3.618 \(\int \frac{1}{(d+e x)^2 (a+b (d+e x)^2+c (d+e x)^4)} \, dx\)

Optimal. Leaf size=195 \[ -\frac{\sqrt{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a e \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a e \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{a e (d+e x)} \]

[Out]

-(1/(a*e*(d + e*x))) - (Sqrt[c]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2
- 4*a*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (Sqrt[c]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

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Rubi [A]  time = 0.286361, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1142, 1123, 1166, 205} \[ -\frac{\sqrt{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a e \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a e \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{a e (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

-(1/(a*e*(d + e*x))) - (Sqrt[c]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2
- 4*a*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (Sqrt[c]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

Rule 1142

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1123

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*x^2 +
c*x^4)^(p + 1))/(a*d*(m + 1)), x] - Dist[1/(a*d^2*(m + 1)), Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p +
5)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && In
tegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{e}\\ &=-\frac{1}{a e (d+e x)}+\frac{\operatorname{Subst}\left (\int \frac{-b-c x^2}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{a e}\\ &=-\frac{1}{a e (d+e x)}-\frac{\left (c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{2 a e}-\frac{\left (c \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx,x,d+e x\right )}{2 a e}\\ &=-\frac{1}{a e (d+e x)}-\frac{\sqrt{c} \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}} e}-\frac{\sqrt{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b+\sqrt{b^2-4 a c}} e}\\ \end{align*}

Mathematica [A]  time = 0.367265, size = 206, normalized size = 1.06 \[ -\frac{\frac{\sqrt{2} \sqrt{c} \left (\sqrt{b^2-4 a c}+b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (\sqrt{b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} (d+e x)}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2}{d+e x}}{2 a e} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(a + b*(d + e*x)^2 + c*(d + e*x)^4)),x]

[Out]

-(2/(d + e*x) + (Sqrt[2]*Sqrt[c]*(b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2
- 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*ArcTan
[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2
*a*e)

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Maple [C]  time = 0.007, size = 168, normalized size = 0.9 \begin{align*}{\frac{1}{2\,ae}\sum _{{\it \_R}={\it RootOf} \left ( c{e}^{4}{{\it \_Z}}^{4}+4\,cd{e}^{3}{{\it \_Z}}^{3}+ \left ( 6\,c{d}^{2}{e}^{2}+b{e}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,c{d}^{3}e+2\,bde \right ){\it \_Z}+c{d}^{4}+b{d}^{2}+a \right ) }{\frac{ \left ( -{{\it \_R}}^{2}c{e}^{2}-2\,{\it \_R}\,cde-c{d}^{2}-b \right ) \ln \left ( x-{\it \_R} \right ) }{2\,c{e}^{3}{{\it \_R}}^{3}+6\,cd{e}^{2}{{\it \_R}}^{2}+6\,c{d}^{2}e{\it \_R}+2\,c{d}^{3}+be{\it \_R}+bd}}}-{\frac{1}{ae \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4),x)

[Out]

1/2/a/e*sum((-_R^2*c*e^2-2*_R*c*d*e-c*d^2-b)/(2*_R^3*c*e^3+6*_R^2*c*d*e^2+6*_R*c*d^2*e+2*c*d^3+_R*b*e+b*d)*ln(
x-_R),_R=RootOf(c*e^4*_Z^4+4*c*d*e^3*_Z^3+(6*c*d^2*e^2+b*e^2)*_Z^2+(4*c*d^3*e+2*b*d*e)*_Z+c*d^4+b*d^2+a))-1/a/
e/(e*x+d)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 2.27964, size = 2753, normalized size = 14.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="fricas")

[Out]

