3.894 \(\int \frac{x^2}{a-b x^2+c x^4} \, dx\)
Optimal. Leaf size=150 \[ \frac{\sqrt{b-\sqrt{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}} \]
[Out]
(Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b
^2 - 4*a*c]) - (Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]
*Sqrt[c]*Sqrt[b^2 - 4*a*c])
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Rubi [A] time = 0.108731, antiderivative size = 150, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used =
{1130, 208} \[ \frac{\sqrt{b-\sqrt{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In]
Int[x^2/(a - b*x^2 + c*x^4),x]
[Out]
(Sqrt[b - Sqrt[b^2 - 4*a*c]]*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b
^2 - 4*a*c]) - (Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]
*Sqrt[c]*Sqrt[b^2 - 4*a*c])
Rule 1130
Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{x^2}{a-b x^2+c x^4} \, dx &=-\left (\frac{1}{2} \left (-1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{-\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx\right )+\frac{1}{2} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{-\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx\\ &=\frac{\sqrt{b-\sqrt{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{b+\sqrt{b^2-4 a c}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}\\ \end{align*}
Mathematica [A] time = 0.107481, size = 137, normalized size = 0.91 \[ \frac{\sqrt{\sqrt{b^2-4 a c}-b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}-b}}\right )-\sqrt{-\sqrt{b^2-4 a c}-b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-\sqrt{b^2-4 a c}-b}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2/(a - b*x^2 + c*x^4),x]
[Out]
(-(Sqrt[-b - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b - Sqrt[b^2 - 4*a*c]]]) + Sqrt[-b + Sqrt[b^2
- 4*a*c]]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a*c])
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Maple [A] time = 0.17, size = 208, normalized size = 1.4 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-{\frac{\sqrt{2}b}{2}\arctan \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-{\frac{\sqrt{2}}{2}{\it Artanh} \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-{\frac{\sqrt{2}b}{2}{\it Artanh} \left ({cx\sqrt{2}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/(c*x^4-b*x^2+a),x)
[Out]
1/2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))-1/2/(-4*a*
c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(x*c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b
-1/2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))-1/2/(-4*a*
c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(x*c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{c x^{4} - b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(c*x^4-b*x^2+a),x, algorithm="maxima")
[Out]
integrate(x^2/(c*x^4 - b*x^2 + a), x)
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Fricas [B] time = 1.52462, size = 1196, normalized size = 7.97 \begin{align*} -\frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{b + \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac{\sqrt{\frac{1}{2}}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{b + \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}} + x\right ) + \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{b + \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\frac{\sqrt{\frac{1}{2}}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{b + \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}} + x\right ) + \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{b - \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\frac{\sqrt{\frac{1}{2}}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{b - \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}} + x\right ) - \frac{1}{2} \, \sqrt{\frac{1}{2}} \sqrt{\frac{b - \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\frac{\sqrt{\frac{1}{2}}{\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt{\frac{b - \frac{b^{2} c - 4 \, a c^{2}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt{b^{2} c^{2} - 4 \, a c^{3}}} + x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(c*x^4-b*x^2+a),x, algorithm="fricas")
[Out]
-1/2*sqrt(1/2)*sqrt((b + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log(sqrt(1/2)*(b^2*c -
4*a*c^2)*sqrt((b + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 4*a*c^3) + x)
+ 1/2*sqrt(1/2)*sqrt((b + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log(-sqrt(1/2)*(b^2*c
- 4*a*c^2)*sqrt((b + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 4*a*c^3) + x
) + 1/2*sqrt(1/2)*sqrt((b - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log(sqrt(1/2)*(b^2*c
- 4*a*c^2)*sqrt((b - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 4*a*c^3) +
x) - 1/2*sqrt(1/2)*sqrt((b - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))*log(-sqrt(1/2)*(b^2
*c - 4*a*c^2)*sqrt((b - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 4*a*c^3)
+ x)
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Sympy [A] time = 0.799969, size = 75, normalized size = 0.5 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (256 a^{2} c^{3} - 128 a b^{2} c^{2} + 16 b^{4} c\right ) + t^{2} \left (16 a b c - 4 b^{3}\right ) + a, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} - 16 t^{3} b^{2} c + 2 t b + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2/(c*x**4-b*x**2+a),x)
[Out]
RootSum(_t**4*(256*a**2*c**3 - 128*a*b**2*c**2 + 16*b**4*c) + _t**2*(16*a*b*c - 4*b**3) + a, Lambda(_t, _t*log
(64*_t**3*a*c**2 - 16*_t**3*b**2*c + 2*_t*b + x)))
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Giac [C] time = 2.63881, size = 5098, normalized size = 33.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(c*x^4-b*x^2+a),x, algorithm="giac")
[Out]
1/2*(3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(
1/2*sqrt(a*c)*b/(a*abs(c)))))^2*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sin(1/2*real_part(ar
ccos(1/2*sqrt(a*c)*b/(a*abs(c))))) - ((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c
