3.2.28 \(\int \frac {x}{(a+8 x-8 x^2+4 x^3-x^4)^2} \, dx\) [128]

Optimal. Leaf size=231 \[ \frac {1+(-1+x)^2}{4 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}+\frac {\left (5+a+(-1+x)^2\right ) (-1+x)}{4 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )}-\frac {\left (10+3 a+\sqrt {4+a}\right ) \tan ^{-1}\left (\frac {-1+x}{\sqrt {1-\sqrt {4+a}}}\right )}{8 (3+a) (4+a)^{3/2} \sqrt {1-\sqrt {4+a}}}+\frac {\left (10+3 a-\sqrt {4+a}\right ) \tan ^{-1}\left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )}{8 (3+a) (4+a)^{3/2} \sqrt {1+\sqrt {4+a}}}+\frac {\tanh ^{-1}\left (\frac {1+(-1+x)^2}{\sqrt {4+a}}\right )}{4 (4+a)^{3/2}} \]

[Out]

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)

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Rubi [A]
time = 0.23, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1694, 1687, 1106, 1180, 210, 1121, 628, 632, 212} \begin {gather*} \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 {\left (3 a+\sqrt {a+4}+10\right ) \text {ArcTan}\left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{8 (a+3) (a+4)^{3/2} \sqrt {1-\sqrt {a+4}}}+\frac {\left (3 a-\sqrt {a+4}+10\right ) \text {ArcTan}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{8 (a+3) (a+4)^{3/2} \sqrt {\sqrt {a+4}+1}}+\frac {(x-1)^2+1}{4 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac {\tanh ^{-1}\left (\frac {(x-1)^2+1}{\sqrt {a+4}}\right )}{4 (a+4)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x]

[Out]

(1 + (-1 + x)^2)/(4*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((5 + a + (-1 + x)^2)*(-1 + x))/(4*(12 + 7*
a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((10 + 3*a + Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a]
]])/(8*(3 + a)*(4 + a)^(3/2)*Sqrt[1 - Sqrt[4 + a]]) + ((10 + 3*a - Sqrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 + Sqrt[
4 + a]]])/(8*(3 + a)*(4 + a)^(3/2)*Sqrt[1 + Sqrt[4 + a]]) + ArcTanh[(1 + (-1 + x)^2)/Sqrt[4 + a]]/(4*(4 + a)^(
3/2))

Rule 210

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])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

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] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1106

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] + Dist[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] /; FreeQ[{a, b, c}, x] && NeQ[b^2 -
4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1121

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 1180

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 1687

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rule 1694

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -d/(4*e) + x)*(a + d^4/(256*e^3)
- b*(d/(8*e)) + (c - 3*(d^2/(8*e)))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0]
&& NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx &=\text {Subst}\left (\int \frac {1+x}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )\\ &=\text {Subst}\left (\int \frac {1}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\right )+\text {Subst}\left (\int \frac {x}{\left (3+a-2 x^2-x^4\right )^2} \, dx,x,-1+x\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 {1}{2} \text {Subst}\left (\int \frac {1}{\left (3+a-2 x-x^2\right )^2} \, dx,x,(-1+x)^2\right )-\frac {\text {Subst}\left (\int \frac {4+2 (3+a)-2 (4+4 (3+a))-2 x^2}{3+a-2 x^2-x^4} \, dx,x,-1+x\right )}{8 \left (12+7 a+a^2\right )}\\ &=\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 {\text {Subst}\left (\int \frac {1}{3+a-2 x-x^2} \, dx,x,(-1+x)^2\right )}{4 (4+a)}-\frac {\left (10+3 a-\sqrt {4+a}\right ) \text {Subst}\left (\int \frac {1}{-1-\sqrt {4+a}-x^2} \, dx,x,-1+x\right )}{8 (3+a) (4+a)^{3/2}}+\frac {\left (10+3 a+\sqrt {4+a}\right ) \text {Subst}\left (\int \frac {1}{-1+\sqrt {4+a}-x^2} \, dx,x,-1+x\right )}{8 (3+a) (4+a)^{3/2}}\\ &=\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 ) \tan ^{-1}\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 ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{8 (3+a) (4+a)^{3/2} \sqrt {1+\sqrt {4+a}}}-\frac {\text {Subst}\left (\int \frac {1}{4 (4+a)-x^2} \, dx,x,-2 \left (1+(-1+x)^2\right )\right )}{2 (4+a)}\\ &=\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 ) \tan ^{-1}\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 ) \tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )}{8 (3+a) (4+a)^{3/2} \sqrt {1+\sqrt {4+a}}}+\frac {\tanh ^{-1}\left (\frac {1+(-1+x)^2}{\sqrt {4+a}}\right )}{4 (4+a)^{3/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.05, size = 166, normalized size = 0.72 \begin {gather*} \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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x]

[Out]

