3.896 \(\int \frac{1}{x^2 (a-b x^2+c x^4)} \, dx\)
Optimal. Leaf size=172 \[ \frac{\sqrt{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{a x} \]
[Out]
-(1/(a*x)) + (Sqrt[c]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqr
t[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b +
Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]])
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Rubi [A] time = 0.203356, antiderivative size = 172, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used =
{1123, 1166, 208} \[ \frac{\sqrt{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} a \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{1}{a x} \]
Antiderivative was successfully verified.
[In]
Int[1/(x^2*(a - b*x^2 + c*x^4)),x]
[Out]
-(1/(a*x)) + (Sqrt[c]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqr
t[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b +
Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]])
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 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{1}{x^2 \left (a-b x^2+c x^4\right )} \, dx &=-\frac{1}{a x}+\frac{\int \frac{b-c x^2}{a-b x^2+c x^4} \, dx}{a}\\ &=-\frac{1}{a x}-\frac{\left (c \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{-\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{2 a}-\frac{\left (c \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{-\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{2 a}\\ &=-\frac{1}{a x}+\frac{\sqrt{c} \left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} a \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}
Mathematica [A] time = 0.396669, size = 199, normalized size = 1.16 \[ -\frac{\frac{\sqrt{2} \sqrt{c} \left (\sqrt{b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-\sqrt{b^2-4 a c}-b}}\right )}{\sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt{2} \sqrt{c} \left (\sqrt{b^2-4 a c}+b\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}-b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}-b}}+\frac{2}{x}}{2 a} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(x^2*(a - b*x^2 + c*x^4)),x]
[Out]
-(2/x + (Sqrt[2]*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b - Sqrt[b^2 - 4*a*c]]])/(S
qrt[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]*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)
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Maple [A] time = 0.178, size = 232, normalized size = 1.4 \begin{align*} -{\frac{c\sqrt{2}}{2\,a}\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{c\sqrt{2}b}{2\,a}\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{c\sqrt{2}}{2\,a}{\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{c\sqrt{2}b}{2\,a}{\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}}}}-{\frac{1}{ax}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^2/(c*x^4-b*x^2+a),x)
[Out]
-1/2*c/a*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*c
/a/(-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*c/a*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*c/a/(-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-1/a/x
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(c*x^4-b*x^2+a),x, algorithm="maxima")
[Out]
-integrate((c*x^2 - b)/(c*x^4 - b*x^2 + a), x)/a - 1/(a*x)
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Fricas [B] time = 1.63054, size = 2267, normalized size = 13.18 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(c*x^4-b*x^2+a),x, algorithm="fricas")
[Out]
1/2*(sqrt(1/2)*a*x*sqrt((b^3 - 3*a*b*c + (a^3*b^2 - 4*a^4*c)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a^7
*c)))/(a^3*b^2 - 4*a^4*c))*log(-2*(b^2*c^2 - a*c^3)*x + sqrt(1/2)*(b^5 - 5*a*b^3*c + 4*a^2*b*c^2 - (a^3*b^4 -
6*a^4*b^2*c + 8*a^5*c^2)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a^7*c)))*sqrt((b^3 - 3*a*b*c + (a^3*b^2
- 4*a^4*c)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))) - sqrt(1/2)*a*x*sqrt(
(b^3 - 3*a*b*c + (a^3*b^2 - 4*a^4*c)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c
))*log(-2*(b^2*c^2 - a*c^3)*x - sqrt(1/2)*(b^5 - 5*a*b^3*c + 4*a^2*b*c^2 - (a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)
*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a^7*c)))*sqrt((b^3 - 3*a*b*c + (a^3*b^2 - 4*a^4*c)*sqrt((b^4 -
2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))) + sqrt(1/2)*a*x*sqrt((b^3 - 3*a*b*c - (a^3*b^
2 - 4*a^4*c)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(-2*(b^2*c^2 - a*c
^3)*x + sqrt(1/2)*(b^5 - 5*a*b^3*c + 4*a^2*b*c^2 + (a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*sqrt((b^4 - 2*a*b^2*c +
a^2*c^2)/(a^6*b^2 - 4*a^7*c)))*sqrt((b^3 - 3*a*b*c - (a^3*b^2 - 4*a^4*c)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(a^
6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))) - sqrt(1/2)*a*x*sqrt((b^3 - 3*a*b*c - (a^3*b^2 - 4*a^4*c)*sqrt((b^4 -
2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log(-2*(b^2*c^2 - a*c^3)*x - sqrt(1/2)*(b^5 -
5*a*b^3*c + 4*a^2*b*c^2 + (a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a
^7*c)))*sqrt((b^3 - 3*a*b*c - (a^3*b^2 - 4*a^4*c)*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/(a^6*b^2 - 4*a^7*c)))/(a^3*
b^2 - 4*a^4*c))) - 2)/(a*x)
