Integrand size = 16, antiderivative size = 93 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^4} \, dx=-\frac {i-a}{3 (i+a) x^3}+\frac {i b}{(i+a)^2 x^2}+\frac {2 b^2}{(1-i a)^3 x}-\frac {2 i b^3 \log (x)}{(i+a)^4}+\frac {2 i b^3 \log (i+a+b x)}{(i+a)^4} \] Output:
-1/3*(I-a)/(I+a)/x^3+I*b/(I+a)^2/x^2+2*b^2/(1-I*a)^3/x-2*I*b^3*ln(x)/(I+a) ^4+2*I*b^3*ln(I+a+b*x)/(I+a)^4
Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^4} \, dx=\frac {(i+a) \left (i+a+i a^2+a^3-3 b x+3 i a b x-6 i b^2 x^2\right )-6 i b^3 x^3 \log (x)+6 i b^3 x^3 \log (i+a+b x)}{3 (i+a)^4 x^3} \] Input:
Integrate[E^((2*I)*ArcTan[a + b*x])/x^4,x]
Output:
((I + a)*(I + a + I*a^2 + a^3 - 3*b*x + (3*I)*a*b*x - (6*I)*b^2*x^2) - (6* I)*b^3*x^3*Log[x] + (6*I)*b^3*x^3*Log[I + a + b*x])/(3*(I + a)^4*x^3)
Time = 0.42 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5618, 86, 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.
\(\displaystyle \int \frac {e^{2 i \arctan (a+b x)}}{x^4} \, dx\) |
\(\Big \downarrow \) 5618 |
\(\displaystyle \int \frac {i a+i b x+1}{x^4 (-i a-i b x+1)}dx\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \int \left (\frac {2 i b^4}{(a+i)^4 (a+b x+i)}-\frac {2 i b^3}{(a+i)^4 x}+\frac {2 i b^2}{(a+i)^3 x^2}-\frac {2 i b}{(a+i)^2 x^3}+\frac {-a+i}{(a+i) x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 i b^3 \log (x)}{(a+i)^4}+\frac {2 i b^3 \log (a+b x+i)}{(a+i)^4}+\frac {2 b^2}{(1-i a)^3 x}+\frac {i b}{(a+i)^2 x^2}-\frac {-a+i}{3 (a+i) x^3}\) |
Input:
Int[E^((2*I)*ArcTan[a + b*x])/x^4,x]
Output:
-1/3*(I - a)/((I + a)*x^3) + (I*b)/((I + a)^2*x^2) + (2*b^2)/((1 - I*a)^3* x) - ((2*I)*b^3*Log[x])/(I + a)^4 + ((2*I)*b^3*Log[I + a + b*x])/(I + a)^4
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[(d + e*x)^m*((1 - I*a*c - I*b*c*x)^(I*(n/2))/(1 + I*a*c + I*b*c*x)^(I*(n/2))), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (81 ) = 162\).
Time = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.87
method | result | size |
default | \(\frac {2 b^{4} \left (\frac {\left (i a^{4} b +4 a^{3} b -6 i a^{2} b -4 a b +i b \right ) \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{2}}+\frac {\left (i a^{5}-10 i a^{3}+5 a^{4}+5 i a -10 a^{2}+1-\frac {\left (i a^{4} b +4 a^{3} b -6 i a^{2} b -4 a b +i b \right ) a}{b}\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}\right )}{\left (a^{2}+1\right )^{4}}-\frac {-a^{2}+2 i a +1}{3 \left (a^{2}+1\right ) x^{3}}+\frac {b \left (i a^{2}+2 a -i\right )}{\left (a^{2}+1\right )^{2} x^{2}}-\frac {2 b^{2} \left (i a^{3}+3 a^{2}-3 i a -1\right )}{\left (a^{2}+1\right )^{3} x}-\frac {2 b^{3} \left (i a^{4}+4 a^{3}-6 i a^{2}-4 a +i\right ) \ln \left (x \right )}{\left (a^{2}+1\right )^{4}}\) | \(267\) |
