3.33 \(\int \frac{d-e x^2}{d^2-f x^2+e^2 x^4} \, dx\)
Optimal. Leaf size=70 \[ \frac{\log \left (x \sqrt{2 d e+f}+d+e x^2\right )}{2 \sqrt{2 d e+f}}-\frac{\log \left (-x \sqrt{2 d e+f}+d+e x^2\right )}{2 \sqrt{2 d e+f}} \]
[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 = 0.0472136, antiderivative size = 70, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, 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 = 0.133033, size = 190, normalized size = 2.71 \[ \frac{\frac{\left (-\sqrt{f^2-4 d^2 e^2}-2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}\right )}{\sqrt{\sqrt{f^2-4 d^2 e^2}-f}}-\frac{\left (\sqrt{f^2-4 d^2 e^2}-2 d e+f\right ) \tan ^{-1}\left (\frac{\sqrt{2} e x}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}\right )}{\sqrt{-\sqrt{f^2-4 d^2 e^2}-f}}}{\sqrt{2} \sqrt{f^2-4 d^2 e^2}} \]
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 - S
qrt[-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 = 0.171, size = 61, normalized size = 0.9 \begin{align*} -{\frac{1}{2}\ln \left ( -e{x}^{2}+x\sqrt{2\,de+f}-d \right ){\frac{1}{\sqrt{2\,de+f}}}}+{\frac{1}{2}\ln \left ( d+e{x}^{2}+x\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]
-1/2/(2*d*e+f)^(1/2)*ln(-e*x^2+x*(2*d*e+f)^(1/2)-d)+1/2*ln(d+e*x^2+x*(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 = 1.28894, size = 378, normalized size = 5.4 \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 = 0.600291, size = 112, normalized size = 1.6 \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 = 1.58361, size = 5387, normalized size = 76.96 \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(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(arcc
os(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_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(arc
cos(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(arc
cos(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_p
art(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_
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(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)/a
bs(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*si
nh(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) + s
qrt(-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(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))))*si
nh(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_pa
rt(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(a
rccos(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(ar
ccos(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*im
ag_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(arcco
s(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(arc
cos(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*r
eal_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(ar
ccos(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)/ab
s(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)