3.525 \(\int \frac{e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx\)
Optimal. Leaf size=73 \[ \frac{\log \left (2 \sqrt{d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{d} \sqrt{f}}-\frac{\log \left (-2 \sqrt{d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{d} \sqrt{f}} \]
[Out]
-Log[e - 2*Sqrt[d]*Sqrt[f]*x + 2*f*x^2]/(4*Sqrt[d]*Sqrt[f]) + Log[e + 2*Sqrt[d]*Sqrt[f]*x + 2*f*x^2]/(4*Sqrt[d
]*Sqrt[f])
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Rubi [A] time = 0.0445108, antiderivative size = 73, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used =
{6, 1164, 628} \[ \frac{\log \left (2 \sqrt{d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{d} \sqrt{f}}-\frac{\log \left (-2 \sqrt{d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
[In]
Int[(e - 2*f*x^2)/(e^2 - 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4),x]
[Out]
-Log[e - 2*Sqrt[d]*Sqrt[f]*x + 2*f*x^2]/(4*Sqrt[d]*Sqrt[f]) + Log[e + 2*Sqrt[d]*Sqrt[f]*x + 2*f*x^2]/(4*Sqrt[d
]*Sqrt[f])
Rule 6
Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] && !FreeQ[v, x]
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{e-2 f x^2}{e^2-4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx &=\int \frac{e-2 f x^2}{e^2+(-4 d+4 e) f x^2+4 f^2 x^4} \, dx\\ &=-\frac{\int \frac{\frac{\sqrt{d}}{\sqrt{f}}+2 x}{-\frac{e}{2 f}-\frac{\sqrt{d} x}{\sqrt{f}}-x^2} \, dx}{4 \sqrt{d} \sqrt{f}}-\frac{\int \frac{\frac{\sqrt{d}}{\sqrt{f}}-2 x}{-\frac{e}{2 f}+\frac{\sqrt{d} x}{\sqrt{f}}-x^2} \, dx}{4 \sqrt{d} \sqrt{f}}\\ &=-\frac{\log \left (e-2 \sqrt{d} \sqrt{f} x+2 f x^2\right )}{4 \sqrt{d} \sqrt{f}}+\frac{\log \left (e+2 \sqrt{d} \sqrt{f} x+2 f x^2\right )}{4 \sqrt{d} \sqrt{f}}\\ \end{align*}
Mathematica [C] time = 0.138129, size = 233, normalized size = 3.19 \[ \frac{-\frac{\left (\sqrt{d} \sqrt{2 e-d}-i d+2 i e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{-i \sqrt{d} \sqrt{2 e-d}-d+e}}\right )}{\sqrt{-i \sqrt{d} \sqrt{2 e-d}-d+e}}-\frac{\left (\sqrt{d} \sqrt{2 e-d}+i d-2 i e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{i \sqrt{d} \sqrt{2 e-d}-d+e}}\right )}{\sqrt{i \sqrt{d} \sqrt{2 e-d}-d+e}}}{2 \sqrt{2} \sqrt{d} \sqrt{f} \sqrt{2 e-d}} \]
Antiderivative was successfully verified.