parallelrisch | \(-\frac {1-6 a b x +2 i a^{7}-18 i a \,b^{2} x^{2}+3 i a^{2} b x -6 b^{2} x^{2}+2 i a +12 a^{2} b^{2} x^{2}-12 a^{3} b x +3 i b x +6 i \ln \left (x \right ) x^{3} a^{4} b^{3}-6 i \ln \left (b x +a +i\right ) x^{3} a^{4} b^{3}-36 i \ln \left (x \right ) x^{3} a^{2} b^{3}+36 i \ln \left (b x +a +i\right ) x^{3} a^{2} b^{3}+24 \ln \left (x \right ) x^{3} a^{3} b^{3}-24 \ln \left (x \right ) x^{3} a \,b^{3}+6 i b^{3} \ln \left (x \right ) x^{3}-6 i b^{3} \ln \left (b x +a +i\right ) x^{3}+6 i a^{5}+2 a^{2}-6 a^{5} b x +18 a^{4} b^{2} x^{2}-2 a^{6}+6 i a^{3}-a^{8}-12 i a^{3} b^{2} x^{2}-3 i a^{4} b x -24 \ln \left (b x +a +i\right ) x^{3} a^{3} b^{3}+24 \ln \left (b x +a +i\right ) x^{3} a \,b^{3}+6 i x^{2} a^{5} b^{2}-3 i x \,a^{6} b}{3 \left (a^{6}+3 a^{4}+3 a^{2}+1\right ) \left (a^{2}+1\right ) x^{3}}\) | \(337\) |
risch | \(\frac {-\frac {2 i b^{2} x^{2}}{\left (a^{2}+2 i a -1\right ) \left (i+a \right )}+\frac {i b x}{a^{2}+2 i a -1}+\frac {-i+a}{3 i+3 a}}{x^{3}}-\frac {b^{3} \ln \left (4 a^{12} b^{2} x^{2}+8 a^{13} b x +4 a^{14}+24 a^{10} b^{2} x^{2}+48 a^{11} b x +28 a^{12}+60 a^{8} b^{2} x^{2}+120 a^{9} b x +84 a^{10}+80 a^{6} b^{2} x^{2}+160 a^{7} b x +140 a^{8}+60 a^{4} b^{2} x^{2}+120 a^{5} b x +140 a^{6}+24 a^{2} b^{2} x^{2}+48 a^{3} b x +84 a^{4}+4 b^{2} x^{2}+8 a b x +28 a^{2}+4\right )}{i a^{4}-4 a^{3}-6 i a^{2}+4 a +i}+\frac {2 i b^{3} \arctan \left (\frac {\left (2 a^{6} b +6 a^{4} b +6 a^{2} b +2 b \right ) x +2 a^{7}+6 a^{5}+6 a^{3}+2 a}{2 a^{6}+6 a^{4}+6 a^{2}+2}\right )}{i a^{4}-4 a^{3}-6 i a^{2}+4 a +i}+\frac {2 b^{3} \ln \left (\left (-2 a^{6} b -6 a^{4} b -6 a^{2} b -2 b \right ) x \right )}{i a^{4}-4 a^{3}-6 i a^{2}+4 a +i}\) | \(400\) |
Input:
int((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^4,x,method=_RETURNVERBOSE)
Output:
2*b^4/(a^2+1)^4*(1/2*(I*a^4*b-6*I*a^2*b+4*a^3*b+I*b-4*a*b)/b^2*ln(b^2*x^2+ 2*a*b*x+a^2+1)+(I*a^5-10*I*a^3+5*a^4+5*I*a-10*a^2+1-(I*a^4*b-6*I*a^2*b+4*a ^3*b+I*b-4*a*b)*a/b)/b*arctan(1/2*(2*b^2*x+2*a*b)/b))-1/3*(2*I*a-a^2+1)/(a ^2+1)/x^3+b*(I*a^2-I+2*a)/(a^2+1)^2/x^2-2*b^2*(I*a^3-3*I*a+3*a^2-1)/(a^2+1 )^3/x-2*b^3*(I*a^4-6*I*a^2+4*a^3+I-4*a)/(a^2+1)^4*ln(x)
Time = 0.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^4} \, dx=\frac {-6 i \, b^{3} x^{3} \log \left (x\right ) + 6 i \, b^{3} x^{3} \log \left (\frac {b x + a + i}{b}\right ) - 6 \, {\left (i \, a - 1\right )} b^{2} x^{2} + a^{4} + 2 i \, a^{3} - 3 \, {\left (-i \, a^{2} + 2 \, a + i\right )} b x + 2 i \, a - 1}{3 \, {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{3}} \] Input:
integrate((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^4,x, algorithm="fricas")
Output:
1/3*(-6*I*b^3*x^3*log(x) + 6*I*b^3*x^3*log((b*x + a + I)/b) - 6*(I*a - 1)* b^2*x^2 + a^4 + 2*I*a^3 - 3*(-I*a^2 + 2*a + I)*b*x + 2*I*a - 1)/((a^4 + 4* I*a^3 - 6*a^2 - 4*I*a + 1)*x^3)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (73) = 146\).
