Integrand size = 24, antiderivative size = 231 \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\frac {1+(-1+x)^2}{4 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}-\frac {\left (10+3 a+\sqrt {4+a}\right ) \arctan \left (\frac {-1+x}{\sqrt {1-\sqrt {4+a}}}\right )}{8 (3+a) (4+a)^{3/2} \sqrt {1-\sqrt {4+a}}}+\frac {\left (10+3 a-\sqrt {4+a}\right ) \arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )}{8 (3+a) (4+a)^{3/2} \sqrt {1+\sqrt {4+a}}}+\frac {\text {arctanh}\left (\frac {1+(-1+x)^2}{\sqrt {4+a}}\right )}{4 (4+a)^{3/2}} \]
1/4*(1+(-1+x)^2)/(4+a)/(3+a-2*(-1+x)^2-(-1+x)^4)+1/4*(5+a+(-1+x)^2)*(-1+x) /(a^2+7*a+12)/(3+a-2*(-1+x)^2-(-1+x)^4)+1/4*arctanh((1+(-1+x)^2)/(4+a)^(1/ 2))/(4+a)^(3/2)-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)+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 = 166, normalized size of antiderivative = 0.72 \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\frac {a+2 x-a x+a x^2+x^3}{4 (3+a) (4+a) \left (a-x \left (-8+8 x-4 x^2+x^3\right )\right )}-\frac {\text {RootSum}\left [a+8 \text {$\#$1}-8 \text {$\#$1}^2+4 \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {6 \log (x-\text {$\#$1})+a \log (x-\text {$\#$1})+4 \log (x-\text {$\#$1}) \text {$\#$1}+2 a \log (x-\text {$\#$1}) \text {$\#$1}+\log (x-\text {$\#$1}) \text {$\#$1}^2}{-2+4 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]}{16 \left (12+7 a+a^2\right )} \]
(a + 2*x - a*x + a*x^2 + x^3)/(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 & , (6*Log[x - #1] + a*Log[x - #1] + 4*Log[x - #1]*#1 + 2*a*Log[x - #1]*#1 + Log[x - #1]*#1^2)/ (-2 + 4*#1 - 3*#1^2 + #1^3) & ]/(16*(12 + 7*a + a^2))
Time = 0.45 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2459, 2202, 1405, 27, 1432, 1086, 1083, 219, 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.
| \(\displaystyle \int \frac {x}{\left (a-x^4+4 x^3-8 x^2+8 x\right )^2} \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {x}{\left (a-(x-1)^4-2 (x-1)^2+3\right )^2}d(x-1)\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)+\int \frac {x-1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)\) |
| \(\Big \downarrow \) 1405 |
| \(\displaystyle -\frac {\int -\frac {2 \left ((x-1)^2+3 a+11\right )}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{8 \left (a^2+7 a+12\right )}+\int \frac {x-1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(x-1)^2+3 a+11}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{4 \left (a^2+7 a+12\right )}+\int \frac {x-1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle \frac {\int \frac {(x-1)^2+3 a+11}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {1}{2} \int \frac {1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^2}d(x-1)^2+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle \frac {\int \frac {(x-1)^2+3 a+11}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {1}{2} \left (\frac {\int \frac {1}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)^2}{2 (a+4)}+\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\right )+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\int \frac {(x-1)^2+3 a+11}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {1}{2} \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\int \frac {1}{4 (a+4)-(x-1)^4}d\left (-2 (x-1)^2-2\right )}{a+4}\right )+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {(x-1)^2+3 a+11}{-(x-1)^4-2 (x-1)^2+a+3}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac {1}{2} \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}\right )\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {1}{2} \left (1-\frac {3 a+10}{\sqrt {a+4}}\right ) \int \frac {1}{-(x-1)^2-\sqrt {a+4}-1}d(x-1)+\frac {1}{2} \left (\frac {3 a+10}{\sqrt {a+4}}+1\right ) \int \frac {1}{-(x-1)^2+\sqrt {a+4}-1}d(x-1)}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac {1}{2} \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\left (\frac {3 a+10}{\sqrt {a+4}}+1\right ) \arctan \left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {1-\sqrt {a+4}}}-\frac {\left (1-\frac {3 a+10}{\sqrt {a+4}}\right ) \arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {\sqrt {a+4}+1}}}{4 \left (a^2+7 a+12\right )}+\frac {(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac {1}{2} \left (\frac {x}{2 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac {\text {arctanh}\left (\frac {-2 (x-1)^2-2}{2 \sqrt {a+4}}\right )}{2 (a+4)^{3/2}}\right )\) |