________________________________________________________________________________________
Sympy [A] time = 1.89738, size = 148, normalized size = 0.86 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (256 a^{5} c^{2} - 128 a^{4} b^{2} c + 16 a^{3} b^{4}\right ) + t^{2} \left (- 48 a^{2} b c^{2} + 28 a b^{3} c - 4 b^{5}\right ) + c^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{5} c^{2} + 48 t^{3} a^{4} b^{2} c - 8 t^{3} a^{3} b^{4} + 10 t a^{2} b c^{2} - 10 t a b^{3} c + 2 t b^{5}}{a c^{3} - b^{2} c^{2}} \right )} \right )\right )} - \frac{1}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**2/(c*x**4-b*x**2+a),x)
[Out]
RootSum(_t**4*(256*a**5*c**2 - 128*a**4*b**2*c + 16*a**3*b**4) + _t**2*(-48*a**2*b*c**2 + 28*a*b**3*c - 4*b**5
) + c**3, Lambda(_t, _t*log(x + (-64*_t**3*a**5*c**2 + 48*_t**3*a**4*b**2*c - 8*_t**3*a**3*b**4 + 10*_t*a**2*b
*c**2 - 10*_t*a*b**3*c + 2*_t*b**5)/(a*c**3 - b**2*c**2)))) - 1/(a*x)
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Giac [C] time = 2.70335, size = 4238, normalized size = 24.64 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(c*x^4-b*x^2+a),x, algorithm="giac")
[Out]
2*(3*(a*c^3)^(3/4)*a*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*cosh(1/2*imag_part(arccos(1/2*sq
rt(a*c)*b/(a*abs(c)))))^3*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) - (a*c^3)^(3/4)*a*cosh(1/2*im
ag_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)*a*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)*a*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)*a*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)*a*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)*a*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*a
bs(c)))))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^3 + (a*c^3)^(3/4)*a*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 - (a*c^3)^(1/4)*a*b
*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))
))) + (a*c^3)^(1/4)*a*b*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))))))*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))))))/(sqrt(b^2 - 4*a*c)*a*b*c*abs(a) + (b^2*c - 4*a*c^2)*a^2) + 2*(3*
(a*c^3)^(3/4)*a*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*abs(c))))) - (a*c^3)^(3/4)*a*cosh(1/2*imag_pa
rt(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)*a*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)*a*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)*a*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*ab
s(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*a
bs(c)))))^2 - 3*(a*c^3)^(3/4)*a*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sin(1/2*real_part(arcc
os(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)
*a*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)*a*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 - (a*c^3)^(1/4)*a*b*c*co
sh(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))))) +
(a*c^3)^(1/4)*a*b*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))))))*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))))))/(sqrt(b^2 - 4*a*c)*a*b*c*abs(a) + (b^2*c - 4*a*c^2)*a^2) + ((a*c^3)^
(3/4)*a*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)*a*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_part(arc
cos(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)
*a*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
)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) + 9*(a*c^3)^(3/4)*a*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(arcco
s(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)*a*
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)))
))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - 9*(a*c^3)^(3/4)*a*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)*a*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)*a*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 - (a*c^3)^(1/4)*a*b*c*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))))) + (a*c^
3)^(1/4)*a*b*c*cos(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))))))*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))/(sqrt(b^2
- 4*a*c)*a*b*c*abs(a) + (b^2*c - 4*a*c^2)*a^2) - ((a*c^3)^(3/4)*a*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)*a*cos(1/2*real_part(a
rccos(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_par
t(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - 3*(a*c^3)^(3/4)*a*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)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a
*abs(c))))) + 9*(a*c^3)^(3/4)*a*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*cosh(1/2*imag_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)))))^2*sinh(1/2*imag_part
(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) + 3*(a*c^3)^(3/4)*a*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)))))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(
c)))))^2 - 9*(a*c^3)^(3/4)*a*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(arcc
os(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - (a*c^3)^(3/4)*a*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)*a*cos(1/2*real_part(arccos(1/2*sqr
t(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 - (a*c^3)^(1/4)*a*b*c*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))))) + (a*c^3)^(1/4)*a*b*c*cos(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))))))*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))/(sqrt(b^2 - 4*a*c)*a*b*c*abs(a) + (b^2*c - 4*a*c^2)*a^2) - 1/
(a*x)