3.27 \(\int \frac{d+e x^2}{d^2+f x^2+e^2 x^4} \, dx\)
Optimal. Leaf size=82 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-f}+2 e x}{\sqrt{2 d e+f}}\right )}{\sqrt{2 d e+f}}-\frac{\tan ^{-1}\left (\frac{\sqrt{2 d e-f}-2 e x}{\sqrt{2 d e+f}}\right )}{\sqrt{2 d e+f}} \]
[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 = 0.109659, antiderivative size = 82, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, 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 = 0.111037, size = 181, normalized size = 2.21 \[ \frac{\frac{\left (\sqrt{f^2-4 d^2 e^2}+2 d e-f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}\right )}{\sqrt{f-\sqrt{f^2-4 d^2 e^2}}}+\frac{\left (\sqrt{f^2-4 d^2 e^2}-2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}\right )}{\sqrt{\sqrt{f^2-4 d^2 e^2}+f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}} \]
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 = 0.204, size = 71, 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 = 1.34911, size = 375, normalized size = 4.57 \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 = 0.42151, size = 122, normalized size = 1.49 \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 = 1.61013, size = 5779, normalized size = 70.48 \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(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*c
osh(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))))*s
inh(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) - sq
rt(-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*re
al_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*re
al_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(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))))^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(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(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*ar
csin(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(arcsi
n(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^(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(arcsi
n(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(arc
sin(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)/abs
(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*p
i + 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*arc
sin(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))))*co
sh(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*si
nh(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) - sqr
t(-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*i
mag_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_part
(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(a
rcsin(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_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(1/4*pi + 1/2*real_part(arcsin(1/2*f*e^(-1)/abs(d))))*cosh(1/2*imag_part(arc
sin(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_pa
rt(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(arc
sin(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)/a
bs(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))*c
os(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)