3.36 \(\int \frac{d+e x^2}{b x^2+c (\frac{d^2}{e^2}+x^4)} \, dx\)
Optimal. Leaf size=130 \[ \frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}+2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right )}{\sqrt{c} \sqrt{b e+2 c d}}-\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}-2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right )}{\sqrt{c} \sqrt{b e+2 c d}} \]
[Out]
-((e^(3/2)*ArcTan[(Sqrt[2*c*d - b*e] - 2*Sqrt[c]*Sqrt[e]*x)/Sqrt[2*c*d + b*e]])/(Sqrt[c]*Sqrt[2*c*d + b*e])) +
(e^(3/2)*ArcTan[(Sqrt[2*c*d - b*e] + 2*Sqrt[c]*Sqrt[e]*x)/Sqrt[2*c*d + b*e]])/(Sqrt[c]*Sqrt[2*c*d + b*e])
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Rubi [A] time = 0.130766, antiderivative size = 130, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used =
{1990, 1161, 618, 204} \[ \frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}+2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right )}{\sqrt{c} \sqrt{b e+2 c d}}-\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}-2 \sqrt{c} \sqrt{e} x}{\sqrt{b e+2 c d}}\right )}{\sqrt{c} \sqrt{b e+2 c d}} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x^2)/(b*x^2 + c*(d^2/e^2 + x^4)),x]
[Out]
-((e^(3/2)*ArcTan[(Sqrt[2*c*d - b*e] - 2*Sqrt[c]*Sqrt[e]*x)/Sqrt[2*c*d + b*e]])/(Sqrt[c]*Sqrt[2*c*d + b*e])) +
(e^(3/2)*ArcTan[(Sqrt[2*c*d - b*e] + 2*Sqrt[c]*Sqrt[e]*x)/Sqrt[2*c*d + b*e]])/(Sqrt[c]*Sqrt[2*c*d + b*e])
Rule 1990
Int[(u_)^(q_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^q*ExpandToSum[v, x]^p, x] /; FreeQ[{p, q}, x] &&
BinomialQ[u, x] && TrinomialQ[v, x] && !(BinomialMatchQ[u, x] && TrinomialMatchQ[v, x])
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}{b x^2+c \left (\frac{d^2}{e^2}+x^4\right )} \, dx &=\int \frac{d+e x^2}{\frac{c d^2}{e^2}+b x^2+c x^4} \, dx\\ &=\frac{e \int \frac{1}{\frac{d}{e}-\frac{\sqrt{2 c d-b e} x}{\sqrt{c} \sqrt{e}}+x^2} \, dx}{2 c}+\frac{e \int \frac{1}{\frac{d}{e}+\frac{\sqrt{2 c d-b e} x}{\sqrt{c} \sqrt{e}}+x^2} \, dx}{2 c}\\ &=-\frac{e \operatorname{Subst}\left (\int \frac{1}{-\frac{b}{c}-\frac{2 d}{e}-x^2} \, dx,x,-\frac{\sqrt{2 c d-b e}}{\sqrt{c} \sqrt{e}}+2 x\right )}{c}-\frac{e \operatorname{Subst}\left (\int \frac{1}{-\frac{b}{c}-\frac{2 d}{e}-x^2} \, dx,x,\frac{\sqrt{2 c d-b e}}{\sqrt{c} \sqrt{e}}+2 x\right )}{c}\\ &=-\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}-2 \sqrt{c} \sqrt{e} x}{\sqrt{2 c d+b e}}\right )}{\sqrt{c} \sqrt{2 c d+b e}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2 c d-b e}+2 \sqrt{c} \sqrt{e} x}{\sqrt{2 c d+b e}}\right )}{\sqrt{c} \sqrt{2 c d+b e}}\\ \end{align*}
Mathematica [A] time = 0.0444802, size = 248, normalized size = 1.91 \[ \frac{e^{3/2} \left (\frac{\left (\sqrt{b^2 e^2-4 c^2 d^2}-b e+2 c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}\right )}{\sqrt{b e-\sqrt{b^2 e^2-4 c^2 d^2}}}+\frac{\left (\sqrt{b^2 e^2-4 c^2 d^2}+b e-2 c d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{e} x}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt{\sqrt{b^2 e^2-4 c^2 d^2}+b e}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2 e^2-4 c^2 d^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x^2)/(b*x^2 + c*(d^2/e^2 + 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])*ArcT
an[(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 = 0.212, size = 582, normalized size = 4.5 \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)/(b*x^2+c*(d^2/e^2+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)*arc
tanh(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}{b x^{2} +{\left (x^{4} + \frac{d^{2}}{e^{2}}\right )} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x^2+d)/(b*x^2+c*(d^2/e^2+x^4)),x, algorithm="maxima")
[Out]
integrate((e*x^2 + d)/(b*x^2 + (x^4 + d^2/e^2)*c), x)
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Fricas [A] time = 1.35055, size = 490, normalized size = 3.77 \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)/(b*x^2+c*(d^2/e^2+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 = 0.783604, size = 160, normalized size = 1.23 \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)/(b*x**2+c*(d**2/e**2+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 = 2.44217, size = 6884, normalized size = 52.95 \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)/(b*x^2+c*(d^2/e^2+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*im
ag_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*im
ag_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*cos
h(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) - s
qrt(-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(arcs
in(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^(1
1/2) - 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(arcs
in(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*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)))))*sinh(1/2*imag_p
art(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*s
inh(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(ar
csin(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*im
ag_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_p
art(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*c
osh(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*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)))))^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*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))*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_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)))))^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) - sq
rt(-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_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(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))*co
s(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)