1/2*(sqrt(1/2)*(a*e^2*x + a*d*e)*sqrt(-((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4
*a^7*c)*e^4)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d
+ sqrt(1/2)*((a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^
4)) - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e)*sqrt(-((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6
*b^2 - 4*a^7*c)*e^4)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))) - sqrt(1/2)*(a*e^2*x + a*d*e)*sqrt(-((a^3*b
^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a
^4*c)*e^2))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d - sqrt(1/2)*((a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c
^2)*e^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e)*sqrt(
-((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) + b^3 - 3*a*b*c)/((a^3*b
^2 - 4*a^4*c)*e^2))) - sqrt(1/2)*(a*e^2*x + a*d*e)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c
^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b
^2*c^2 - a*c^3)*d + sqrt(1/2)*((a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*
b^2 - 4*a^7*c)*e^4)) + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c
+ a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))) + sqrt(1/2)*(a*e^2*x + a*d*
e)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - b^3 + 3*a*b*c)/
((a^3*b^2 - 4*a^4*c)*e^2))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d - sqrt(1/2)*((a^3*b^4 - 6*a^4*
b^2*c + 8*a^5*c^2)*e^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) + (b^5 - 5*a*b^3*c + 4*a^2*
b*c^2)*e)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - b^3 + 3*
a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))) - 2)/(a*e^2*x + a*d*e)

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Sympy [A]  time = 3.66293, size = 211, normalized size = 1.08 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (256 a^{5} c^{2} e^{4} - 128 a^{4} b^{2} c e^{4} + 16 a^{3} b^{4} e^{4}\right ) + t^{2} \left (48 a^{2} b c^{2} e^{2} - 28 a b^{3} c e^{2} + 4 b^{5} e^{2}\right ) + c^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{5} c^{2} e^{3} + 48 t^{3} a^{4} b^{2} c e^{3} - 8 t^{3} a^{3} b^{4} e^{3} - 10 t a^{2} b c^{2} e + 10 t a b^{3} c e - 2 t b^{5} e + a c^{3} d - b^{2} c^{2} d}{a c^{3} e - b^{2} c^{2} e} \right )} \right )\right )} - \frac{1}{a d e + a e^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(a+b*(e*x+d)**2+c*(e*x+d)**4),x)

[Out]

RootSum(_t**4*(256*a**5*c**2*e**4 - 128*a**4*b**2*c*e**4 + 16*a**3*b**4*e**4) + _t**2*(48*a**2*b*c**2*e**2 - 2
8*a*b**3*c*e**2 + 4*b**5*e**2) + c**3, Lambda(_t, _t*log(x + (-64*_t**3*a**5*c**2*e**3 + 48*_t**3*a**4*b**2*c*
e**3 - 8*_t**3*a**3*b**4*e**3 - 10*_t*a**2*b*c**2*e + 10*_t*a*b**3*c*e - 2*_t*b**5*e + a*c**3*d - b**2*c**2*d)
/(a*c**3*e - b**2*c**2*e)))) - 1/(a*d*e + a*e**2*x)

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Giac [C]  time = 3.04139, size = 4884, normalized size = 25.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(a+b*(e*x+d)^2+c*(e*x+d)^4),x, algorithm="giac")

[Out]