Time = 0.56 (sec) , antiderivative size = 286, normalized size of antiderivative = 3.08 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^4} \, dx=- \frac {2 i b^{3} \log {\left (- \frac {2 a^{5} b^{3}}{\left (a + i\right )^{4}} - \frac {10 i a^{4} b^{3}}{\left (a + i\right )^{4}} + \frac {20 a^{3} b^{3}}{\left (a + i\right )^{4}} + \frac {20 i a^{2} b^{3}}{\left (a + i\right )^{4}} + 2 a b^{3} - \frac {10 a b^{3}}{\left (a + i\right )^{4}} + 4 b^{4} x + 2 i b^{3} - \frac {2 i b^{3}}{\left (a + i\right )^{4}} \right )}}{\left (a + i\right )^{4}} + \frac {2 i b^{3} \log {\left (\frac {2 a^{5} b^{3}}{\left (a + i\right )^{4}} + \frac {10 i a^{4} b^{3}}{\left (a + i\right )^{4}} - \frac {20 a^{3} b^{3}}{\left (a + i\right )^{4}} - \frac {20 i a^{2} b^{3}}{\left (a + i\right )^{4}} + 2 a b^{3} + \frac {10 a b^{3}}{\left (a + i\right )^{4}} + 4 b^{4} x + 2 i b^{3} + \frac {2 i b^{3}}{\left (a + i\right )^{4}} \right )}}{\left (a + i\right )^{4}} - \frac {- a^{3} - i a^{2} - a + 6 i b^{2} x^{2} + x \left (- 3 i a b + 3 b\right ) - i}{x^{3} \cdot \left (3 a^{3} + 9 i a^{2} - 9 a - 3 i\right )} \] Input:
integrate((1+I*(b*x+a))**2/(1+(b*x+a)**2)/x**4,x)
Output:
-2*I*b**3*log(-2*a**5*b**3/(a + I)**4 - 10*I*a**4*b**3/(a + I)**4 + 20*a** 3*b**3/(a + I)**4 + 20*I*a**2*b**3/(a + I)**4 + 2*a*b**3 - 10*a*b**3/(a + I)**4 + 4*b**4*x + 2*I*b**3 - 2*I*b**3/(a + I)**4)/(a + I)**4 + 2*I*b**3*l og(2*a**5*b**3/(a + I)**4 + 10*I*a**4*b**3/(a + I)**4 - 20*a**3*b**3/(a + I)**4 - 20*I*a**2*b**3/(a + I)**4 + 2*a*b**3 + 10*a*b**3/(a + I)**4 + 4*b* *4*x + 2*I*b**3 + 2*I*b**3/(a + I)**4)/(a + I)**4 - (-a**3 - I*a**2 - a + 6*I*b**2*x**2 + x*(-3*I*a*b + 3*b) - I)/(x**3*(3*a**3 + 9*I*a**2 - 9*a - 3 *I))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (69) = 138\).