((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)) + (x/(2*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ArcTan h[(-2 - 2*(-1 + x)^2)/(2*Sqrt[4 + a])]/(2*(4 + a)^(3/2)))/2
3.2.28.3.1 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)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
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[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
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_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.68
| method | result | size |
| default | \(\frac {\frac {x^{3}}{4 a^{2}+28 a +48}+\frac {a \,x^{2}}{4 \left (4+a \right ) \left (3+a \right )}-\frac {\left (a -2\right ) x}{4 \left (a^{2}+7 a +12\right )}+\frac {a}{4 a^{2}+28 a +48}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (6+\textit {\_R}^{2}+2 \left (a +2\right ) \textit {\_R} +a \right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}}{16 a^{2}+112 a +192}\) | \(158\) |
| risch | \(\frac {\frac {x^{3}}{4 a^{2}+28 a +48}+\frac {a \,x^{2}}{4 \left (4+a \right ) \left (3+a \right )}-\frac {\left (a -2\right ) x}{4 \left (a^{2}+7 a +12\right )}+\frac {a}{4 a^{2}+28 a +48}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (\frac {\textit {\_R}^{2}}{a^{2}+7 a +12}+\frac {2 \left (a +2\right ) \textit {\_R}}{a^{2}+7 a +12}+\frac {6+a}{a^{2}+7 a +12}\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{16}\) | \(181\) |
(1/4/(a^2+7*a+12)*x^3+1/4*a/(4+a)/(3+a)*x^2-1/4*(a-2)/(a^2+7*a+12)*x+1/4/( a^2+7*a+12)*a)/(-x^4+4*x^3-8*x^2+a+8*x)+1/16/(a^2+7*a+12)*sum((6+_R^2+2*(a +2)*_R+a)/(-_R^3+3*_R^2-4*_R+2)*ln(x-_R),_R=RootOf(_Z^4-4*_Z^3+8*_Z^2-8*_Z -a))
Timed out. \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (197) = 394\).
Time = 16.40 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.33 \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\frac {- a x^{2} - a - x^{3} + x \left (a - 2\right )}{- 4 a^{3} - 28 a^{2} - 48 a + x^{4} \cdot \left (4 a^{2} + 28 a + 48\right ) + x^{3} \left (- 16 a^{2} - 112 a - 192\right ) + x^{2} \cdot \left (32 a^{2} + 224 a + 384\right ) + x \left (- 32 a^{2} - 224 a - 384\right )} + \operatorname {RootSum} {\left (t^{4} \cdot \left (65536 a^{9} + 2162688 a^{8} + 31653888 a^{7} + 269680640 a^{6} + 1473773568 a^{5} + 5357174784 a^{4} + 12952010752 a^{3} + 20082327552 a^{2} + 18119393280 a + 7247757312\right ) + t^{2} \left (- 2048 a^{6} - 50688 a^{5} - 520704 a^{4} - 2842624 a^{3} - 8699904 a^{2} - 14155776 a - 9568256\right ) + t \left (1152 a^{4} + 17792 a^{3} + 102912 a^{2} + 264192 a + 253952\right ) + 16 a^{3} - 57 a^{2} - 984 a - 2064, \left ( t \mapsto t \log {\left (x + \frac {98304 t^{3} a^{12} + 3948544 t^{3} a^{11} + 72196096 t^{3} a^{10} + 793837568 t^{3} a^{9} + 5839372288 t^{3} a^{8} + 30226464768 t^{3} a^{7} + 112668450816 t^{3} a^{6} + 303864643584 t^{3} a^{5} + 586157391872 t^{3} a^{4} + 784017129472 t^{3} a^{3} + 683648483328 t^{3} a^{2} + 343136010240 t^{3} a + 72477573120 t^{3} + 30208 t^{2} a^{10} + 986624 t^{2} a^{9} + 14420992 t^{2} a^{8} + 124156928 t^{2} a^{7} + 696815104 t^{2} a^{6} + 2661758464 t^{2} a^{5} + 7001485312 t^{2} a^{4} + 12506562560 t^{2} a^{3} + 14494924800 t^{2} a^{2} + 9820569600 t^{2} a + 2944401408 t^{2} - 1536 t a^{9} - 52048 t a^{8} - 757040 t a^{7} - 6200656 t a^{6} - 31380496 t a^{5} - 100736416 t a^{4} - 200813696 t a^{3} - 228144640 t a^{2} - 114632704 t a - 2490368 t + 248 a^{7} + 6797 a^{6} + 71132 a^{5} + 369745 a^{4} + 987758 a^{3} + 1128896 a^{2} - 129568 a - 956416}{576 a^{7} + 10985 a^{6} + 88746 a^{5} + 396609 a^{4} + 1076268 a^{3} + 1826304 a^{2} + 1867776 a + 917504} \right )} \right )\right )} \]