[In]
Integrate[(e - 2*f*x^2)/(e^2 - 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4),x]
[Out]
(-((((-I)*d + (2*I)*e + Sqrt[d]*Sqrt[-d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[-d + e - I*Sqrt[d]*Sqrt[-d + 2
*e]]])/Sqrt[-d + e - I*Sqrt[d]*Sqrt[-d + 2*e]]) - ((I*d - (2*I)*e + Sqrt[d]*Sqrt[-d + 2*e])*ArcTan[(Sqrt[2]*Sq
rt[f]*x)/Sqrt[-d + e + I*Sqrt[d]*Sqrt[-d + 2*e]]])/Sqrt[-d + e + I*Sqrt[d]*Sqrt[-d + 2*e]])/(2*Sqrt[2]*Sqrt[d]
*Sqrt[-d + 2*e]*Sqrt[f])
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Maple [B] time = 0.048, size = 394, normalized size = 5.4 \begin{align*}{\frac{f\sqrt{2}d}{4}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{-df+fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}{\frac{1}{\sqrt{-df+fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}}-{\frac{f\sqrt{2}e}{2}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{-df+fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}{\frac{1}{\sqrt{-df+fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}}-{\frac{\sqrt{2}}{4}\arctan \left ({fx\sqrt{2}{\frac{1}{\sqrt{-df+fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{-df+fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}}+{\frac{f\sqrt{2}d}{4}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{df-fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}{\frac{1}{\sqrt{df-fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}}-{\frac{f\sqrt{2}e}{2}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{df-fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}{\frac{1}{\sqrt{df-fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}}+{\frac{\sqrt{2}}{4}{\it Artanh} \left ({fx\sqrt{2}{\frac{1}{\sqrt{df-fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}} \right ){\frac{1}{\sqrt{df-fe+\sqrt{{f}^{2}d \left ( d-2\,e \right ) }}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-2*f*x^2+e)/(4*f^2*x^4-4*d*f*x^2+4*e*f*x^2+e^2),x)
[Out]
1/4*f/(f^2*d*(d-2*e))^(1/2)*2^(1/2)/(-d*f+f*e+(f^2*d*(d-2*e))^(1/2))^(1/2)*arctan(f*x*2^(1/2)/(-d*f+f*e+(f^2*d
*(d-2*e))^(1/2))^(1/2))*d-1/2*f/(f^2*d*(d-2*e))^(1/2)*2^(1/2)/(-d*f+f*e+(f^2*d*(d-2*e))^(1/2))^(1/2)*arctan(f*
x*2^(1/2)/(-d*f+f*e+(f^2*d*(d-2*e))^(1/2))^(1/2))*e-1/4*2^(1/2)/(-d*f+f*e+(f^2*d*(d-2*e))^(1/2))^(1/2)*arctan(
f*x*2^(1/2)/(-d*f+f*e+(f^2*d*(d-2*e))^(1/2))^(1/2))+1/4*f/(f^2*d*(d-2*e))^(1/2)*2^(1/2)/(d*f-f*e+(f^2*d*(d-2*e
))^(1/2))^(1/2)*arctanh(f*x*2^(1/2)/(d*f-f*e+(f^2*d*(d-2*e))^(1/2))^(1/2))*d-1/2*f/(f^2*d*(d-2*e))^(1/2)*2^(1/
2)/(d*f-f*e+(f^2*d*(d-2*e))^(1/2))^(1/2)*arctanh(f*x*2^(1/2)/(d*f-f*e+(f^2*d*(d-2*e))^(1/2))^(1/2))*e+1/4*2^(1
/2)/(d*f-f*e+(f^2*d*(d-2*e))^(1/2))^(1/2)*arctanh(f*x*2^(1/2)/(d*f-f*e+(f^2*d*(d-2*e))^(1/2))^(1/2))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{2 \, f x^{2} - e}{4 \, f^{2} x^{4} - 4 \, d f x^{2} + 4 \, e f x^{2} + e^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-2*f*x^2+e)/(4*f^2*x^4-4*d*f*x^2+4*e*f*x^2+e^2),x, algorithm="maxima")
[Out]
-integrate((2*f*x^2 - e)/(4*f^2*x^4 - 4*d*f*x^2 + 4*e*f*x^2 + e^2), x)
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Fricas [A] time = 1.45365, size = 327, normalized size = 4.48 \begin{align*} \left [\frac{\sqrt{d f} \log \left (\frac{4 \, f^{2} x^{4} + 4 \,{\left (d + e\right )} f x^{2} + e^{2} + 4 \,{\left (2 \, f x^{3} + e x\right )} \sqrt{d f}}{4 \, f^{2} x^{4} - 4 \,{\left (d - e\right )} f x^{2} + e^{2}}\right )}{4 \, d f}, -\frac{\sqrt{-d f} \arctan \left (\frac{\sqrt{-d f} x}{d}\right ) - \sqrt{-d f} \arctan \left (\frac{{\left (2 \, f x^{3} -{\left (2 \, d - e\right )} x\right )} \sqrt{-d f}}{d e}\right )}{2 \, d f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-2*f*x^2+e)/(4*f^2*x^4-4*d*f*x^2+4*e*f*x^2+e^2),x, algorithm="fricas")
[Out]
[1/4*sqrt(d*f)*log((4*f^2*x^4 + 4*(d + e)*f*x^2 + e^2 + 4*(2*f*x^3 + e*x)*sqrt(d*f))/(4*f^2*x^4 - 4*(d - e)*f*
x^2 + e^2))/(d*f), -1/2*(sqrt(-d*f)*arctan(sqrt(-d*f)*x/d) - sqrt(-d*f)*arctan((2*f*x^3 - (2*d - e)*x)*sqrt(-d
*f)/(d*e)))/(d*f)]
