Integrand size = 22, antiderivative size = 169 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}-\frac {\left (10+3 a+\sqrt {4+a}\right ) \arctan \left (\frac {-1+x}{\sqrt {1-\sqrt {4+a}}}\right )}{8 (3+a) (4+a)^{3/2} \sqrt {1-\sqrt {4+a}}}+\frac {\left (10+3 a-\sqrt {4+a}\right ) \arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )}{8 (3+a) (4+a)^{3/2} \sqrt {1+\sqrt {4+a}}} \]
1/4*(5+a+(-1+x)^2)*(-1+x)/(a^2+7*a+12)/(3+a-2*(-1+x)^2-(-1+x)^4)-1/8*arcta n((-1+x)/(1-(4+a)^(1/2))^(1/2))*(10+3*a+(4+a)^(1/2))/(3+a)/(4+a)^(3/2)/(1- (4+a)^(1/2))^(1/2)+1/8*arctan((-1+x)/(1+(4+a)^(1/2))^(1/2))*(10+3*a-(4+a)^ (1/2))/(3+a)/(4+a)^(3/2)/(1+(4+a)^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\frac {(-1+x) \left (6+a-2 x+x^2\right )}{4 (3+a) (4+a) \left (a-x \left (-8+8 x-4 x^2+x^3\right )\right )}-\frac {\text {RootSum}\left [a+8 \text {$\#$1}-8 \text {$\#$1}^2+4 \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {12 \log (x-\text {$\#$1})+3 a \log (x-\text {$\#$1})-2 \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2}{-2+4 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]}{16 \left (12+7 a+a^2\right )} \]
((-1 + x)*(6 + a - 2*x + x^2))/(4*(3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4*x^2 + x^3))) - RootSum[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , (12*Log[x - #1] + 3*a*Log[x - #1] - 2*Log[x - #1]*#1 + Log[x - #1]*#1^2)/(-2 + 4*#1 - 3*#1 ^2 + #1^3) & ]/(16*(12 + 7*a + a^2))
Time = 0.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2458, 1405, 27, 1480, 217}
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.
| \(\displaystyle \int \frac {1}{\left (a-x^4+4 x^3-8 x^2+8 x\right )^2} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \frac {1}{\left (a-(x-1)^4-2 (x-1)^2+3\right )^2}d(x-1)\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle \frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\int -\frac {2 \left ((x-1)^2+3 a+11\right )}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{8 \left (a^2+7 a+12\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(x-1)^2+3 a+11}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {1}{2} \left (1-\frac {3 a+10}{\sqrt {a+4}}\right ) \int \frac {1}{-(x-1)^2-\sqrt {a+4}-1}d(x-1)+\frac {1}{2} \left (\frac {3 a+10}{\sqrt {a+4}}+1\right ) \int \frac {1}{-(x-1)^2+\sqrt {a+4}-1}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\left (\frac {3 a+10}{\sqrt {a+4}}+1\right ) \arctan \left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {1-\sqrt {a+4}}}-\frac {\left (1-\frac {3 a+10}{\sqrt {a+4}}\right ) \arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {\sqrt {a+4}+1}}}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
((5 + a + (-1 + x)^2)*(-1 + x))/(4*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (-1/2*((1 + (10 + 3*a)/Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[ 1 - Sqrt[4 + a]]])/Sqrt[1 - Sqrt[4 + a]] - ((1 - (10 + 3*a)/Sqrt[4 + a])*A rcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]])/(2*Sqrt[1 + Sqrt[4 + a]]))/(4*(12 + 7*a + a^2))
3.2.21.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 - 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) ), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(b^2 - 2*a*c + 2*(p + 1)*( b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(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]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {\frac {x^{3}}{4 a^{2}+28 a +48}-\frac {3 x^{2}}{4 \left (4+a \right ) \left (3+a \right )}+\frac {\left (a +8\right ) x}{4 a^{2}+28 a +48}-\frac {6+a}{4 \left (a^{2}+7 a +12\right )}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} +3 a +12\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}}{16 \left (4+a \right ) \left (3+a \right )}\) | \(158\) |
| risch | \(\frac {\frac {x^{3}}{4 a^{2}+28 a +48}-\frac {3 x^{2}}{4 \left (4+a \right ) \left (3+a \right )}+\frac {\left (a +8\right ) x}{4 a^{2}+28 a +48}-\frac {6+a}{4 \left (a^{2}+7 a +12\right )}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (\frac {\textit {\_R}^{2}}{a^{2}+7 a +12}-\frac {2 \textit {\_R}}{a^{2}+7 a +12}+\frac {3}{3+a}\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{16}\) | \(172\) |
(1/4/(a^2+7*a+12)*x^3-3/4/(4+a)/(3+a)*x^2+1/4*(a+8)/(a^2+7*a+12)*x-1/4*(6+ a)/(a^2+7*a+12))/(-x^4+4*x^3-8*x^2+a+8*x)+1/16/(4+a)/(3+a)*sum((_R^2-2*_R+ 3*a+12)/(-_R^3+3*_R^2-4*_R+2)*ln(x-_R),_R=RootOf(_Z^4-4*_Z^3+8*_Z^2-8*_Z-a ))
Leaf count of result is larger than twice the leaf count of optimal. 1948 vs. \(2 (139) = 278\).