(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))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.03, size = 158, normalized size = 0.68

method	result	size
default	\(\frac {\frac {x^{3}}{4 a^{2}+28 a +48}+\frac {a \,x^{2}}{4 \left (4+a \right ) \left (3+a \right )}-\frac {\left (a -2\right ) x}{4 \left (a^{2}+7 a +12\right )}+\frac {a}{4 a^{2}+28 a +48}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (6+\textit {\_R}^{2}+2 \left (a +2\right ) \textit {\_R} +a \right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}}{16 a^{2}+112 a +192}\)	\(158\)
risch	\(\frac {\frac {x^{3}}{4 a^{2}+28 a +48}+\frac {a \,x^{2}}{4 \left (4+a \right ) \left (3+a \right )}-\frac {\left (a -2\right ) x}{4 \left (a^{2}+7 a +12\right )}+\frac {a}{4 a^{2}+28 a +48}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\left (\frac {\textit {\_R}^{2}}{a^{2}+7 a +12}+\frac {2 \left (a +2\right ) \textit {\_R}}{a^{2}+7 a +12}+\frac {6+a}{a^{2}+7 a +12}\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{16}\)	\(181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^4+4*x^3-8*x^2+a+8*x)^2,x,method=_RETURNVERBOSE)

[Out]

(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/(a^2+7*a+12))/(-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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^4+4*x^3-8*x^2+a+8*x)^2,x, algorithm="maxima")

[Out]

-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)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^4+4*x^3-8*x^2+a+8*x)^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (197) = 394\).
time = 18.78, size = 539, normalized size = 2.33 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**4+4*x**3-8*x**2+a+8*x)**2,x)

[Out]

(-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 + 12952010752*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 - 1
4155776*a - 9568256) + _t*(1152*a**4 + 17792*a**3 + 102912*a**2 + 264192*a + 253952) + 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 + 724775
73120*_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 - 3138
0496*_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))))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^4+4*x^3-8*x^2+a+8*x)^2,x, algorithm="giac")

[Out]

integrate(x/(x^4 - 4*x^3 + 8*x^2 - a - 8*x)^2, x)

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Mupad [B]
time = 2.82, size = 1167, normalized size = 5.05 \begin {gather*} \left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (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\right )\,\left (\frac {336\,a^3+3600\,a^2+12800\,a+15104}{64\,\left (a^5+18\,a^4+129\,a^3+460\,a^2+816\,a+576\right )}+\mathrm {root}\left (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\right )\,\left (\mathrm {root}\left (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\right )\,\left (\frac {4096\,a^6+90112\,a^5+823296\,a^4+3997696\,a^3+10878976\,a^2+15728640\,a+9437184}{64\,\left (a^5+18\,a^4+129\,a^3+460\,a^2+816\,a+576\right )}-\frac {x\,\left (1024\,a^6+22528\,a^5+205824\,a^4+999424\,a^3+2719744\,a^2+3932160\,a+2359296\right )}{16\,\left (a^5+18\,a^4+129\,a^3+460\,a^2+816\,a+576\right )}\right )-\frac {1280\,a^5+23552\,a^4+172800\,a^3+631808\,a^2+1150976\,a+835584}{64\,\left (a^5+18\,a^4+129\,a^3+460\,a^2+816\,a+576\right )}+\frac {x\,\left (128\,a^5+2304\,a^4+16512\,a^3+58880\,a^2+104448\,a+73728\right )}{16\,\left (a^5+18\,a^4+129\,a^3+460\,a^2+816\,a+576\right )}\right )-\frac {x\,\left (20\,a^3+228\,a^2+864\,a+1088\right )}{16\,\left (a^5+18\,a^4+129\,a^3+460\,a^2+816\,a+576\right )}\right )+\frac {4\,a^2+35\,a+68}{64\,\left (a^5+18\,a^4+129\,a^3+460\,a^2+816\,a+576\right )}+\frac {x\,\left (2\,a^2+9\,a+8\right )}{16\,\left (a^5+18\,a^4+129\,a^3+460\,a^2+816\,a+576\right )}\right )\,\mathrm {root}\left (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\right )\right )+\frac {\frac {x^3}{4\,\left (a^2+7\,a+12\right )}+\frac {a}{4\,\left (a+3\right )\,\left (a+4\right )}-\frac {x\,\left (a-2\right )}{4\,\left (a+3\right )\,\left (a+4\right )}+\frac {a\,x^2}{4\,\left (a+3\right )\,\left (a+4\right )}}{-x^4+4\,x^3-8\,x^2+8\,x+a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x)

[Out]

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 + 269
680640*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)*(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 + 72
47757312*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)*((15728640*a + 10878976*a^2 + 3997696*a^3 + 823296*a^4 + 90112*a^5 + 4096*a^6 + 9437184)/(64*(81
6*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576)) - (x*(3932160*a + 2719744*a^2 + 999424*a^3 + 205824*a^4 + 22528
*a^5 + 1024*a^6 + 2359296))/(16*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576))) - (1150976*a + 631808*a^2 +
172800*a^3 + 23552*a^4 + 1280*a^5 + 835584)/(64*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576)) + (x*(10444
8*a + 58880*a^2 + 16512*a^3 + 2304*a^4 + 128*a^5 + 73728))/(16*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576
))) - (x*(864*a + 228*a^2 + 20*a^3 + 1088))/(16*(816*a + 460*a^2 + 129*a^3 + 18*a^4 + a^5 + 576))) + (x*(9*a +
2*a^2 + 8))/(16*(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 + 535717478
4*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 + 2
53952*z - 984*a - 57*a^2 + 16*a^3 - 2064, z, k), k, 1, 4) + (x^3/(4*(7*a + a^2 + 12)) + a/(4*(a + 3)*(a + 4))
- (x*(a - 2))/(4*(a + 3)*(a + 4)) + (a*x^2)/(4*(a + 3)*(a + 4)))/(a + 8*x - 8*x^2 + 4*x^3 - x^4)

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