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

Optimal. Leaf size=189 \[ \frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.210849, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1673, 12, 1093, 205, 1107, 618, 206} \[ \frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + b*x^2 + c*x^4),x]

[Out]

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

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1093

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

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.272544, size = 194, normalized size = 1.03 \[ \frac{\frac{2 \sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}}}-\frac{2 \sqrt{2} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b}}+e \left (\log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )-\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )\right )}{2 \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.089, size = 231, normalized size = 1.2 \begin{align*} -{\frac{e}{8\,ac-2\,{b}^{2}}\sqrt{-4\,ac+{b}^{2}}\ln \left ( -2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}-b \right ) }+2\,{\frac{c\sqrt{-4\,ac+{b}^{2}}d\sqrt{2}}{ \left ( 8\,ac-2\,{b}^{2} \right ) \sqrt{ \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) c}}{\it Artanh} \left ({\frac{cx\sqrt{2}}{\sqrt{ \left ( \sqrt{-4\,ac+{b}^{2}}-b \right ) c}}} \right ) }+{\frac{e}{8\,ac-2\,{b}^{2}}\sqrt{-4\,ac+{b}^{2}}\ln \left ( 2\,c{x}^{2}+\sqrt{-4\,ac+{b}^{2}}+b \right ) }+2\,{\frac{c\sqrt{-4\,ac+{b}^{2}}d\sqrt{2}}{ \left ( 8\,ac-2\,{b}^{2} \right ) \sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}\arctan \left ({\frac{cx\sqrt{2}}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-4*a*c+b^2)^(1/2)/(8*a*c-2*b^2)*e*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)+2*c*(-4*a*c+b^2)^(1/2)/(8*a*c-2*b^2)*d*2
^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2)*arctanh(c*x*2^(1/2)/(((-4*a*c+b^2)^(1/2)-b)*c)^(1/2))+(-4*a*c+b^2)^(1/
2)/(8*a*c-2*b^2)*e*ln(2*c*x^2+(-4*a*c+b^2)^(1/2)+b)+2*c*(-4*a*c+b^2)^(1/2)/(8*a*c-2*b^2)*d*2^(1/2)/((b+(-4*a*c
+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x + d}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Sympy [B]  time = 93.8135, size = 471, normalized size = 2.49 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (256 a^{3} c^{2} - 128 a^{2} b^{2} c + 16 a b^{4}\right ) + t^{2} \left (32 a^{2} c e^{2} - 8 a b^{2} e^{2} - 16 a b c d^{2} + 4 b^{3} d^{2}\right ) + t \left (- 16 a c d^{2} e + 4 b^{2} d^{2} e\right ) + a e^{4} + b d^{2} e^{2} + c d^{4}, \left ( t \mapsto t \log{\left (x + \frac{- 512 t^{3} a^{4} c^{2} e^{2} + 256 t^{3} a^{3} b^{2} c e^{2} - 128 t^{3} a^{3} b c^{2} d^{2} - 32 t^{3} a^{2} b^{4} e^{2} + 64 t^{3} a^{2} b^{3} c d^{2} - 8 t^{3} a b^{5} d^{2} - 64 t^{2} a^{3} b c e^{3} - 64 t^{2} a^{3} c^{2} d^{2} e + 16 t^{2} a^{2} b^{3} e^{3} + 4 t^{2} a b^{4} d^{2} e - 32 t a^{3} c e^{4} + 8 t a^{2} b^{2} e^{4} + 24 t a^{2} b c d^{2} e^{2} - 16 t a^{2} c^{2} d^{4} - 6 t a b^{3} d^{2} e^{2} + 12 t a b^{2} c d^{4} - 2 t b^{4} d^{4} - 4 a^{2} b e^{5} + 20 a^{2} c d^{2} e^{3} - 5 a b^{2} d^{2} e^{3} + 8 a b c d^{4} e - b^{3} d^{4} e}{16 a^{2} c d e^{4} + 8 a b c d^{3} e^{2} - 4 a c^{2} d^{5} + b^{2} c d^{5}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**4*(256*a**3*c**2 - 128*a**2*b**2*c + 16*a*b**4) + _t**2*(32*a**2*c*e**2 - 8*a*b**2*e**2 - 16*a*b*c
*d**2 + 4*b**3*d**2) + _t*(-16*a*c*d**2*e + 4*b**2*d**2*e) + a*e**4 + b*d**2*e**2 + c*d**4, Lambda(_t, _t*log(
x + (-512*_t**3*a**4*c**2*e**2 + 256*_t**3*a**3*b**2*c*e**2 - 128*_t**3*a**3*b*c**2*d**2 - 32*_t**3*a**2*b**4*
e**2 + 64*_t**3*a**2*b**3*c*d**2 - 8*_t**3*a*b**5*d**2 - 64*_t**2*a**3*b*c*e**3 - 64*_t**2*a**3*c**2*d**2*e +
16*_t**2*a**2*b**3*e**3 + 4*_t**2*a*b**4*d**2*e - 32*_t*a**3*c*e**4 + 8*_t*a**2*b**2*e**4 + 24*_t*a**2*b*c*d**
2*e**2 - 16*_t*a**2*c**2*d**4 - 6*_t*a*b**3*d**2*e**2 + 12*_t*a*b**2*c*d**4 - 2*_t*b**4*d**4 - 4*a**2*b*e**5 +
20*a**2*c*d**2*e**3 - 5*a*b**2*d**2*e**3 + 8*a*b*c*d**4*e - b**3*d**4*e)/(16*a**2*c*d*e**4 + 8*a*b*c*d**3*e**
2 - 4*a*c**2*d**5 + b**2*c*d**5))))

