}+2 x\right )}{e}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{2 d e+f}-2 e x}{\sqrt{2 d e-f}}\right )}{\sqrt{2 d e-f}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2 d e+f}+2 e x}{\sqrt{2 d e-f}}\right )}{\sqrt{2 d e-f}}\\ \end{align*}
Mathematica [B] time = 0.107449, size = 189, normalized size = 2.2 \[ \frac{\frac{\left (\sqrt{f^2-4 d^2 e^2}+2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}\right )}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}+\frac{\left (\sqrt{f^2-4 d^2 e^2}-2 d e-f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}\right )}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}} \]
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 = 0.175, size = 75, normalized size = 0.9 \begin{align*} -{\arctan \left ({ \left ( -2\,ex+\sqrt{2\,de+f} \right ){\frac{1}{\sqrt{2\,de-f}}}} \right ){\frac{1}{\sqrt{2\,de-f}}}}+{\arctan \left ({ \left ( 2\,ex+\sqrt{2\,de+f} \right ){\frac{1}{\sqrt{2\,de-f}}}} \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]
-arctan((-2*e*x+(2*d*e+f)^(1/2))/(2*d*e-f)^(1/2))/(2*d*e-f)^(1/2)+arctan((2*e*x+(2*d*e+f)^(1/2))/(2*d*e-f)^(1/
2))/(2*d*e-f)^(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} - 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 = 1.39025, size = 379, normalized size = 4.41 \begin{align*} \left [-\frac{\sqrt{-2 \, d e + f} \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 \,{\left (2 \, d e - f\right )}}, -\frac{\sqrt{2 \, d e - f} \arctan \left (-\frac{e x}{\sqrt{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*sqrt(-2*d*e + f)*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))/(2*d*e - f), -(sqrt(2*d*e - f)*arctan(-e*x/sqrt(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 = 0.424422, size = 121, 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 = 1.59795, size = 5387, normalized size = 62.64 \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))*c
os(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))^3*e*sin(1/2*
real_part(arccos(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(arccos(1/2*f*e^(-1)/abs(d))))^3*e*sin(1/2*real_part(arcco
s(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*f*e^(-1
)/abs(d))))^2*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(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))*c
osh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))^2*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^3*sinh(1/2
*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arcc
os(1/2*f*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(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(arccos(1/2*f*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d)))
)^3*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*e*sin(1/2
*real_part(arccos(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(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(1/2*real_part(arcc
os(1/2*f*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arccos(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(arccos(1/2*f
*e^(-1)/abs(d))))*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d)
)))*sinh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d)))))*arctan(((d^2)^(1/4)*cos(1/2*arccos(1/2*f*e^(-1)/abs(d)))
*e^(-1/2) + x)*e^(1/2)/((d^2)^(1/4)*sin(1/2*arccos(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(1/2*
real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))^3*e*sin(1/2*real_pa
rt(arccos(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(arccos(1/2*f*e^(-1)/abs(d))))^3*e*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d
))))^2*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(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(arccos(1/2*f*e^(-1)/abs(d))))^2*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^3*sinh(1/2*imag_p
art(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*cosh(1/2*imag_part(arccos(1/2*
f*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(1/2*f*e^(-1)/ab
s(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(arccos(1/2*f*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^3*sin
h(1/2*imag_part(arccos(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) + sq
rt(-4*d^2*e^2 + f^2)*(d^2)^(3/4)*f*e^(9/2))*cos(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*e*sin(1/2*real_p
art(arccos(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))^3 + (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))*e*sin(1/2*real_part(arccos(1/2*
f*e^(-1)/abs(d))))^3*sinh(1/2*imag_part(arccos(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(arccos(1/2*f*e^(-1)
/abs(d))))*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*sin
h(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d)))))*arctan(-((d^2)^(1/4)*cos(1/2*arccos(1/2*f*e^(-1)/abs(d)))*e^(-1
/2) - x)*e^(1/2)/((d^2)^(1/4)*sin(1/2*arccos(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(1/2*real_par
t(arccos(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/
2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))^3*e*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))^2
*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_
part(arccos(1/2*f*e^(-1)/abs(d))))^2*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(ar
ccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arccos(1/2*f*e^(-
1)/abs(d))))*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*
cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*sinh(1/2*
imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^3*e*sinh(1/2*imag_part(arc
cos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*f*e^(-
1)/abs(d))))^2*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d)
)))*cosh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*sinh(1/2*
imag_part(arccos(1/2*f*e^(-1)/abs(d)))))*log(2*(d^2)^(1/4)*x*cos(1/2*arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*ima
g_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos
(1/2*f*e^(-1)/abs(d))))^3*e*sin(1/2*real_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1
)/abs(d))))^3*cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))^2*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))^2*e*sin(1/2
*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arccos(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(1/2*real_part(arcc
os(1/2*f*e^(-1)/abs(d))))^3*cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))))*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d))
))*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arccos(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(1/2*re
al_part(arccos(1/2*f*e^(-1)/abs(d))))^3*e*sinh(1/2*imag_part(arccos(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(1/2*real_part(arc
cos(1/2*f*e^(-1)/abs(d))))*e*sin(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))^2*sinh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arccos(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(1/2*real_part(arccos(1/2*f*e^(-1)/abs(d))))*sinh(1/2*imag_part(arccos(1/2*f*e^(-1)/abs(d)))))*log(-
2*(d^2)^(1/4)*x*cos(1/2*arccos(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)