3.34 \(\int \frac{d-e x^2}{\frac{c d^2}{e^2}+b x^2+c x^4} \, dx\)
Optimal. Leaf size=134 \[ \frac{e^{3/2} \log \left (\sqrt{e} x \sqrt{2 c d-b e}+\sqrt{c} d+\sqrt{c} e x^2\right )}{2 \sqrt{c} \sqrt{2 c d-b e}}-\frac{e^{3/2} \log \left (-\sqrt{e} x \sqrt{2 c d-b e}+\sqrt{c} d+\sqrt{c} e x^2\right )}{2 \sqrt{c} \sqrt{2 c d-b e}} \]
[Out]
-(e^(3/2)*Log[Sqrt[c]*d - Sqrt[e]*Sqrt[2*c*d - b*e]*x + Sqrt[c]*e*x^2])/(2*Sqrt[c]*Sqrt[2*c*d - b*e]) + (e^(3/
2)*Log[Sqrt[c]*d + Sqrt[e]*Sqrt[2*c*d - b*e]*x + Sqrt[c]*e*x^2])/(2*Sqrt[c]*Sqrt[2*c*d - b*e])
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Rubi [A] time = 0.100717, antiderivative size = 134, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used =
{1164, 628} \[ \frac{e^{3/2} \log \left (\sqrt{e} x \sqrt{2 c d-b e}+\sqrt{c} d+\sqrt{c} e x^2\right )}{2 \sqrt{c} \sqrt{2 c d-b e}}-\frac{e^{3/2} \log \left (-\sqrt{e} x \sqrt{2 c d-b e}+\sqrt{c} d+\sqrt{c} e x^2\right )}{2 \sqrt{c} \sqrt{2 c d-b e}} \]
Antiderivative was successfully verified.
[In]
Int[(d - e*x^2)/((c*d^2)/e^2 + b*x^2 + c*x^4),x]
[Out]
-(e^(3/2)*Log[Sqrt[c]*d - Sqrt[e]*Sqrt[2*c*d - b*e]*x + Sqrt[c]*e*x^2])/(2*Sqrt[c]*Sqrt[2*c*d - b*e]) + (e^(3/
2)*Log[Sqrt[c]*d + Sqrt[e]*Sqrt[2*c*d - b*e]*x + Sqrt[c]*e*x^2])/(2*Sqrt[c]*Sqrt[2*c*d - b*e])
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}{\frac{c d^2}{e^2}+b x^2+c x^4} \, dx &=-\frac{e^{3/2} \int \frac{\frac{\sqrt{2 c d-b e}}{\sqrt{c} \sqrt{e}}+2 x}{-\frac{d}{e}-\frac{\sqrt{2 c d-b e} x}{\sqrt{c} \sqrt{e}}-x^2} \, dx}{2 \sqrt{c} \sqrt{2 c d-b e}}-\frac{e^{3/2} \int \frac{\frac{\sqrt{2 c d-b e}}{\sqrt{c} \sqrt{e}}-2 x}{-\frac{d}{e}+\frac{\sqrt{2 c d-b e} x}{\sqrt{c} \sqrt{e}}-x^2} \, dx}{2 \sqrt{c} \sqrt{2 c d-b e}}\\ &=-\frac{e^{3/2} \log \left (\sqrt{c} d-\sqrt{e} \sqrt{2 c d-b e} x+\sqrt{c} e x^2\right )}{2 \sqrt{c} \sqrt{2 c d-b e}}+\frac{e^{3/2} \log \left (\sqrt{c} d+\sqrt{e} \sqrt{2 c d-b e} x+\sqrt{c} e x^2\right )}{2 \sqrt{c} \sqrt{2 c d-b e}}\\ \end{align*}
Mathematica [A] time = 0.164182, size = 250, normalized size = 1.87 \[ \frac{e^{3/2} \left (-\frac{\left (\sqrt{b^2 e^2-4 c^2 d^2}-b e-2 c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}\right )}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}-\frac{\left (\sqrt{b^2 e^2-4 c^2 d^2}+b e+2 c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2 e^2-4 c^2 d^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d - e*x^2)/((c*d^2)/e^2 + b*x^2 + c*x^4),x]
[Out]
(e^(3/2)*(-(((-2*c*d - b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e - Sqrt[-4
*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e - Sqrt[-4*c^2*d^2 + b^2*e^2]]) - ((2*c*d + b*e + Sqrt[-4*c^2*d^2 + b^2*e^2])*A
rcTan[(Sqrt[2]*Sqrt[c]*Sqrt[e]*x)/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^2]]])/Sqrt[b*e + Sqrt[-4*c^2*d^2 + b^2*e^
2]]))/(Sqrt[2]*Sqrt[c]*Sqrt[-4*c^2*d^2 + b^2*e^2])
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Maple [B] time = 0.275, size = 582, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x)
[Out]
-1/2*e^4/(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2)*2^(1/2)/((-b*e^2+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*ar
ctanh(c*e*x*2^(1/2)/((-b*e^2+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))*b-e^3*c/(e^2*(b*e-2*c*d)*(b*e+2*c*
d))^(1/2)*2^(1/2)/((-b*e^2+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*arctanh(c*e*x*2^(1/2)/((-b*e^2+(e^2*(
b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))*d+1/2*e^2*2^(1/2)/((-b*e^2+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1
/2)*arctanh(c*e*x*2^(1/2)/((-b*e^2+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))-1/2*e^4/(e^2*(b*e-2*c*d)*(b*
e+2*c*d))^(1/2)*2^(1/2)/((b*e^2+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*arctan(c*e*x*2^(1/2)/((b*e^2+(e^
2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))*b-e^3*c/(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2)*2^(1/2)/((b*e^2+(e^2*(
b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*arctan(c*e*x*2^(1/2)/((b*e^2+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(
