Integrand size = 22, antiderivative size = 97 \[ \int \frac {1}{x^3 \left (a-b+2 a x^2+a x^4\right )} \, dx=-\frac {1}{2 (a-b) x^2}-\frac {\sqrt {a} (a+b) \text {arctanh}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 (a-b)^2 \sqrt {b}}-\frac {2 a \log (x)}{(a-b)^2}+\frac {a \log \left (a-b+2 a x^2+a x^4\right )}{2 (a-b)^2} \] Output:
-1/2/(a-b)/x^2-1/2*a^(1/2)*(a+b)*arctanh(a^(1/2)*(x^2+1)/b^(1/2))/(a-b)^2/ b^(1/2)-2*a*ln(x)/(a-b)^2+1/2*a*ln(a*x^4+2*a*x^2+a-b)/(a-b)^2
Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.51 \[ \int \frac {1}{x^3 \left (a-b+2 a x^2+a x^4\right )} \, dx=\frac {-8 a \sqrt {b} x^2 \log (x)+\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^2 x^2 \log \left (-\sqrt {b}+\sqrt {a} \left (1+x^2\right )\right )-\left (\sqrt {a}-\sqrt {b}\right ) \left (2 \left (\sqrt {a} \sqrt {b}+b\right )+\left (a x^2-\sqrt {a} \sqrt {b} x^2\right ) \log \left (\sqrt {b}+\sqrt {a} \left (1+x^2\right )\right )\right )}{4 (a-b)^2 \sqrt {b} x^2} \] Input:
Integrate[1/(x^3*(a - b + 2*a*x^2 + a*x^4)),x]
Output:
(-8*a*Sqrt[b]*x^2*Log[x] + Sqrt[a]*(Sqrt[a] + Sqrt[b])^2*x^2*Log[-Sqrt[b] + Sqrt[a]*(1 + x^2)] - (Sqrt[a] - Sqrt[b])*(2*(Sqrt[a]*Sqrt[b] + b) + (a*x ^2 - Sqrt[a]*Sqrt[b]*x^2)*Log[Sqrt[b] + Sqrt[a]*(1 + x^2)]))/(4*(a - b)^2* Sqrt[b]*x^2)
Time = 0.53 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1434, 1145, 25, 27, 1200, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^3 \left (a x^4+2 a x^2+a-b\right )} \, dx\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (a x^4+2 a x^2+a-b\right )}dx^2\) |
| \(\Big \downarrow \) 1145 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {a \left (x^2+2\right )}{x^2 \left (a x^4+2 a x^2+a-b\right )}dx^2}{a-b}-\frac {1}{x^2 (a-b)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {a \left (x^2+2\right )}{x^2 \left (a x^4+2 a x^2+a-b\right )}dx^2}{a-b}-\frac {1}{x^2 (a-b)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a \int \frac {x^2+2}{x^2 \left (a x^4+2 a x^2+a-b\right )}dx^2}{a-b}-\frac {1}{x^2 (a-b)}\right )\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a \int \left (\frac {-2 a x^2-3 a-b}{(a-b) \left (a x^4+2 a x^2+a-b\right )}+\frac {2}{(a-b) x^2}\right )dx^2}{a-b}-\frac {1}{x^2 (a-b)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {a \left (\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} (a-b)}+\frac {2 \log \left (x^2\right )}{a-b}-\frac {\log \left (a x^4+2 a x^2+a-b\right )}{a-b}\right )}{a-b}-\frac {1}{x^2 (a-b)}\right )\) |
Input:
Int[1/(x^3*(a - b + 2*a*x^2 + a*x^4)),x]
Output:
(-(1/((a - b)*x^2)) - (a*(((a + b)*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/( Sqrt[a]*(a - b)*Sqrt[b]) + (2*Log[x^2])/(a - b) - Log[a - b + 2*a*x^2 + a* x^4]/(a - b)))/(a - b))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp [1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[m, -1]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85
| method | result | size |
| default | \(-\frac {1}{2 \left (a -b \right ) x^{2}}-\frac {2 a \ln \left (x \right )}{\left (a -b \right )^{2}}+\frac {a \left (\ln \left (a \,x^{4}+2 a \,x^{2}+a -b \right )-\frac {\left (a +b \right ) \operatorname {arctanh}\left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{\sqrt {a b}}\right )}{2 \left (a -b \right )^{2}}\) | \(82\) |
| risch | \(-\frac {1}{2 \left (a -b \right ) x^{2}}-\frac {2 a \ln \left (x \right )}{a^{2}-2 a b +b^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{2} b -2 b^{2} a +b^{3}\right ) \textit {\_Z}^{2}-4 a b \textit {\_Z} -a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{3}-3 a^{2} b +9 b^{2} a -5 b^{3}\right ) \textit {\_R}^{2}+\left (-8 a^{2}+8 a b \right ) \textit {\_R} +4 a \right ) x^{2}+\left (-a^{3}+3 a^{2} b -3 b^{2} a +b^{3}\right ) \textit {\_R}^{2}+\left (-7 a^{2}+6 a b +b^{2}\right ) \textit {\_R} +8 a \right )\right )}{4}\) | \(160\) |
Input:
int(1/x^3/(a*x^4+2*a*x^2+a-b),x,method=_RETURNVERBOSE)
Output:
-1/2/(a-b)/x^2-2*a*ln(x)/(a-b)^2+1/2/(a-b)^2*a*(ln(a*x^4+2*a*x^2+a-b)-(a+b )/(a*b)^(1/2)*arctanh(1/2*(2*a*x^2+2*a)/(a*b)^(1/2)))
Time = 0.08 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.10 \[ \int \frac {1}{x^3 \left (a-b+2 a x^2+a x^4\right )} \, dx=\left [\frac {{\left (a + b\right )} x^{2} \sqrt {\frac {a}{b}} \log \left (\frac {a x^{4} + 2 \, a x^{2} - 2 \, {\left (b x^{2} + b\right )} \sqrt {\frac {a}{b}} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right ) + 2 \, a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 8 \, a x^{2} \log \left (x\right ) - 2 \, a + 2 \, b}{4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} x^{2}}, \frac {{\left (a + b\right )} x^{2} \sqrt {-\frac {a}{b}} \arctan \left ({\left (x^{2} + 1\right )} \sqrt {-\frac {a}{b}}\right ) + a x^{2} \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, a x^{2} \log \left (x\right ) - a + b}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} x^{2}}\right ] \] Input:
integrate(1/x^3/(a*x^4+2*a*x^2+a-b),x, algorithm="fricas")
Output:
[1/4*((a + b)*x^2*sqrt(a/b)*log((a*x^4 + 2*a*x^2 - 2*(b*x^2 + b)*sqrt(a/b) + a + b)/(a*x^4 + 2*a*x^2 + a - b)) + 2*a*x^2*log(a*x^4 + 2*a*x^2 + a - b ) - 8*a*x^2*log(x) - 2*a + 2*b)/((a^2 - 2*a*b + b^2)*x^2), 1/2*((a + b)*x^ 2*sqrt(-a/b)*arctan((x^2 + 1)*sqrt(-a/b)) + a*x^2*log(a*x^4 + 2*a*x^2 + a - b) - 4*a*x^2*log(x) - a + b)/((a^2 - 2*a*b + b^2)*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (83) = 166\).