Time = 0.27 (sec) , antiderivative size = 1948, normalized size of antiderivative = 11.53 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\text {Too large to display} \]
-1/16*(4*x^3 - ((a^2 + 7*a + 12)*x^4 - 4*(a^2 + 7*a + 12)*x^3 - a^3 + 8*(a ^2 + 7*a + 12)*x^2 - 7*a^2 - 8*(a^2 + 7*a + 12)*x - 12*a)*sqrt((15*a^2 + ( a^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728)*sqrt((81*a^2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^4 + 111105*a^3 + 156492*a^2 + 128304*a + 46656)) + 105*a + 184)/(a^6 + 21*a^ 5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728))*log(-81*a^2 + (81*a^2 + 567*a + 992)*x + (27*a^4 + 408*a^3 + 2309*a^2 - 2*(2*a^7 + 49*a^6 + 513*a ^5 + 2975*a^4 + 10321*a^3 + 21420*a^2 + 24624*a + 12096)*sqrt((81*a^2 + 55 8*a + 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^4 + 11 1105*a^3 + 156492*a^2 + 128304*a + 46656)) + 5800*a + 5456)*sqrt((15*a^2 + (a^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728)*sqrt((81*a^ 2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^ 4 + 111105*a^3 + 156492*a^2 + 128304*a + 46656)) + 105*a + 184)/(a^6 + 21* a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728)) - 567*a - 992) + ((a^ 2 + 7*a + 12)*x^4 - 4*(a^2 + 7*a + 12)*x^3 - a^3 + 8*(a^2 + 7*a + 12)*x^2 - 7*a^2 - 8*(a^2 + 7*a + 12)*x - 12*a)*sqrt((15*a^2 + (a^6 + 21*a^5 + 183* a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728)*sqrt((81*a^2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^4 + 111105*a^3 + 1564 92*a^2 + 128304*a + 46656)) + 105*a + 184)/(a^6 + 21*a^5 + 183*a^4 + 847*a ^3 + 2196*a^2 + 3024*a + 1728))*log(-81*a^2 + (81*a^2 + 567*a + 992)*x ...
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (144) = 288\).