)*b)*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs
(c)))))^3 - 9*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(
arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sin(1/2*real_
part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) + 3*((a*c^3)
^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*
b/(a*abs(c)))))^2*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sinh(1/2*imag_part(arccos(1/2*sqrt(
a*c)*b/(a*abs(c))))) + 9*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2
*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sin(
1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 -
3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cosh(1/2*imag_part(arccos(1/2
*sqrt(a*c)*b/(a*abs(c)))))*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sinh(1/2*imag_part(arccos(
1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - 3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)
*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c
)))))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3 + ((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (
a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sinh(1/2*imag_part(
arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3)*arctan(((a/c)^(1/4)*cos(1/2*arccos(1/2*sqrt(a*c)*b/(a*abs(c)))) + x)/(
(a/c)^(1/4)*sin(1/2*arccos(1/2*sqrt(a*c)*b/(a*abs(c))))))/(a*b^2*c^3 - 4*a^2*c^4) + 1/2*(3*((a*c^3)^(3/4)*b^2
- 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))
)))^2*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*ab
s(c))))) - ((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cosh(1/2*imag_part(ar
ccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3 - 9*((a*c^3)^(3/
4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*
abs(c)))))^2*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*
b/(a*abs(c)))))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) + 3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/
4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sin(1/2*
real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) + 9*(
(a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt
(a*c)*b/(a*abs(c)))))^2*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sin(1/2*real_part(arccos(1/2*s
qrt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - 3*((a*c^3)^(3/4)*b^2 - 4*
(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))
*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))
)))^2 - 3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arcc
os(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(a
rccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3 + ((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*
a*c)*b)*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*
abs(c)))))^3)*arctan(-((a/c)^(1/4)*cos(1/2*arccos(1/2*sqrt(a*c)*b/(a*abs(c)))) - x)/((a/c)^(1/4)*sin(1/2*arcco
s(1/2*sqrt(a*c)*b/(a*abs(c))))))/(a*b^2*c^3 - 4*a^2*c^4) + 1/4*(((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*
c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*cosh(1/2*imag_part(ar
ccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3 - 3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4
*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*a
bs(c)))))^3*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - 3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*
a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*cosh(1/2*ima
g_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) + 9*((a*
c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*
c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sin(1/2*real_part(arccos(1/2*sqrt
(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) + 3*((a*c^3)^(3/4)*b^2 - 4*(a*
c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*c
osh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))
^2 - 9*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(
1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sin(1/2*real_part(arccos
(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - ((a*c^3)^(3/4)*b^
2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c
)))))^3*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3 + 3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c
- (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sin(1/2*real_part
(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3)*log(2*x*(a/
c)^(1/4)*cos(1/2*arccos(1/2*sqrt(a*c)*b/(a*abs(c)))) + x^2 + sqrt(a/c))/(a*b^2*c^3 - 4*a^2*c^4) - 1/4*(((a*c^3
)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*
b/(a*abs(c)))))^3*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3 - 3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)
^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/
2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 -
3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*s
qrt(a*c)*b/(a*abs(c)))))^3*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arccos
(1/2*sqrt(a*c)*b/(a*abs(c))))) + 9*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*
b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))
)))^2*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*ab
s(c))))) + 3*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(a
rccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sinh(1/2*imag_pa
rt(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - 9*((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^
2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b
/(a*abs(c)))))*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c
)*b/(a*abs(c)))))^2 - ((a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*re
al_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3 + 3*(
(a*c^3)^(3/4)*b^2 - 4*(a*c^3)^(3/4)*a*c - (a*c^3)^(3/4)*sqrt(b^2 - 4*a*c)*b)*cos(1/2*real_part(arccos(1/2*sqrt
(a*c)*b/(a*abs(c)))))*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_part(arccos(1/2*s
qrt(a*c)*b/(a*abs(c)))))^3)*log(-2*x*(a/c)^(1/4)*cos(1/2*arccos(1/2*sqrt(a*c)*b/(a*abs(c)))) + x^2 + sqrt(a/c)
)/(a*b^2*c^3 - 4*a^2*c^4)