3.128 \(\int \frac{x}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx\)

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

[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))

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Rubi [A]  time = 0.739641, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{(1-x) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(1-x)^4-2 (1-x)^2+3\right )}+\frac{(x-1)^2+1}{4 (a+4) \left (a-(1-x)^4-2 (1-x)^2+3\right )}+\frac{\left (3 a+\sqrt{a+4}+10\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{a+4}}}\right )}{8 (a+3) (a+4)^{3/2} \sqrt{1-\sqrt{a+4}}}-\frac{\left (3 a-\sqrt{a+4}+10\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )}{8 (a+3) (a+4)^{3/2} \sqrt{\sqrt{a+4}+1}}+\frac{\tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{4 (a+4)^{3/2}} \]

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 + Sq
rt[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))

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Rubi in Sympy [A]  time = 82.0614, size = 184, normalized size = 0.8 \[ - \frac{\operatorname{atanh}{\left (\frac{- \left (x - 1\right )^{2} - 1}{\sqrt{a + 4}} \right )}}{4 \left (a + 4\right )^{\frac{3}{2}}} + \frac{\left (x - 1\right ) \left (2 a + \left (2 a + 10\right ) \left (x - 1\right ) + 2 \left (x - 1\right )^{3} + 2 \left (x - 1\right )^{2} + 10\right )}{8 \left (a + 3\right ) \left (a + 4\right ) \left (a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3\right )} + \frac{\left (3 a - \sqrt{a + 4} + 10\right ) \operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}}{8 \left (a + 3\right ) \left (a + 4\right )^{\frac{3}{2}} \sqrt{\sqrt{a + 4} + 1}} - \frac{\left (3 a + \sqrt{a + 4} + 10\right ) \operatorname{atan}{\left (\frac{x - 1}{\sqrt{- \sqrt{a + 4} + 1}} \right )}}{8 \left (a + 3\right ) \left (a + 4\right )^{\frac{3}{2}} \sqrt{- \sqrt{a + 4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-x**4+4*x**3-8*x**2+a+8*x)**2,x)

[Out]

-atanh((-(x - 1)**2 - 1)/sqrt(a + 4))/(4*(a + 4)**(3/2)) + (x - 1)*(2*a + (2*a +
10)*(x - 1) + 2*(x - 1)**3 + 2*(x - 1)**2 + 10)/(8*(a + 3)*(a + 4)*(a - (x - 1)
**4 - 2*(x - 1)**2 + 3)) + (3*a - sqrt(a + 4) + 10)*atan((x - 1)/sqrt(sqrt(a + 4
) + 1))/(8*(a + 3)*(a + 4)**(3/2)*sqrt(sqrt(a + 4) + 1)) - (3*a + sqrt(a + 4) +
10)*atan((x - 1)/sqrt(-sqrt(a + 4) + 1))/(8*(a + 3)*(a + 4)**(3/2)*sqrt(-sqrt(a
+ 4) + 1))

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Mathematica [C]  time = 0.0986098, size = 166, normalized size = 0.72 \[ \frac{a x^2-a x+a+x^3+2 x}{4 (a+3) (a+4) \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )}-\frac{\text{RootSum}\left [-\text{$\#$1}^4+4 \text{$\#$1}^3-8 \text{$\#$1}^2+8 \text{$\#$1}+a\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})+2 \text{$\#$1} a \log (x-\text{$\#$1})+a \log (x-\text{$\#$1})+4 \text{$\#$1} \log (x-\text{$\#$1})+6 \log (x-\text{$\#$1})}{\text{$\#$1}^3-3 \text{$\#$1}^2+4 \text{$\#$1}-2}\&\right ]}{16 \left (a^2+7 a+12\right )} \]

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]  time = 0.025, size = 162, normalized size = 0.7 \[{\frac{1}{{x}^{4}-4\,{x}^{3}+8\,{x}^{2}-a-8\,x} \left ( -{\frac{{x}^{3}}{4\,{a}^{2}+28\,a+48}}-{\frac{a{x}^{2}}{4\,{a}^{2}+28\,a+48}}+{\frac{ \left ( a-2 \right ) x}{4\,{a}^{2}+28\,a+48}}-{\frac{a}{4\,{a}^{2}+28\,a+48}} \right ) }+{\frac{1}{16}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-8\,{\it \_Z}-a \right ) }{\frac{ \left ( -6-{{\it \_R}}^{2}+2\, \left ( -a-2 \right ){\it \_R}-a \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ({a}^{2}+7\,a+12 \right ) \left ({{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+4\,{\it \_R}-2 \right ) }}} \]