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Sympy [A] time = 0.635968, size = 63, normalized size = 0.86 \begin{align*} - \frac{\sqrt{\frac{1}{d f}} \log{\left (- d x \sqrt{\frac{1}{d f}} + \frac{e}{2 f} + x^{2} \right )}}{4} + \frac{\sqrt{\frac{1}{d f}} \log{\left (d x \sqrt{\frac{1}{d f}} + \frac{e}{2 f} + x^{2} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-2*f*x**2+e)/(4*f**2*x**4-4*d*f*x**2+4*e*f*x**2+e**2),x)
[Out]
-sqrt(1/(d*f))*log(-d*x*sqrt(1/(d*f)) + e/(2*f) + x**2)/4 + sqrt(1/(d*f))*log(d*x*sqrt(1/(d*f)) + e/(2*f) + x*
*2)/4
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Giac [C] time = 2.22849, size = 4694, normalized size = 64.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-2*f*x^2+e)/(4*f^2*x^4-4*d*f*x^2+4*e*f*x^2+e^2),x, algorithm="giac")
[Out]
1/8*(3*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*cosh(1/2*imag_part(arcco
s(d*f*e^(-1)/abs(f) - f/abs(f))))^3*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) - 4^(3/4)
*(f^2)^(3/4)*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*e^(3/2)*sin(1/2*real_part(arccos(d*f*
e^(-1)/abs(f) - f/abs(f))))^3 - 9*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))
^2*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs
(f) - f/abs(f))))*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) + 3*4^(3/4)*(f^2)^(3/4)*cosh(1/2*i
mag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f
))))^3*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) + 9*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arc
cos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sin(1/
2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2
- 3*4^(3/4)*(f^2)^(3/4)*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sin(1/2*real_part(a
rccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2 - 3*4^(3/4
)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*e^(3/2)*sin(1/2*real_part(arccos(d*f*
e^(-1)/abs(f) - f/abs(f))))*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3 + 4^(3/4)*(f^2)^(3/4)*
e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f)
- f/abs(f))))^3 - 2*4^(1/4)*(f^2)^(1/4)*f*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*s
in(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) + 2*4^(1/4)*(f^2)^(1/4)*f*e^(3/2)*sin(1/2*real_part(ar
ccos(d*f*e^(-1)/abs(f) - f/abs(f))))*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))))*arctan(4*(1/4)
^(3/4)*((1/4)^(1/4)*(f^(-2))^(1/4)*cos(1/2*arccos((d*f - f*e)*e^(-1)/abs(f)))*e^(1/2) + x)*(f^2)^(1/4)*e^(-1/2
)/sin(1/2*arccos((d*f - f*e)*e^(-1)/abs(f))))/((d^2 - 2*d*e)*f^2 + sqrt(d^2 - 2*d*e)*(d*f - f*e)*abs(f)) + 1/8
*(3*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*cosh(1/2*imag_part(arccos(d
*f*e^(-1)/abs(f) - f/abs(f))))^3*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) - 4^(3/4)*(f
^2)^(3/4)*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(
-1)/abs(f) - f/abs(f))))^3 - 9*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*
cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f)
- f/abs(f))))*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) + 3*4^(3/4)*(f^2)^(3/4)*cosh(1/2*imag
_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))
)^3*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) + 9*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos
(d*f*e^(-1)/abs(f) - f/abs(f))))^2*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sin(1/2*r
eal_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2 -
3*4^(3/4)*(f^2)^(3/4)*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sin(1/2*real_part(arcc
os(d*f*e^(-1)/abs(f) - f/abs(f))))^3*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2 - 3*4^(3/4)*(
f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(
-1)/abs(f) - f/abs(f))))*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3 + 4^(3/4)*(f^2)^(3/4)*e^(
3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) -
f/abs(f))))^3 - 2*4^(1/4)*(f^2)^(1/4)*f*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sin(
1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) + 2*4^(1/4)*(f^2)^(1/4)*f*e^(3/2)*sin(1/2*real_part(arcco