1/2))*d-1/2*e^2*2^(1/2)/((b*e^2+(e^2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2)*arctan(c*e*x*2^(1/2)/((b*e^2+(e^
2*(b*e-2*c*d)*(b*e+2*c*d))^(1/2))*c)^(1/2))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{e x^{2} - d}{c x^{4} + b x^{2} + \frac{c d^{2}}{e^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x, algorithm="maxima")
[Out]
-integrate((e*x^2 - d)/(c*x^4 + b*x^2 + c*d^2/e^2), x)
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Fricas [A] time = 1.40643, size = 494, normalized size = 3.69 \begin{align*} \left [\frac{1}{2} \, e \sqrt{\frac{e}{2 \, c^{2} d - b c e}} \log \left (\frac{c e^{2} x^{4} + c d^{2} +{\left (4 \, c d e - b e^{2}\right )} x^{2} + 2 \,{\left ({\left (2 \, c^{2} d e - b c e^{2}\right )} x^{3} +{\left (2 \, c^{2} d^{2} - b c d e\right )} x\right )} \sqrt{\frac{e}{2 \, c^{2} d - b c e}}}{c e^{2} x^{4} + b e^{2} x^{2} + c d^{2}}\right ), -e \sqrt{-\frac{e}{2 \, c^{2} d - b c e}} \arctan \left (c x \sqrt{-\frac{e}{2 \, c^{2} d - b c e}}\right ) + e \sqrt{-\frac{e}{2 \, c^{2} d - b c e}} \arctan \left (\frac{{\left (c e x^{3} -{\left (c d - b e\right )} x\right )} \sqrt{-\frac{e}{2 \, c^{2} d - b c e}}}{d}\right )\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e*x^2+d)/(c*d^2/e^2+b*x^2+c*x^4),x, algorithm="fricas")
[Out]
[1/2*e*sqrt(e/(2*c^2*d - b*c*e))*log((c*e^2*x^4 + c*d^2 + (4*c*d*e - b*e^2)*x^2 + 2*((2*c^2*d*e - b*c*e^2)*x^3
+ (2*c^2*d^2 - b*c*d*e)*x)*sqrt(e/(2*c^2*d - b*c*e)))/(c*e^2*x^4 + b*e^2*x^2 + c*d^2)), -e*sqrt(-e/(2*c^2*d -
b*c*e))*arctan(c*x*sqrt(-e/(2*c^2*d - b*c*e))) + e*sqrt(-e/(2*c^2*d - b*c*e))*arctan((c*e*x^3 - (c*d - b*e)*x
)*sqrt(-e/(2*c^2*d - b*c*e))/d)]
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Sympy [A] time = 0.795899, size = 158, normalized size = 1.18 \begin{align*} \frac{\sqrt{- \frac{e^{3}}{c \left (b e - 2 c d\right )}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (- b e \sqrt{- \frac{e^{3}}{c \left (b e - 2 c d\right )}} + 2 c d \sqrt{- \frac{e^{3}}{c \left (b e - 2 c d\right )}}\right )}{e^{2}} \right )}}{2} - \frac{\sqrt{- \frac{e^{3}}{c \left (b e - 2 c d\right )}} \log{\left (\frac{d}{e} + x^{2} + \frac{x \left (b e \sqrt{- \frac{e^{3}}{c \left (b e - 2 c d\right )}} - 2 c d \sqrt{- \frac{e^{3}}{c \left (b e - 2 c d\right )}}\right )}{e^{2}} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-e*x**2+d)/(c*d**2/e**2+b*x**2+c*x**4),x)
[Out]
sqrt(-e**3/(c*(b*e - 2*c*d)))*log(d/e + x**2 + x*(-b*e*sqrt(-e**3/(c*(b*e - 2*c*d))) + 2*c*d*sqrt(-e**3/(c*(b*
e - 2*c*d))))/e**2)/2 - sqrt(-e**3/(c*(b*e - 2*c*d)))*log(d/e + x**2 + x*(b*e*sqrt(-e**3/(c*(b*e - 2*c*d))) -
2*c*d*sqrt(-e**3/(c*(b*e - 2*c*d))))/e**2)/2
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Giac [C] time = 2.44361, size = 6884, normalized size = 51.37 \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)/(c*d^2/e^2+b*x^2+c*x^4),x, algorithm="giac")
[Out]
-1/2*(3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)
^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2*cosh(1/2*imag_part(arcsin(1/2*b*e/(c
*abs(d)))))^3*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d))))) - (4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2
*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cosh(1/2*imag_part(arcsin(1/
2*b*e/(c*abs(d)))))^3*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3 - 9*(4*c^3*(d^2)^(3/4)*d^2*e
^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2
*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2*e*sin(5/4*pi + 1/2
*real_part(arcsin(1/2*b*e/(c*abs(d)))))*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))))) + 3*(4*c^3*(d^2)^(3/4)
*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cosh(1/2*i
mag_part(arcsin(1/2*b*e/(c*abs(d)))))^2*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*sinh(1/2*i
mag_part(arcsin(1/2*b*e/(c*abs(d))))) + 9*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-
4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2*co
sh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*sinh(1
/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2 - 3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) -
sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))*e*sin(
5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2 - 3*(4
*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(
9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c
*abs(d)))))*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^3 + (4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(
3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*e*sin(5/4*pi + 1/2*real_part(arcsin(1/
2*b*e/(c*abs(d)))))^3*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^3 - (4*c^3*(d^2)^(1/4)*d^3*e^(11/2) - b^
2*c*(d^2)^(1/4)*d*e^(15/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(1/4)*d*e^(11/2))*cosh(1/2*imag_part(arc
sin(1/2*b*e/(c*abs(d)))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d))))) + (4*c^3*(d^2)^(1/4)*d^3*e^(
11/2) - b^2*c*(d^2)^(1/4)*d*e^(15/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(1/4)*d*e^(11/2))*sin(5/4*pi +
1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))))))*arctan(-((d^2)^(1
/4)*cos(5/4*pi + 1/2*arcsin(1/2*b*e/(c*abs(d))))*e^(-1/2) - x)*e^(1/2)/((d^2)^(1/4)*sin(5/4*pi + 1/2*arcsin(1/
2*b*e/(c*abs(d))))))/(4*c^4*d^4*e^4 - b^2*c^2*d^2*e^6) - 1/2*(3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(
3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*
b*e/(c*abs(d)))))^2*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^3*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*
b*e/(c*abs(d))))) - (4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^
4)*b*c*(d^2)^(3/4)*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^3*e*sin(1/4*pi + 1/2*real_part(arc
sin(1/2*b*e/(c*abs(d)))))^3 - 9*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*
e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2*cosh(1/2*ima
g_part(arcsin(1/2*b*e/(c*abs(d)))))^2*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*sinh(1/2*imag_
part(arcsin(1/2*b*e/(c*abs(d))))) + 3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^
2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2*e*sin(1/4*pi +
1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))))) + 9*(4*c^3*(d^2)
^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(
1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))*e*sin(1/
4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2 - 3*(4*c^3
*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2)
)*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*
sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2 - 3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13
/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs
(d)))))^2*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d
)))))^3 + (4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^
2)^(3/4)*e^(9/2))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*sinh(1/2*imag_part(arcsin(1/2*b*
e/(c*abs(d)))))^3 - (4*c^3*(d^2)^(1/4)*d^3*e^(11/2) - b^2*c*(d^2)^(1/4)*d*e^(15/2) - sqrt(-4*c^2*d^2*e^2 + b^2
*e^4)*b*c*(d^2)^(1/4)*d*e^(11/2))*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))*sin(1/4*pi + 1/2*real_part(a
rcsin(1/2*b*e/(c*abs(d))))) + (4*c^3*(d^2)^(1/4)*d^3*e^(11/2) - b^2*c*(d^2)^(1/4)*d*e^(15/2) - sqrt(-4*c^2*d^2
*e^2 + b^2*e^4)*b*c*(d^2)^(1/4)*d*e^(11/2))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*sinh(1/2*i
mag_part(arcsin(1/2*b*e/(c*abs(d))))))*arctan(-((d^2)^(1/4)*cos(1/4*pi + 1/2*arcsin(1/2*b*e/(c*abs(d))))*e^(-1
/2) - x)*e^(1/2)/((d^2)^(1/4)*sin(1/4*pi + 1/2*arcsin(1/2*b*e/(c*abs(d))))))/(4*c^4*d^4*e^4 - b^2*c^2*d^2*e^6)
+ 1/4*((4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)
^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*cosh(1/2*imag_part(arcsin(1/2*b*e/(c
*abs(d)))))^3*e - 3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^
4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*cosh(1/2*imag_part(arcsin(
1/2*b*e/(c*abs(d)))))^3*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2 - 3*(4*c^3*(d^2)^(3/4)*d^2
*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1
/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2*e*sinh(1/2*imag_
part(arcsin(1/2*b*e/(c*abs(d))))) + 9*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^