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Giac [C]  time = 1.97199, size = 3996, normalized size = 21.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(sqrt(a*c)*b^2*c + 4*sqrt(a*c)*a*c^2 + sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cos(5/4*pi + 1/2*real_part(arcs
in(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*e*sin(5/4*pi + 1/2*
real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))) - 4*(sqrt(a*c)*b^2*c + 4*sqrt(a*c)*a*c^2 - sqrt(b^2 - 4*a*c)*sq
rt(a*c)*b*c)*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arcsin(1/2*sqr
t(a*c)*b/(a*abs(c)))))*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(ar
csin(1/2*sqrt(a*c)*b/(a*abs(c))))) - 2*(sqrt(a*c)*b^2*c - 4*sqrt(a*c)*a*c^2 + sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)
*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(
a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 + ((a*c^3)^(1/4)*b^2*c - 4*(a*c
^3)^(1/4)*a*c^2 + (a*c^3)^(1/4)*sqrt(b^2 - 4*a*c)*b*c)*d*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))
))*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))) - ((a*c^3)^(1/4)*b^2*c - 4*(a*c^3)^(1/4)*a*
c^2 + (a*c^3)^(1/4)*sqrt(b^2 - 4*a*c)*b*c)*d*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*s
inh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))))*arctan(-((a/c)^(1/4)*cos(5/4*pi + 1/2*arcsin(1/2*sqrt(
a*c)*b/(a*abs(c)))) - x)/((a/c)^(1/4)*sin(5/4*pi + 1/2*arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))))/(a*b^2*c^2 - 4*a^
2*c^3) + 1/2*(2*(sqrt(a*c)*b^2*c - 4*sqrt(a*c)*a*c^2 - sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cos(1/4*pi + 1/2*real_
part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*e*sin(1/4*
pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))) + 4*(sqrt(a*c)*b^2*c - 4*sqrt(a*c)*a*c^2 - sqrt(b^2 -
4*a*c)*sqrt(a*c)*b*c)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arcsi
n(1/2*sqrt(a*c)*b/(a*abs(c)))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*sinh(1/2*ima
g_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))) + 2*(sqrt(a*c)*b^2*c + 4*sqrt(a*c)*a*c^2 + sqrt(b^2 - 4*a*c)*sqrt(
a*c)*b*c)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*e*sin(1/4*pi + 1/2*real_part(arcsin(
1/2*sqrt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 + ((a*c^3)^(1/4)*b^2*c
- 4*(a*c^3)^(1/4)*a*c^2 + (a*c^3)^(1/4)*sqrt(b^2 - 4*a*c)*b*c)*d*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a
*abs(c)))))*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))) - ((a*c^3)^(1/4)*b^2*c - 4*(a*c^3)
^(1/4)*a*c^2 + (a*c^3)^(1/4)*sqrt(b^2 - 4*a*c)*b*c)*d*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs
(c)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))))*arctan(-((a/c)^(1/4)*cos(1/4*pi + 1/2*arcsin(
1/2*sqrt(a*c)*b/(a*abs(c)))) - x)/((a/c)^(1/4)*sin(1/4*pi + 1/2*arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))))/(a*b^2*c
^2 - 4*a^2*c^3) + 1/4*((sqrt(a*c)*b^2*c + 4*sqrt(a*c)*a*c^2 + sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cos(5/4*pi + 1/
2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*e
+ (sqrt(a*c)*b^2*c - 4*sqrt(a*c)*a*c^2 - sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cosh(1/2*imag_part(arcsin(1/2*sqrt(