Time = 0.12 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.83 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^4} \, dx=\frac {2 \, {\left (a^{4} - 4 i \, a^{3} - 6 \, a^{2} + 4 i \, a + 1\right )} b^{3} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1} + \frac {{\left (i \, a^{4} + 4 \, a^{3} - 6 i \, a^{2} - 4 \, a + i\right )} b^{3} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1} - \frac {2 \, {\left (i \, a^{4} + 4 \, a^{3} - 6 i \, a^{2} - 4 \, a + i\right )} b^{3} \log \left (x\right )}{a^{8} + 4 \, a^{6} + 6 \, a^{4} + 4 \, a^{2} + 1} + \frac {a^{6} - 2 i \, a^{5} + 6 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} b^{2} x^{2} + a^{4} - 4 i \, a^{3} + 3 \, {\left (i \, a^{4} + 2 \, a^{3} + 2 \, a - i\right )} b x - a^{2} - 2 i \, a - 1}{3 \, {\left (a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1\right )} x^{3}} \] Input:
integrate((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^4,x, algorithm="maxima")
Output:
2*(a^4 - 4*I*a^3 - 6*a^2 + 4*I*a + 1)*b^3*arctan((b^2*x + a*b)/b)/(a^8 + 4 *a^6 + 6*a^4 + 4*a^2 + 1) + (I*a^4 + 4*a^3 - 6*I*a^2 - 4*a + I)*b^3*log(b^ 2*x^2 + 2*a*b*x + a^2 + 1)/(a^8 + 4*a^6 + 6*a^4 + 4*a^2 + 1) - 2*(I*a^4 + 4*a^3 - 6*I*a^2 - 4*a + I)*b^3*log(x)/(a^8 + 4*a^6 + 6*a^4 + 4*a^2 + 1) + 1/3*(a^6 - 2*I*a^5 + 6*(-I*a^3 - 3*a^2 + 3*I*a + 1)*b^2*x^2 + a^4 - 4*I*a^ 3 + 3*(I*a^4 + 2*a^3 + 2*a - I)*b*x - a^2 - 2*I*a - 1)/((a^6 + 3*a^4 + 3*a ^2 + 1)*x^3)
Time = 0.13 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.35 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^4} \, dx=\frac {2 \, b^{4} \log \left (b x + a + i\right )}{-i \, a^{4} b + 4 \, a^{3} b + 6 i \, a^{2} b - 4 \, a b - i \, b} + \frac {2 \, b^{3} \log \left ({\left | x \right |}\right )}{i \, a^{4} - 4 \, a^{3} - 6 i \, a^{2} + 4 \, a + i} + \frac {a^{4} + 2 i \, a^{3} - 6 i \, {\left (a b^{2} + i \, b^{2}\right )} x^{2} + 3 i \, {\left (a^{2} b + 2 i \, a b - b\right )} x + 2 i \, a - 1}{3 \, {\left (a + i\right )}^{4} x^{3}} \] Input:
integrate((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^4,x, algorithm="giac")
Output:
2*b^4*log(b*x + a + I)/(-I*a^4*b + 4*a^3*b + 6*I*a^2*b - 4*a*b - I*b) + 2* b^3*log(abs(x))/(I*a^4 - 4*a^3 - 6*I*a^2 + 4*a + I) + 1/3*(a^4 + 2*I*a^3 - 6*I*(a*b^2 + I*b^2)*x^2 + 3*I*(a^2*b + 2*I*a*b - b)*x + 2*I*a - 1)/((a + I)^4*x^3)
Time = 23.46 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.14 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^4} \, dx=\frac {\frac {a-\mathrm {i}}{3\,\left (a+1{}\mathrm {i}\right )}-\frac {b^2\,x^2\,2{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^3}+\frac {b\,x\,1{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^2}}{x^3}+\frac {b^3\,\mathrm {atanh}\left (\frac {a^4+a^3\,4{}\mathrm {i}-6\,a^2-a\,4{}\mathrm {i}+1}{{\left (a+1{}\mathrm {i}\right )}^4}-\frac {x\,\left (2\,a^{12}\,b^2+12\,a^{10}\,b^2+30\,a^8\,b^2+40\,a^6\,b^2+30\,a^4\,b^2+12\,a^2\,b^2+2\,b^2\right )}{{\left (a+1{}\mathrm {i}\right )}^4\,\left (-b\,a^9+3{}\mathrm {i}\,b\,a^8+8{}\mathrm {i}\,b\,a^6+6\,b\,a^5+6{}\mathrm {i}\,b\,a^4+8\,b\,a^3+3\,b\,a-b\,1{}\mathrm {i}\right )}\right )\,4{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^4} \] Input:
int((a*1i + b*x*1i + 1)^2/(x^4*((a + b*x)^2 + 1)),x)
Output:
((a - 1i)/(3*(a + 1i)) - (b^2*x^2*2i)/(a + 1i)^3 + (b*x*1i)/(a + 1i)^2)/x^ 3 + (b^3*atanh((a^3*4i - 6*a^2 - a*4i + a^4 + 1)/(a + 1i)^4 - (x*(2*b^2 + 12*a^2*b^2 + 30*a^4*b^2 + 40*a^6*b^2 + 30*a^8*b^2 + 12*a^10*b^2 + 2*a^12*b ^2))/((a + 1i)^4*(3*a*b - b*1i + 8*a^3*b + a^4*b*6i + 6*a^5*b + a^6*b*8i + a^8*b*3i - a^9*b)))*4i)/(a + 1i)^4
Time = 0.16 (sec) , antiderivative size = 464, normalized size of antiderivative = 4.99 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^4} \, dx=\frac {-1+2 a^{6}+6 \mathit {atan} \left (b x +a \right ) b^{3} x^{3}+6 a^{5} b x -6 a^{5} i -18 \,\mathrm {log}\left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{2} b^{3} i \,x^{3}-6 \,\mathrm {log}\left (x \right ) a^{4} b^{3} i \,x^{3}+36 \,\mathrm {log}\left (x \right ) a^{2} b^{3} i \,x^{3}+12 a^{3} b^{2} i \,x^{2}+18 a \,b^{2} i \,x^{2}-6 a^{5} b^{2} i \,x^{2}-3 b i x -18 a^{4} b^{2} x^{2}+6 b^{2} x^{2}+3 a^{4} b i x -2 a^{7} i +6 \mathit {atan} \left (b x +a \right ) a^{4} b^{3} x^{3}-36 \mathit {atan} \left (b x +a \right ) a^{2} b^{3} x^{3}+12 \,\mathrm {log}\left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{3} b^{3} x^{3}-12 \,\mathrm {log}\left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a \,b^{3} x^{3}+3 \,\mathrm {log}\left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) b^{3} i \,x^{3}-24 \,\mathrm {log}\left (x \right ) a^{3} b^{3} x^{3}+24 \,\mathrm {log}\left (x \right ) a \,b^{3} x^{3}-6 \,\mathrm {log}\left (x \right ) b^{3} i \,x^{3}+3 a^{6} b i x -2 a^{2}-3 a^{2} b i x -6 a^{3} i -2 a i +12 a^{3} b x -12 a^{2} b^{2} x^{2}+6 a b x -24 \mathit {atan} \left (b x +a \right ) a^{3} b^{3} i \,x^{3}+24 \mathit {atan} \left (b x +a \right ) a \,b^{3} i \,x^{3}+3 \,\mathrm {log}\left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{4} b^{3} i \,x^{3}+a^{8}}{3 x^{3} \left (a^{8}+4 a^{6}+6 a^{4}+4 a^{2}+1\right )} \] Input:
int((1+I*(b*x+a))^2/(1+(b*x+a)^2)/x^4,x)
Output:
(6*atan(a + b*x)*a**4*b**3*x**3 - 24*atan(a + b*x)*a**3*b**3*i*x**3 - 36*a tan(a + b*x)*a**2*b**3*x**3 + 24*atan(a + b*x)*a*b**3*i*x**3 + 6*atan(a + b*x)*b**3*x**3 + 3*log(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**4*b**3*i*x**3 + 12*log(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**3*b**3*x**3 - 18*log(a**2 + 2*a* b*x + b**2*x**2 + 1)*a**2*b**3*i*x**3 - 12*log(a**2 + 2*a*b*x + b**2*x**2 + 1)*a*b**3*x**3 + 3*log(a**2 + 2*a*b*x + b**2*x**2 + 1)*b**3*i*x**3 - 6*l og(x)*a**4*b**3*i*x**3 - 24*log(x)*a**3*b**3*x**3 + 36*log(x)*a**2*b**3*i* x**3 + 24*log(x)*a*b**3*x**3 - 6*log(x)*b**3*i*x**3 + a**8 - 2*a**7*i + 3* a**6*b*i*x + 2*a**6 - 6*a**5*b**2*i*x**2 + 6*a**5*b*x - 6*a**5*i - 18*a**4 *b**2*x**2 + 3*a**4*b*i*x + 12*a**3*b**2*i*x**2 + 12*a**3*b*x - 6*a**3*i - 12*a**2*b**2*x**2 - 3*a**2*b*i*x - 2*a**2 + 18*a*b**2*i*x**2 + 6*a*b*x - 2*a*i + 6*b**2*x**2 - 3*b*i*x - 1)/(3*x**3*(a**8 + 4*a**6 + 6*a**4 + 4*a** 2 + 1))