(-a*x**2 - a - x**3 + x*(a - 2))/(-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 + 12 952010752*a**3 + 20082327552*a**2 + 18119393280*a + 7247757312) + _t**2*(- 2048*a**6 - 50688*a**5 - 520704*a**4 - 2842624*a**3 - 8699904*a**2 - 14155 776*a - 9568256) + _t*(1152*a**4 + 17792*a**3 + 102912*a**2 + 264192*a + 2 53952) + 16*a**3 - 57*a**2 - 984*a - 2064, Lambda(_t, _t*log(x + (98304*_t **3*a**12 + 3948544*_t**3*a**11 + 72196096*_t**3*a**10 + 793837568*_t**3*a **9 + 5839372288*_t**3*a**8 + 30226464768*_t**3*a**7 + 112668450816*_t**3* a**6 + 303864643584*_t**3*a**5 + 586157391872*_t**3*a**4 + 784017129472*_t **3*a**3 + 683648483328*_t**3*a**2 + 343136010240*_t**3*a + 72477573120*_t **3 + 30208*_t**2*a**10 + 986624*_t**2*a**9 + 14420992*_t**2*a**8 + 124156 928*_t**2*a**7 + 696815104*_t**2*a**6 + 2661758464*_t**2*a**5 + 7001485312 *_t**2*a**4 + 12506562560*_t**2*a**3 + 14494924800*_t**2*a**2 + 9820569600 *_t**2*a + 2944401408*_t**2 - 1536*_t*a**9 - 52048*_t*a**8 - 757040*_t*a** 7 - 6200656*_t*a**6 - 31380496*_t*a**5 - 100736416*_t*a**4 - 200813696*_t* a**3 - 228144640*_t*a**2 - 114632704*_t*a - 2490368*_t + 248*a**7 + 6797*a **6 + 71132*a**5 + 369745*a**4 + 987758*a**3 + 1128896*a**2 - 129568*a - 9 56416)/(576*a**7 + 10985*a**6 + 88746*a**5 + 396609*a**4 + 1076268*a**3...
\[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\int { \frac {x}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{2}} \,d x } \]
-1/4*(a*x^2 + x^3 - (a - 2)*x + a)/((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((2*(a + 2)*x + x^2 + a + 6)/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), x)/(a^2 + 7*a + 12)
\[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\int { \frac {x}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{2}} \,d x } \]
Time = 10.33 (sec) , antiderivative size = 1167, normalized size of antiderivative = 5.05 \[ \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx=\text {Too large to display} \]
symsum(log((35*a + 4*a^2 + 68)/(64*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a ^5 + 576)) - root(12952010752*a^3*z^4 + 31653888*a^7*z^4 + 2162688*a^8*z^4 + 65536*a^9*z^4 + 18119393280*a*z^4 + 20082327552*a^2*z^4 + 1473773568*a^ 5*z^4 + 5357174784*a^4*z^4 + 269680640*a^6*z^4 + 7247757312*z^4 - 8699904* a^2*z^2 - 2842624*a^3*z^2 - 520704*a^4*z^2 - 50688*a^5*z^2 - 2048*a^6*z^2 - 14155776*a*z^2 - 9568256*z^2 + 102912*a^2*z + 17792*a^3*z + 1152*a^4*z + 264192*a*z + 253952*z - 984*a - 57*a^2 + 16*a^3 - 2064, z, k)*((12800*a + 3600*a^2 + 336*a^3 + 15104)/(64*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576)) + root(12952010752*a^3*z^4 + 31653888*a^7*z^4 + 2162688*a^8*z^4 + 65536*a^9*z^4 + 18119393280*a*z^4 + 20082327552*a^2*z^4 + 1473773568*a^5* z^4 + 5357174784*a^4*z^4 + 269680640*a^6*z^4 + 7247757312*z^4 - 8699904*a^ 2*z^2 - 2842624*a^3*z^2 - 520704*a^4*z^2 - 50688*a^5*z^2 - 2048*a^6*z^2 - 14155776*a*z^2 - 9568256*z^2 + 102912*a^2*z + 17792*a^3*z + 1152*a^4*z + 2 64192*a*z + 253952*z - 984*a - 57*a^2 + 16*a^3 - 2064, z, k)*(root(1295201 0752*a^3*z^4 + 31653888*a^7*z^4 + 2162688*a^8*z^4 + 65536*a^9*z^4 + 181193 93280*a*z^4 + 20082327552*a^2*z^4 + 1473773568*a^5*z^4 + 5357174784*a^4*z^ 4 + 269680640*a^6*z^4 + 7247757312*z^4 - 8699904*a^2*z^2 - 2842624*a^3*z^2 - 520704*a^4*z^2 - 50688*a^5*z^2 - 2048*a^6*z^2 - 14155776*a*z^2 - 956825 6*z^2 + 102912*a^2*z + 17792*a^3*z + 1152*a^4*z + 264192*a*z + 253952*z - 984*a - 57*a^2 + 16*a^3 - 2064, z, k)*((15728640*a + 10878976*a^2 + 399...