3.893 \(\int \frac{x^4}{a-b x^2+c x^4} \, dx\)

Optimal. Leaf size=179 \[ -\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x}{c} \]

[Out]

x/c - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2
]*c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sq
rt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.364524, antiderivative size = 179, 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 = {1122, 1166, 208} \[ -\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{2} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{2} c^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x}{c} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a - b*x^2 + c*x^4),x]

[Out]

x/c - ((b - (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2
]*c^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*x)/Sq
rt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

Rule 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d^3*(d*x)^(m - 3)*(a + b*
x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[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{x^4}{a-b x^2+c x^4} \, dx &=\frac{x}{c}-\frac{\int \frac{a-b x^2}{a-b x^2+c x^4} \, dx}{c}\\ &=\frac{x}{c}+\frac{\left (b-\frac{b^2-2 a c}{\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}{2 c}+\frac{\left (b+\frac{b^2-2 a c}{\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}{2 c}\\ &=\frac{x}{c}-\frac{\left (b-\frac{b^2-2 a c}{\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} c^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\left (b+\frac{b^2-2 a c}{\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} c^{3/2} \sqrt{b+\sqrt{b^2-4 a c}}}\\ \end{align*}

Mathematica [A]  time = 0.131441, size = 208, normalized size = 1.16 \[ \frac{\left (b \sqrt{b^2-4 a c}-2 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{-\sqrt{b^2-4 a c}-b}}\right )}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{-\sqrt{b^2-4 a c}-b}}+\frac{\left (b \sqrt{b^2-4 a c}+2 a c-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}-b}}\right )}{\sqrt{2} c^{3/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}-b}}+\frac{x}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a - b*x^2 + c*x^4),x]

[Out]

x/c + ((b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*
c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[-b - Sqrt[b^2 - 4*a*c]]) + ((-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2
]*Sqrt[c]*x)/Sqrt[-b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b^2 - 4*a*c]*Sqrt[-b + Sqrt[b^2 - 4*a*c]])

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Maple [B]  time = 0.183, size = 343, normalized size = 1.9 \begin{align*}{\frac{x}{c}}+{\frac{\sqrt{2}b}{2\,c}\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}}}}+{\sqrt{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{\sqrt{2}{b}^{2}}{2\,c}\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}b}{2\,c}{\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}}}}+{\sqrt{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{\sqrt{2}{b}^{2}}{2\,c}{\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^4/(c*x^4-b*x^2+a),x)

[Out]

x/c+1/2/c*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/
(-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))*a-1/2/c/(-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^2-1/2/c*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/(-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))*a-1/2/c/(-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^2

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

[Out]

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

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Fricas [B]  time = 1.61687, size = 2157, normalized size = 12.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.68431, size = 129, normalized size = 0.72 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (256 a^{2} c^{5} - 128 a b^{2} c^{4} + 16 b^{4} c^{3}\right ) + t^{2} \left (- 48 a^{2} b c^{2} + 28 a b^{3} c - 4 b^{5}\right ) + a^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 32 t^{3} a b c^{4} + 8 t^{3} b^{3} c^{3} - 4 t a^{2} c^{2} + 8 t a b^{2} c - 2 t b^{4}}{a^{2} c - a b^{2}} \right )} \right )\right )} + \frac{x}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**4*(256*a**2*c**5 - 128*a*b**2*c**4 + 16*b**4*c**3) + _t**2*(-48*a**2*b*c**2 + 28*a*b**3*c - 4*b**5
) + a**3, Lambda(_t, _t*log(x + (-32*_t**3*a*b*c**4 + 8*_t**3*b**3*c**3 - 4*_t*a**2*c**2 + 8*_t*a*b**2*c - 2*_
t*b**4)/(a**2*c - a*b**2)))) + x/c

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Giac [C]  time = 2.62217, size = 4250, normalized size = 23.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(3*(a*c^3)^(3/4)*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*s
qrt(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)*b*cosh(1/2*i
mag_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*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*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sin(1/2*rea
l_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*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*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sin(1/2*real_par
t(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*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*sin(1/2*real_part(arcco
s(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*
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)
)))) + (a*c^3)^(1/4)*a*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))))))*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)*b*c^2*abs(c) + (b^2*c - 4*a*c^2)*c^2) - 2*(3
*(a*c^3)^(3/4)*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*abs(c))))) - (a*c^3)^(3/4)*b*cosh(1/2*imag_p
art(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*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*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sin(1/2*real_par
t(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*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*a
bs(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*cosh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*sin(1/2*real_part(arc
cos(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*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*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*c^2*c
osh(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*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*sqr
t(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)*b*c^2*abs(c) + (b^2*c - 4*a*c^2)*c^2) - ((a*c^3)
^(3/4)*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*cos(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))*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)))))^2 - 3*(a*c^3)^(3/4
)*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)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) + 9*(a*c^3)^(3/4)*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(arcc
os(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
*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)*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*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*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*c^2*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*c^2*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)*b*c^2*abs(c) + (b^2*c - 4*a*c^2)*c^2) + ((a*c^3)^(3/4)*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*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_pa
rt(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - 3*(a*c^3)^(3/4)*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)))))^2*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(
a*abs(c))))) + 9*(a*c^3)^(3/4)*b*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)))))^2*sin(1/2*real_part(arccos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2*sinh(1/2*imag_par
t(arccos(1/2*sqrt(a*c)*b/(a*abs(c))))) + 3*(a*c^3)^(3/4)*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)))))*sinh(1/2*imag_part(arccos(1/2*sqrt(a*c)*b/(a*abs
(c)))))^2 - 9*(a*c^3)^(3/4)*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(arc
cos(1/2*sqrt(a*c)*b/(a*abs(c)))))^2 - (a*c^3)^(3/4)*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*cos(1/2*real_part(arccos(1/2*sq
rt(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*c^2*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*c^2*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*arcco
s(1/2*sqrt(a*c)*b/(a*abs(c)))) + x^2 + sqrt(a/c))/(sqrt(b^2 - 4*a*c)*b*c^2*abs(c) + (b^2*c - 4*a*c^2)*c^2) + x
/c