3.524 \(\int \frac{e-2 f x^2}{e^2+4 d f x^2+4 e f x^2+4 f^2 x^4} \, dx\)
Optimal. Leaf size=81 \[ \frac{\log \left (2 \sqrt{-d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{-d} \sqrt{f}}-\frac{\log \left (-2 \sqrt{-d} \sqrt{f} x+e+2 f x^2\right )}{4 \sqrt{-d} \sqrt{f}} \]
[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*Sqr
t[-d]*Sqrt[f])
________________________________________________________________________________________
Rubi [A] time = 0.0583478, antiderivative size = 81, 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*Sqr
t[-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+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 [B] time = 0.120562, size = 191, normalized size = 2.36 \[ \frac{-\frac{\left (\sqrt{d} \sqrt{d+2 e}-d-2 e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{-\sqrt{d} \sqrt{d+2 e}+d+e}}\right )}{\sqrt{-\sqrt{d} \sqrt{d+2 e}+d+e}}-\frac{\left (\sqrt{d} \sqrt{d+2 e}+d+2 e\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{f} x}{\sqrt{\sqrt{d} \sqrt{d+2 e}+d+e}}\right )}{\sqrt{\sqrt{d} \sqrt{d+2 e}+d+e}}}{2 \sqrt{2} \sqrt{d} \sqrt{f} \sqrt{d+2 e}} \]
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]
(-(((-d - 2*e + Sqrt[d]*Sqrt[d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[d + e - Sqrt[d]*Sqrt[d + 2*e]]])/Sqrt[d
+ e - Sqrt[d]*Sqrt[d + 2*e]]) - ((d + 2*e + Sqrt[d]*Sqrt[d + 2*e])*ArcTan[(Sqrt[2]*Sqrt[f]*x)/Sqrt[d + e + Sq
rt[d]*Sqrt[d + 2*e]]])/Sqrt[d + e + Sqrt[d]*Sqrt[d + 2*e]])/(2*Sqrt[2]*Sqrt[d]*Sqrt[d + 2*e]*Sqrt[f])
________________________________________________________________________________________
Maple [B] time = 0.051, size = 394, normalized size = 4.9 \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))
________________________________________________________________________________________
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)
________________________________________________________________________________________
Fricas [A] time = 1.41653, size = 325, normalized size = 4.01 \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)]
________________________________________________________________________________________
Sympy [A] time = 0.617299, size = 70, 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
________________________________________________________________________________________
Giac [C] time = 2.26747, size = 4845, normalized size = 59.81 \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(arcc
os(-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)*c
osh(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*r
eal_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*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)*si
n(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))))^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*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))))^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(a
rccos(-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(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)))))*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_pa
rt(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*real_part(arccos(-d*f*e^(-1)/abs(f) - f/abs(f))))*sinh(1/2*imag_par
t(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(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))))^2 - 3*4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arccos(-d*f*e^(-1)/abs(f) - f/ab
s(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_pa
rt(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(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)))))*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/abs(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*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(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*cos
h(1/2*imag_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))))^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(arccos(-d*f*e^(-1)/abs(f) - f/abs(f))))^2 - 4^(3/4)*(f^2)^(3/4)*cos(1/2*real_part(arcco
s(-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)/a
bs(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)/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_pa
rt(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))))^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(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_pa
rt(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*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(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*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_par
t(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))