s(d*f*e^(-1)/abs(f) - f/abs(f))))*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))))*arctan(-4*(1/4)^(
3/4)*((1/4)^(1/4)*(f^(-2))^(1/4)*cos(1/2*arccos((d*f - f*e)*e^(-1)/abs(f)))*e^(1/2) - x)*(f^2)^(1/4)*e^(-1/2)/
sin(1/2*arccos((d*f - f*e)*e^(-1)/abs(f))))/((d^2 - 2*d*e)*f^2 + sqrt(d^2 - 2*d*e)*(d*f - f*e)*abs(f)) + 1/16*
(4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*cosh(1/2*imag_part(arccos(d*f*
e^(-1)/abs(f) - f/abs(f))))^3*e^(3/2) - 3*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/a
bs(f))))*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-
1)/abs(f) - f/abs(f))))^2 - 3*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*c
osh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*e^(3/2)*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f)
- f/abs(f)))) + 9*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*cosh(1/2*imag_
part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))
^2*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) + 3*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(
d*f*e^(-1)/abs(f) - f/abs(f))))^3*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sinh(1/2*i
mag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2 - 9*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/
abs(f) - f/abs(f))))*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sin(1/2*real_part(arcco
s(d*f*e^(-1)/abs(f) - f/abs(f))))^2*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2 - 4^(3/4)*(f^2
)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*e^(3/2)*sinh(1/2*imag_part(arccos(d*f*e^(-1
)/abs(f) - f/abs(f))))^3 + 3*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3
/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f
/abs(f))))^3 - 2*4^(1/4)*(f^2)^(1/4)*f*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*cosh(1/2*imag_
part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2) + 2*4^(1/4)*(f^2)^(1/4)*f*cos(1/2*real_part(arccos(d*f*e^(
-1)/abs(f) - f/abs(f))))*e^(3/2)*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))))*log(2*(1/4)^(1/4)*
(f^(-2))^(1/4)*x*cos(1/2*arccos((d*f - f*e)*e^(-1)/abs(f)))*e^(1/2) + x^2 + 1/2*sqrt(f^(-2))*e)/((d^2 - 2*d*e)
*f^2 + sqrt(d^2 - 2*d*e)*(d*f - f*e)*abs(f)) - 1/16*(4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/a
bs(f) - f/abs(f))))^3*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*e^(3/2) - 3*4^(3/4)*(f^2)^(3
/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/a
bs(f))))^3*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2 - 3*4^(3/4)*(f^2)^(3/4)*cos(1/2*
real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2
*e^(3/2)*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f)))) + 9*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(a
rccos(d*f*e^(-1)/abs(f) - f/abs(f))))*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*e^(3/2)*sin(
1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))
)) + 3*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3*cosh(1/2*imag_part(arcco
s(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2 - 9*4^(3
/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*cosh(1/2*imag_part(arccos(d*f*e^(-1)/
abs(f) - f/abs(f))))*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^2*sinh(1/2*imag_part(arc
cos(d*f*e^(-1)/abs(f) - f/abs(f))))^2 - 4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs
(f))))^3*e^(3/2)*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3 + 3*4^(3/4)*(f^2)^(3/4)*cos(1/2*r
eal_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sin(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))
))^2*sinh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))^3 - 2*4^(1/4)*(f^2)^(1/4)*f*cos(1/2*real_part(a
rccos(d*f*e^(-1)/abs(f) - f/abs(f))))*cosh(1/2*imag_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2) + 2*4^
(1/4)*(f^2)^(1/4)*f*cos(1/2*real_part(arccos(d*f*e^(-1)/abs(f) - f/abs(f))))*e^(3/2)*sinh(1/2*imag_part(arccos
(d*f*e^(-1)/abs(f) - f/abs(f)))))*log(-2*(1/4)^(1/4)*(f^(-2))^(1/4)*x*cos(1/2*arccos((d*f - f*e)*e^(-1)/abs(f)
))*e^(1/2) + x^2 + 1/2*sqrt(f^(-2))*e)/((d^2 - 2*d*e)*f^2 + sqrt(d^2 - 2*d*e)*(d*f - f*e)*abs(f))