2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*cosh(1/2
*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2*sinh(1/2
*imag_part(arcsin(1/2*b*e/(c*abs(d))))) + 3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt
(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*
cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))*e*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2 - 9*(4*c^3
*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2)
)*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))*e*si
n(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2 - (4
*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(
9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*e*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))
)))^3 + 3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^
2)^(3/4)*e^(9/2))*cos(5/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*e*sin(5/4*pi + 1/2*real_part(arcsin(
1/2*b*e/(c*abs(d)))))^2*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^3 - (4*c^3*(d^2)^(1/4)*d^3*e^(11/2) -
b^2*c*(d^2)^(1/4)*d*e^(15/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(1/4)*d*e^(11/2))*cos(5/4*pi + 1/2*rea
l_part(arcsin(1/2*b*e/(c*abs(d)))))*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))))) + (4*c^3*(d^2)^(1/4)*d^3*e
^(11/2) - b^2*c*(d^2)^(1/4)*d*e^(15/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(1/4)*d*e^(11/2))*cos(5/4*pi
+ 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))))))*log(-2*(d^2)^(1
/4)*x*cos(5/4*pi + 1/2*arcsin(1/2*b*e/(c*abs(d))))*e^(-1/2) + x^2 + sqrt(d^2)*e^(-1))/(4*c^4*d^4*e^4 - b^2*c^2
*d^2*e^6) + 1/4*((4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*
b*c*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*cosh(1/2*imag_part(arcsin(1
/2*b*e/(c*abs(d)))))^3*e - 3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2
+ b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*cosh(1/2*imag_par
t(arcsin(1/2*b*e/(c*abs(d)))))^3*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2 - 3*(4*c^3*(d^2)^
(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(1
/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2*e*sinh(
1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))))) + 9*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - s
qrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))
*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2
*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))))) + 3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/
2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(
d)))))^3*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))*e*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^2 -
9*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(3/4
)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)
))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^2*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))
))^2 - (4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^
(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))^3*e*sinh(1/2*imag_part(arcsin(1/2*b*e/(
c*abs(d)))))^3 + 3*(4*c^3*(d^2)^(3/4)*d^2*e^(9/2) - b^2*c*(d^2)^(3/4)*e^(13/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4
)*b*c*(d^2)^(3/4)*e^(9/2))*cos(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*e*sin(1/4*pi + 1/2*real_par
t(arcsin(1/2*b*e/(c*abs(d)))))^2*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d)))))^3 - (4*c^3*(d^2)^(1/4)*d^3*e^
(11/2) - b^2*c*(d^2)^(1/4)*d*e^(15/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(1/4)*d*e^(11/2))*cos(1/4*pi
+ 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*cosh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))))) + (4*c^3*(d^2)^(1
/4)*d^3*e^(11/2) - b^2*c*(d^2)^(1/4)*d*e^(15/2) - sqrt(-4*c^2*d^2*e^2 + b^2*e^4)*b*c*(d^2)^(1/4)*d*e^(11/2))*c
os(1/4*pi + 1/2*real_part(arcsin(1/2*b*e/(c*abs(d)))))*sinh(1/2*imag_part(arcsin(1/2*b*e/(c*abs(d))))))*log(-2
*(d^2)^(1/4)*x*cos(1/4*pi + 1/2*arcsin(1/2*b*e/(c*abs(d))))*e^(-1/2) + x^2 + sqrt(d^2)*e^(-1))/(4*c^4*d^4*e^4
- b^2*c^2*d^2*e^6)