Time = 13.10 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.84 \[ \int \frac {1}{x^3 \left (a-b+2 a x^2+a x^4\right )} \, dx=- \frac {2 a \log {\left (x \right )}}{\left (a - b\right )^{2}} + \left (\frac {a}{2 \left (a - b\right )^{2}} - \frac {\sqrt {a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- 4 a^{2} b \left (\frac {a}{2 \left (a - b\right )^{2}} - \frac {\sqrt {a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac {a}{2 \left (a - b\right )^{2}} - \frac {\sqrt {a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) + 3 a b - 4 b^{3} \left (\frac {a}{2 \left (a - b\right )^{2}} - \frac {\sqrt {a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right )}{a^{2} + a b} \right )} + \left (\frac {a}{2 \left (a - b\right )^{2}} + \frac {\sqrt {a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- 4 a^{2} b \left (\frac {a}{2 \left (a - b\right )^{2}} + \frac {\sqrt {a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) + a^{2} + 8 a b^{2} \left (\frac {a}{2 \left (a - b\right )^{2}} + \frac {\sqrt {a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right ) + 3 a b - 4 b^{3} \left (\frac {a}{2 \left (a - b\right )^{2}} + \frac {\sqrt {a b} \left (a + b\right )}{4 b \left (a^{2} - 2 a b + b^{2}\right )}\right )}{a^{2} + a b} \right )} - \frac {1}{x^{2} \cdot \left (2 a - 2 b\right )} \] Input:
integrate(1/x**3/(a*x**4+2*a*x**2+a-b),x)
Output:
-2*a*log(x)/(a - b)**2 + (a/(2*(a - b)**2) - sqrt(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b**2)))*log(x**2 + (-4*a**2*b*(a/(2*(a - b)**2) - sqrt(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b**2))) + a**2 + 8*a*b**2*(a/(2*(a - b)**2) - sqr t(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b**2))) + 3*a*b - 4*b**3*(a/(2*(a - b) **2) - sqrt(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b**2))))/(a**2 + a*b)) + (a/ (2*(a - b)**2) + sqrt(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b**2)))*log(x**2 + (-4*a**2*b*(a/(2*(a - b)**2) + sqrt(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b** 2))) + a**2 + 8*a*b**2*(a/(2*(a - b)**2) + sqrt(a*b)*(a + b)/(4*b*(a**2 - 2*a*b + b**2))) + 3*a*b - 4*b**3*(a/(2*(a - b)**2) + sqrt(a*b)*(a + b)/(4* b*(a**2 - 2*a*b + b**2))))/(a**2 + a*b)) - 1/(x**2*(2*a - 2*b))
Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^3 \left (a-b+2 a x^2+a x^4\right )} \, dx=\frac {a \log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} - \frac {a \log \left (x^{2}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {{\left (a^{2} + a b\right )} \log \left (\frac {a x^{2} + a - \sqrt {a b}}{a x^{2} + a + \sqrt {a b}}\right )}{4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a b}} - \frac {1}{2 \, {\left (a - b\right )} x^{2}} \] Input:
integrate(1/x^3/(a*x^4+2*a*x^2+a-b),x, algorithm="maxima")
Output:
1/2*a*log(a*x^4 + 2*a*x^2 + a - b)/(a^2 - 2*a*b + b^2) - a*log(x^2)/(a^2 - 2*a*b + b^2) + 1/4*(a^2 + a*b)*log((a*x^2 + a - sqrt(a*b))/(a*x^2 + a + s qrt(a*b)))/((a^2 - 2*a*b + b^2)*sqrt(a*b)) - 1/2/((a - b)*x^2)
Time = 0.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^3 \left (a-b+2 a x^2+a x^4\right )} \, dx=\frac {a \log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} - \frac {a \log \left (x^{2}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {{\left (a^{2} + a b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {-a b}}\right )}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a b}} + \frac {2 \, a x^{2} - a + b}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} x^{2}} \] Input:
integrate(1/x^3/(a*x^4+2*a*x^2+a-b),x, algorithm="giac")
Output:
1/2*a*log(a*x^4 + 2*a*x^2 + a - b)/(a^2 - 2*a*b + b^2) - a*log(x^2)/(a^2 - 2*a*b + b^2) + 1/2*(a^2 + a*b)*arctan((a*x^2 + a)/sqrt(-a*b))/((a^2 - 2*a *b + b^2)*sqrt(-a*b)) + 1/2*(2*a*x^2 - a + b)/((a^2 - 2*a*b + b^2)*x^2)
Time = 19.51 (sec) , antiderivative size = 389, normalized size of antiderivative = 4.01 \[ \int \frac {1}{x^3 \left (a-b+2 a x^2+a x^4\right )} \, dx=\frac {\ln \left (100\,a\,{\left (a\,b\right )}^{7/2}-198\,b\,{\left (a\,b\right )}^{7/2}-a^3\,{\left (a\,b\right )}^{5/2}+100\,b^3\,{\left (a\,b\right )}^{5/2}-b^5\,{\left (a\,b\right )}^{3/2}+a^2\,b^6-100\,a^3\,b^5+198\,a^4\,b^4-100\,a^5\,b^3+a^6\,b^2+a^2\,b^6\,x^2-100\,a^3\,b^5\,x^2+198\,a^4\,b^4\,x^2-100\,a^5\,b^3\,x^2+a^6\,b^2\,x^2\right )\,\left (\frac {a\,\sqrt {a\,b}}{4}+b\,\left (\frac {a}{2}+\frac {\sqrt {a\,b}}{4}\right )\right )}{a^2\,b-2\,a\,b^2+b^3}-\frac {2\,a\,\ln \left (x\right )}{a^2-2\,a\,b+b^2}-\frac {\ln \left (198\,b\,{\left (a\,b\right )}^{7/2}-100\,a\,{\left (a\,b\right )}^{7/2}+a^3\,{\left (a\,b\right )}^{5/2}-100\,b^3\,{\left (a\,b\right )}^{5/2}+b^5\,{\left (a\,b\right )}^{3/2}+a^2\,b^6-100\,a^3\,b^5+198\,a^4\,b^4-100\,a^5\,b^3+a^6\,b^2+a^2\,b^6\,x^2-100\,a^3\,b^5\,x^2+198\,a^4\,b^4\,x^2-100\,a^5\,b^3\,x^2+a^6\,b^2\,x^2\right )\,\left (\frac {a\,\sqrt {a\,b}}{4}-b\,\left (\frac {a}{2}-\frac {\sqrt {a\,b}}{4}\right )\right )}{a^2\,b-2\,a\,b^2+b^3}-\frac {1}{2\,x^2\,\left (a-b\right )} \] Input:
int(1/(x^3*(a - b + 2*a*x^2 + a*x^4)),x)
Output:
(log(100*a*(a*b)^(7/2) - 198*b*(a*b)^(7/2) - a^3*(a*b)^(5/2) + 100*b^3*(a* b)^(5/2) - b^5*(a*b)^(3/2) + a^2*b^6 - 100*a^3*b^5 + 198*a^4*b^4 - 100*a^5 *b^3 + a^6*b^2 + a^2*b^6*x^2 - 100*a^3*b^5*x^2 + 198*a^4*b^4*x^2 - 100*a^5 *b^3*x^2 + a^6*b^2*x^2)*((a*(a*b)^(1/2))/4 + b*(a/2 + (a*b)^(1/2)/4)))/(a^ 2*b - 2*a*b^2 + b^3) - (2*a*log(x))/(a^2 - 2*a*b + b^2) - (log(198*b*(a*b) ^(7/2) - 100*a*(a*b)^(7/2) + a^3*(a*b)^(5/2) - 100*b^3*(a*b)^(5/2) + b^5*( a*b)^(3/2) + a^2*b^6 - 100*a^3*b^5 + 198*a^4*b^4 - 100*a^5*b^3 + a^6*b^2 + a^2*b^6*x^2 - 100*a^3*b^5*x^2 + 198*a^4*b^4*x^2 - 100*a^5*b^3*x^2 + a^6*b ^2*x^2)*((a*(a*b)^(1/2))/4 - b*(a/2 - (a*b)^(1/2)/4)))/(a^2*b - 2*a*b^2 + b^3) - 1/(2*x^2*(a - b))
Time = 0.17 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.66 \[ \int \frac {1}{x^3 \left (a-b+2 a x^2+a x^4\right )} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a \,x^{2}+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b \,x^{2}+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a \,x^{2}+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b \,x^{2}-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right ) a \,x^{2}-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right ) b \,x^{2}+2 \,\mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a b \,x^{2}+2 \,\mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) a b \,x^{2}+2 \,\mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right ) a b \,x^{2}-8 \,\mathrm {log}\left (x \right ) a b \,x^{2}-2 a b +2 b^{2}}{4 b \,x^{2} \left (a^{2}-2 a b +b^{2}\right )} \] Input:
int(1/x^3/(a*x^4+2*a*x^2+a-b),x)
Output:
(sqrt(b)*sqrt(a)*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a*x**2 + sq rt(b)*sqrt(a)*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*b*x**2 + sqrt( b)*sqrt(a)*log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a*x**2 + sqrt(b)*sqr t(a)*log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*b*x**2 - sqrt(b)*sqrt(a)*l og(sqrt(b)*sqrt(a) + a*x**2 + a)*a*x**2 - sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt (a) + a*x**2 + a)*b*x**2 + 2*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x) *a*b*x**2 + 2*log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*a*b*x**2 + 2*log( sqrt(b)*sqrt(a) + a*x**2 + a)*a*b*x**2 - 8*log(x)*a*b*x**2 - 2*a*b + 2*b** 2)/(4*b*x**2*(a**2 - 2*a*b + b**2))