Time = 3.34 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.74 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\frac {a - x^{3} + 3 x^{2} + x \left (- a - 8\right ) + 6}{- 4 a^{3} - 28 a^{2} - 48 a + x^{4} \cdot \left (4 a^{2} + 28 a + 48\right ) + x^{3} \left (- 16 a^{2} - 112 a - 192\right ) + x^{2} \cdot \left (32 a^{2} + 224 a + 384\right ) + x \left (- 32 a^{2} - 224 a - 384\right )} + \operatorname {RootSum} {\left (t^{4} \cdot \left (65536 a^{9} + 2162688 a^{8} + 31653888 a^{7} + 269680640 a^{6} + 1473773568 a^{5} + 5357174784 a^{4} + 12952010752 a^{3} + 20082327552 a^{2} + 18119393280 a + 7247757312\right ) + t^{2} \left (- 7680 a^{5} - 145920 a^{4} - 1107968 a^{3} - 4202496 a^{2} - 7962624 a - 6029312\right ) - 81 a^{2} - 576 a - 1024, \left ( t \mapsto t \log {\left (x + \frac {- 16384 t^{3} a^{7} - 401408 t^{3} a^{6} - 4202496 t^{3} a^{5} - 24371200 t^{3} a^{4} - 84549632 t^{3} a^{3} - 175472640 t^{3} a^{2} - 201719808 t^{3} a - 99090432 t^{3} + 432 t a^{4} + 7488 t a^{3} + 47024 t a^{2} + 128096 t a + 128512 t - 81 a^{2} - 567 a - 992}{81 a^{2} + 567 a + 992} \right )} \right )\right )} \]
(a - x**3 + 3*x**2 + x*(-a - 8) + 6)/(-4*a**3 - 28*a**2 - 48*a + x**4*(4*a **2 + 28*a + 48) + x**3*(-16*a**2 - 112*a - 192) + x**2*(32*a**2 + 224*a + 384) + x*(-32*a**2 - 224*a - 384)) + RootSum(_t**4*(65536*a**9 + 2162688* a**8 + 31653888*a**7 + 269680640*a**6 + 1473773568*a**5 + 5357174784*a**4 + 12952010752*a**3 + 20082327552*a**2 + 18119393280*a + 7247757312) + _t** 2*(-7680*a**5 - 145920*a**4 - 1107968*a**3 - 4202496*a**2 - 7962624*a - 60 29312) - 81*a**2 - 576*a - 1024, Lambda(_t, _t*log(x + (-16384*_t**3*a**7 - 401408*_t**3*a**6 - 4202496*_t**3*a**5 - 24371200*_t**3*a**4 - 84549632* _t**3*a**3 - 175472640*_t**3*a**2 - 201719808*_t**3*a - 99090432*_t**3 + 4 32*_t*a**4 + 7488*_t*a**3 + 47024*_t*a**2 + 128096*_t*a + 128512*_t - 81*a **2 - 567*a - 992)/(81*a**2 + 567*a + 992))))
\[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\int { \frac {1}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{2}} \,d x } \]
-1/4*(x^3 + (a + 8)*x - 3*x^2 - a - 6)/((a^2 + 7*a + 12)*x^4 - 4*(a^2 + 7* a + 12)*x^3 - a^3 + 8*(a^2 + 7*a + 12)*x^2 - 7*a^2 - 8*(a^2 + 7*a + 12)*x - 12*a) - 1/4*integrate((x^2 + 3*a - 2*x + 12)/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), x)/(a^2 + 7*a + 12)
Leaf count of result is larger than twice the leaf count of optimal. 8503 vs. \(2 (139) = 278\).
Time = 6.37 (sec) , antiderivative size = 8503, normalized size of antiderivative = 50.31 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\text {Too large to display} \]