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)

[Out]

(-1/4/(a^2+7*a+12)*x^3-1/4*a/(a^2+7*a+12)*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*sum((-6-_R^2+2*(-a-2)*_R-a)/(a^2+7*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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{a x^{2} + x^{3} -{\left (a - 2\right )} x + a}{4 \,{\left ({\left (a^{2} + 7 \, a + 12\right )} x^{4} - 4 \,{\left (a^{2} + 7 \, a + 12\right )} x^{3} - a^{3} + 8 \,{\left (a^{2} + 7 \, a + 12\right )} x^{2} - 7 \, a^{2} - 8 \,{\left (a^{2} + 7 \, a + 12\right )} x - 12 \, a\right )}} - \frac{\int \frac{2 \,{\left (a + 2\right )} x + x^{2} + a + 6}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x}}{4 \,{\left (a^{2} + 7 \, a + 12\right )}} \]

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*in
tegrate((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(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError

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Sympy [A]  time = 45.0025, size = 539, normalized size = 2.33 \[ - \frac{a x^{2} + a + x^{3} + x \left (- a + 2\right )}{- 4 a^{3} - 28 a^{2} - 48 a + x^{4} \left (4 a^{2} + 28 a + 48\right ) + x^{3} \left (- 16 a^{2} - 112 a - 192\right ) + x^{2} \left (32 a^{2} + 224 a + 384\right ) + x \left (- 32 a^{2} - 224 a - 384\right )} + \operatorname{RootSum}{\left (t^{4} \left (65536 a^{9} + 2162688 a^{8} + 31653888 a^{7} + 269680640 a^{6} + 1473773568 a^{5} + 5357174784 a^{4} + 12952010752 a^{3} + 20082327552 a^{2} + 18119393280 a + 7247757312\right ) + t^{2} \left (- 2048 a^{6} - 50688 a^{5} - 520704 a^{4} - 2842624 a^{3} - 8699904 a^{2} - 14155776 a - 9568256\right ) + t \left (1152 a^{4} + 17792 a^{3} + 102912 a^{2} + 264192 a + 253952\right ) + 16 a^{3} - 57 a^{2} - 984 a - 2064, \left ( t \mapsto t \log{\left (x + \frac{98304 t^{3} a^{12} + 3948544 t^{3} a^{11} + 72196096 t^{3} a^{10} + 793837568 t^{3} a^{9} + 5839372288 t^{3} a^{8} + 30226464768 t^{3} a^{7} + 112668450816 t^{3} a^{6} + 303864643584 t^{3} a^{5} + 586157391872 t^{3} a^{4} + 784017129472 t^{3} a^{3} + 683648483328 t^{3} a^{2} + 343136010240 t^{3} a + 72477573120 t^{3} + 30208 t^{2} a^{10} + 986624 t^{2} a^{9} + 14420992 t^{2} a^{8} + 124156928 t^{2} a^{7} + 696815104 t^{2} a^{6} + 2661758464 t^{2} a^{5} + 7001485312 t^{2} a^{4} + 12506562560 t^{2} a^{3} + 14494924800 t^{2} a^{2} + 9820569600 t^{2} a + 2944401408 t^{2} - 1536 t a^{9} - 52048 t a^{8} - 757040 t a^{7} - 6200656 t a^{6} - 31380496 t a^{5} - 100736416 t a^{4} - 200813696 t a^{3} - 228144640 t a^{2} - 114632704 t a - 2490368 t + 248 a^{7} + 6797 a^{6} + 71132 a^{5} + 369745 a^{4} + 987758 a^{3} + 1128896 a^{2} - 129568 a - 956416}{576 a^{7} + 10985 a^{6} + 88746 a^{5} + 396609 a^{4} + 1076268 a^{3} + 1826304 a^{2} + 1867776 a + 917504} \right )} \right )\right )} \]

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 + 200823
27552*a**2 + 18119393280*a + 7247757312) + _t**2*(-2048*a**6 - 50688*a**5 - 5207
04*a**4 - 2842624*a**3 - 8699904*a**2 - 14155776*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 + 72477
573120*_t**3 + 30208*_t**2*a**10 + 986624*_t**2*a**9 + 14420992*_t**2*a**8 + 124
156928*_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 + 8
8746*a**5 + 396609*a**4 + 1076268*a**3 + 1826304*a**2 + 1867776*a + 917504))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{2}}\,{d x} \]

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)