a*c)*b/(a*abs(c)))))^2*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 + 2*(sqrt(a*c)*b^2*
c - 4*sqrt(a*c)*a*c^2 - sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*
abs(c)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*e*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)
*b/(a*abs(c))))) - 2*(sqrt(a*c)*b^2*c - 4*sqrt(a*c)*a*c^2 - sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cosh(1/2*imag_par
t(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*e*sin(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sin
h(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))) - (sqrt(a*c)*b^2*c - 4*sqrt(a*c)*a*c^2 + sqrt(b^2 - 4*a*c
)*sqrt(a*c)*b*c)*cos(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*e*sinh(1/2*imag_part(arcsin
(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - (sqrt(a*c)*b^2*c + 4*sqrt(a*c)*a*c^2 - sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*e*s
in(5/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*
abs(c)))))^2 - ((a*c^3)^(1/4)*b^2*c - 4*(a*c^3)^(1/4)*a*c^2 + (a*c^3)^(1/4)*sqrt(b^2 - 4*a*c)*b*c)*d*cos(5/4*p
i + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))
+ ((a*c^3)^(1/4)*b^2*c - 4*(a*c^3)^(1/4)*a*c^2 + (a*c^3)^(1/4)*sqrt(b^2 - 4*a*c)*b*c)*d*cos(5/4*pi + 1/2*real
_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))))*log(-2*x*(
a/c)^(1/4)*cos(5/4*pi + 1/2*arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))) + x^2 + sqrt(a/c))/(a*b^2*c^2 - 4*a^2*c^3) + 1
/4*((sqrt(a*c)*b^2*c - 4*sqrt(a*c)*a*c^2 - sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cos(1/4*pi + 1/2*real_part(arcsin(
1/2*sqrt(a*c)*b/(a*abs(c)))))^2*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*e + (sqrt(a*c)*b^2*c
- 4*sqrt(a*c)*a*c^2 - sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))
)^2*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 + 2*(sqrt(a*c)*b^2*c - 4*sqrt(a*c)*a*c
^2 - sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*cosh(1
/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*e*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))) +
2*(sqrt(a*c)*b^2*c + 4*sqrt(a*c)*a*c^2 - sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*cosh(1/2*imag_part(arcsin(1/2*sqrt(a
*c)*b/(a*abs(c)))))*e*sin(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arc
sin(1/2*sqrt(a*c)*b/(a*abs(c))))) + (sqrt(a*c)*b^2*c - 4*sqrt(a*c)*a*c^2 + sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*co
s(1/4*pi + 1/2*real_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*e*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a
*abs(c)))))^2 + (sqrt(a*c)*b^2*c + 4*sqrt(a*c)*a*c^2 + sqrt(b^2 - 4*a*c)*sqrt(a*c)*b*c)*e*sin(1/4*pi + 1/2*rea
l_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - ((a*
c^3)^(1/4)*b^2*c - 4*(a*c^3)^(1/4)*a*c^2 + (a*c^3)^(1/4)*sqrt(b^2 - 4*a*c)*b*c)*d*cos(1/4*pi + 1/2*real_part(a
rcsin(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))) + ((a*c^3)^(1/4)*b
^2*c - 4*(a*c^3)^(1/4)*a*c^2 + (a*c^3)^(1/4)*sqrt(b^2 - 4*a*c)*b*c)*d*cos(1/4*pi + 1/2*real_part(arcsin(1/2*sq
rt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(arcsin(1/2*sqrt(a*c)*b/(a*abs(c))))))*log(-2*x*(a/c)^(1/4)*cos(1/4*
pi + 1/2*arcsin(1/2*sqrt(a*c)*b/(a*abs(c)))) + x^2 + sqrt(a/c))/(a*b^2*c^2 - 4*a^2*c^3)