1/16*(sqrt((15*a^3 + 165*a^2 + (9*a^3 + 103*a^2 + 392*a + 496)*sqrt(a + 4) + 604*a + 736)/(a^3 + 11*a^2 + 40*a + 48))*log(abs(243*sqrt(a + 4)*a^10 + 324*a^10 + 8640*sqrt(a + 4)*a^9 + 81*sqrt(15*a^4 + 210*a^3 + 1099*a^2 + ( 9*a^4 + 130*a^3 + 701*a^2 + 1672*a + 1488)*sqrt(a + 4) + 2548*a + 2208)*a^ 8*x + 11466*a^9 + 81*sqrt(15*a^4 + 210*a^3 + 1099*a^2 + (9*a^4 + 130*a^3 + 701*a^2 + 1672*a + 1488)*sqrt(a + 4) + 2548*a + 2208)*sqrt(a + 4)*a^7*x - 81*sqrt(15*a^4 + 210*a^3 + 1099*a^2 + (9*a^4 + 130*a^3 + 701*a^2 + 1672*a + 1488)*sqrt(a + 4) + 2548*a + 2208)*a^8 + 138027*sqrt(a + 4)*a^8 + 2340* sqrt(15*a^4 + 210*a^3 + 1099*a^2 + (9*a^4 + 130*a^3 + 701*a^2 + 1672*a + 1 488)*sqrt(a + 4) + 2548*a + 2208)*a^7*x - 81*sqrt(15*a^4 + 210*a^3 + 1099* a^2 + (9*a^4 + 130*a^3 + 701*a^2 + 1672*a + 1488)*sqrt(a + 4) + 2548*a + 2 208)*sqrt(a + 4)*a^7 + 182314*a^8 + 2016*sqrt(15*a^4 + 210*a^3 + 1099*a^2 + (9*a^4 + 130*a^3 + 701*a^2 + 1672*a + 1488)*sqrt(a + 4) + 2548*a + 2208) *sqrt(a + 4)*a^6*x - 2340*sqrt(15*a^4 + 210*a^3 + 1099*a^2 + (9*a^4 + 130* a^3 + 701*a^2 + 1672*a + 1488)*sqrt(a + 4) + 2548*a + 2208)*a^7 + 1304648* sqrt(a + 4)*a^7 + 29518*sqrt(15*a^4 + 210*a^3 + 1099*a^2 + (9*a^4 + 130*a^ 3 + 701*a^2 + 1672*a + 1488)*sqrt(a + 4) + 2548*a + 2208)*a^6*x - 2016*sqr t(15*a^4 + 210*a^3 + 1099*a^2 + (9*a^4 + 130*a^3 + 701*a^2 + 1672*a + 1488 )*sqrt(a + 4) + 2548*a + 2208)*sqrt(a + 4)*a^6 + 1715172*a^7 + 21454*sqrt( 15*a^4 + 210*a^3 + 1099*a^2 + (9*a^4 + 130*a^3 + 701*a^2 + 1672*a + 148...
Time = 11.03 (sec) , antiderivative size = 4591, normalized size of antiderivative = 27.17 \[ \int \frac {1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\text {Too large to display} \]
atan(-(((15552*a - 9*a*((a + 4)^9)^(1/2) - 31*((a + 4)^9)^(1/2) + 8208*a^2 + 2164*a^3 + 285*a^4 + 15*a^5 + 11776)/(256*(276480*a + 306432*a^2 + 1976 32*a^3 + 81744*a^4 + 22488*a^5 + 4115*a^6 + 483*a^7 + 33*a^8 + a^9 + 11059 2)))^(1/2)*((((15728640*a + 10878976*a^2 + 3997696*a^3 + 823296*a^4 + 9011 2*a^5 + 4096*a^6 + 9437184)/(64*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576)) - (x*(208896*a + 117760*a^2 + 33024*a^3 + 4608*a^4 + 256*a^5 + 147 456))/(4*(168*a + 73*a^2 + 14*a^3 + a^4 + 144)))*((15552*a - 9*a*((a + 4)^ 9)^(1/2) - 31*((a + 4)^9)^(1/2) + 8208*a^2 + 2164*a^3 + 285*a^4 + 15*a^5 + 11776)/(256*(276480*a + 306432*a^2 + 197632*a^3 + 81744*a^4 + 22488*a^5 + 4115*a^6 + 483*a^7 + 33*a^8 + a^9 + 110592)))^(1/2) - (733184*a + 396288* a^2 + 106752*a^3 + 14336*a^4 + 768*a^5 + 540672)/(64*(816*a + 460*a^2 + 12 9*a^3 + 18*a^4 + a^5 + 576)))*((15552*a - 9*a*((a + 4)^9)^(1/2) - 31*((a + 4)^9)^(1/2) + 8208*a^2 + 2164*a^3 + 285*a^4 + 15*a^5 + 11776)/(256*(27648 0*a + 306432*a^2 + 197632*a^3 + 81744*a^4 + 22488*a^5 + 4115*a^6 + 483*a^7 + 33*a^8 + a^9 + 110592)))^(1/2) + (5568*a + 1552*a^2 + 144*a^3 + 6656)/( 64*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576)) - (x*(61*a + 9*a^2 + 104))/(4*(168*a + 73*a^2 + 14*a^3 + a^4 + 144)))*1i + ((15552*a - 9*a*((a + 4)^9)^(1/2) - 31*((a + 4)^9)^(1/2) + 8208*a^2 + 2164*a^3 + 285*a^4 + 15* a^5 + 11776)/(256*(276480*a + 306432*a^2 + 197632*a^3 + 81744*a^4 + 22488* a^5 + 4115*a^6 + 483*a^7 + 33*a^8 + a^9 + 110592)))^(1/2)*((((15728640*...