-2*(3*(a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*cosh(1/2*imag_part
(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))
) - (a^3*c)^(3/4)*b*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sin(5/4*pi + 1/2*real_part(a
rcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3 - 9*(a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*
b*abs(a)/(a^2*c))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sin(5/4*pi + 1/2*real_part
(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) + 3*(a^3
*c)^(3/4)*b*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/
2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) + 9*(a^3*c)^(3/4
)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(a
*c)*b*abs(a)/(a^2*c))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_part
(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2 - 3*(a^3*c)^(3/4)*b*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(
a)/(a^2*c))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sinh(1/2*imag_part(arcsin(
1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2 - 3*(a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(
a)/(a^2*c))))^2*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_part(arcsin(
1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3 + (a^3*c)^(3/4)*b*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)
/(a^2*c))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3 + (a^3*c)^(1/4)*a^2*c*cosh(1/2*ima
g_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*
c)))) - (a^3*c)^(1/4)*a^2*c*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_
part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))))*arctan(-((c/a)^(1/4)*cos(5/4*pi + 1/2*arcsin(1/2*sqrt(a*c)*b*ab
s(a)/(a^2*c)))*e^(-1) + e^(-1)/(x*e + d))*e/((c/a)^(1/4)*sin(5/4*pi + 1/2*arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c
)))))/(sqrt(b^2 - 4*a*c)*a^2*b*abs(a)*e - (a*b^2*e - 4*a^2*c*e)*a^2) - 2*(3*(a^3*c)^(3/4)*b*cos(1/4*pi + 1/2*r
eal_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))
)^3*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) - (a^3*c)^(3/4)*b*cosh(1/2*imag_part(a
rcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^
3 - 9*(a^3*c)^(3/4)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*cosh(1/2*imag_part
(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))
)*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) + 3*(a^3*c)^(3/4)*b*cosh(1/2*imag_part(arcsin(1/
2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sinh(1
/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) + 9*(a^3*c)^(3/4)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/
2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sin(1/4*pi + 1/2
*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))
)^2 - 3*(a^3*c)^(3/4)*b*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sin(1/4*pi + 1/2*real_part
(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2 - 3*
(a^3*c)^(3/4)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sin(1/4*pi + 1/2*real_pa
rt(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3 + (a
^3*c)^(3/4)*b*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sinh(1/2*imag_part(arcsin(
1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3 + (a^3*c)^(1/4)*a^2*c*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^
2*c))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) - (a^3*c)^(1/4)*a^2*c*sin(1/4*pi +
1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*
c)))))*arctan(-((c/a)^(1/4)*cos(1/4*pi + 1/2*arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))*e^(-1) + e^(-1)/(x*e + d)
)*e/((c/a)^(1/4)*sin(1/4*pi + 1/2*arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))))/(sqrt(b^2 - 4*a*c)*a^2*b*abs(a)*e
- (a*b^2*e - 4*a^2*c*e)*a^2) + ((a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*
c))))^3*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3 - 3*(a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*rea
l_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*
sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2 - 3*(a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*rea
l_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^
2*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) + 9*(a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*real_part(a
rcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sin(5/4*
pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)
/(a^2*c)))) + 3*(a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*cosh(1/2
*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))
)^2 - 9*(a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*cosh(1/2*imag_part
(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^
2*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2 - (a^3*c)^(3/4)*b*cos(5/4*pi + 1/2*real_part(a
rcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3 + 3*(a
^3*c)^(3/4)*b*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sin(5/4*pi + 1/2*real_part(a
rcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3 + (a^3
*c)^(1/4)*a^2*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*cosh(1/2*imag_part(arcsin(
1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) - (a^3*c)^(1/4)*a^2*c*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(
a)/(a^2*c))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))))*log(sqrt(c/a)*e^(-2) + 2*(c/a)^(1/4
)*cos(5/4*pi + 1/2*arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))*e^(-2)/(x*e + d) + e^(-2)/(x*e + d)^2)/(sqrt(b^2 -
4*a*c)*a^2*b*abs(a)*e - (a*b^2*e - 4*a^2*c*e)*a^2) + ((a^3*c)^(3/4)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sq
rt(a*c)*b*abs(a)/(a^2*c))))^3*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3 - 3*(a^3*c)^(3/4)*
b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*
b*abs(a)/(a^2*c))))^3*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2 - 3*(a^3*c)^(3/4)*
b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c
)*b*abs(a)/(a^2*c))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) + 9*(a^3*c)^(3/4)*b*cos(1/
4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)
/(a^2*c))))^2*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sinh(1/2*imag_part(arcsin(
1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) + 3*(a^3*c)^(3/4)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)
/(a^2*c))))^3*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a
*c)*b*abs(a)/(a^2*c))))^2 - 9*(a^3*c)^(3/4)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)
)))*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c
)*b*abs(a)/(a^2*c))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2 - (a^3*c)^(3/4)*b*cos(1/
4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^3*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(
a)/(a^2*c))))^3 + 3*(a^3*c)^(3/4)*b*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sin(1/
4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(
a)/(a^2*c))))^3 + (a^3*c)^(1/4)*a^2*c*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*cosh
(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))) - (a^3*c)^(1/4)*a^2*c*cos(1/4*pi + 1/2*real_part(arcsi
n(1/2*sqrt(a*c)*b*abs(a)/(a^2*c))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))))*log(sqrt(c/a)
*e^(-2) + 2*(c/a)^(1/4)*cos(1/4*pi + 1/2*arcsin(1/2*sqrt(a*c)*b*abs(a)/(a^2*c)))*e^(-2)/(x*e + d) + e^(-2)/(x*
e + d)^2)/(sqrt(b^2 - 4*a*c)*a^2*b*abs(a)*e - (a*b^2*e - 4*a^2*c*e)*a^2) - e^(-1)/((x*e + d)*a)