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

Optimal. Leaf size=349 \[ \frac{(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}-\frac{3 \left (7 a^2+\left (4 \sqrt{a+4}+47\right ) a+14 \sqrt{a+4}+80\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{64 (a+3)^2 (a+4)^{5/2} \sqrt{1-\sqrt{a+4}}}-\frac{3 \left (-\frac{7 a^2+47 a+80}{\sqrt{a+4}}+4 a+14\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{64 (a+3)^2 (a+4)^2 \sqrt{\sqrt{a+4}+1}}+\frac{3 \left ((x-1)^2+1\right )}{16 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac{(x-1)^2+1}{8 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}+\frac{(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{32 (a+3)^2 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )}+\frac{3 \tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{16 (a+4)^{5/2}} \]

[Out]

(1 + (-1 + x)^2)/(8*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^2) + (3*(1 + (-1
+ x)^2))/(16*(4 + a)^2*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((5 + a + (-1 + x
)^2)*(-1 + x))/(8*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^2) + (((6
+ a)*(25 + 7*a) + 6*(7 + 2*a)*(-1 + x)^2)*(-1 + x))/(32*(3 + a)^2*(4 + a)^2*(3
+ a - 2*(-1 + x)^2 - (-1 + x)^4)) - (3*(80 + 7*a^2 + 14*Sqrt[4 + a] + a*(47 + 4*
Sqrt[4 + a]))*ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^(5/2
)*Sqrt[1 - Sqrt[4 + a]]) - (3*(14 + 4*a - (80 + 47*a + 7*a^2)/Sqrt[4 + a])*ArcTa
n[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^2*Sqrt[1 + Sqrt[4 + a]]
) + (3*ArcTanh[(1 + (-1 + x)^2)/Sqrt[4 + a]])/(16*(4 + a)^(5/2))

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Rubi [A]  time = 1.51275, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{(1-x) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(1-x)^4-2 (1-x)^2+3\right )^2}+\frac{3 \left (7 a^2+\left (4 \sqrt{a+4}+47\right ) a+14 \sqrt{a+4}+80\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{a+4}}}\right )}{64 (a+3)^2 (a+4)^{5/2} \sqrt{1-\sqrt{a+4}}}+\frac{3 \left (-\frac{7 a^2+47 a+80}{\sqrt{a+4}}+4 a+14\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )}{64 (a+3)^2 (a+4)^2 \sqrt{\sqrt{a+4}+1}}+\frac{3 \left ((x-1)^2+1\right )}{16 (a+4)^2 \left (a-(1-x)^4-2 (1-x)^2+3\right )}+\frac{(x-1)^2+1}{8 (a+4) \left (a-(1-x)^4-2 (1-x)^2+3\right )^2}-\frac{(1-x) \left (6 (2 a+7) (1-x)^2+(a+6) (7 a+25)\right )}{32 (a+3)^2 (a+4)^2 \left (a-(1-x)^4-2 (1-x)^2+3\right )}+\frac{3 \tanh ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a+4}}\right )}{16 (a+4)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(1 + (-1 + x)^2)/(8*(4 + a)*(3 + a - 2*(1 - x)^2 - (1 - x)^4)^2) + (3*(1 + (-1 +
x)^2))/(16*(4 + a)^2*(3 + a - 2*(1 - x)^2 - (1 - x)^4)) - (((6 + a)*(25 + 7*a)
+ 6*(7 + 2*a)*(1 - x)^2)*(1 - x))/(32*(3 + a)^2*(4 + a)^2*(3 + a - 2*(1 - x)^2 -
(1 - x)^4)) - ((5 + a + (-1 + x)^2)*(1 - x))/(8*(12 + 7*a + a^2)*(3 + a - 2*(1
- x)^2 - (1 - x)^4)^2) + (3*(80 + 7*a^2 + 14*Sqrt[4 + a] + a*(47 + 4*Sqrt[4 + a]
))*ArcTan[(1 - x)/Sqrt[1 - Sqrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^(5/2)*Sqrt[1 - S
qrt[4 + a]]) + (3*(14 + 4*a - (80 + 47*a + 7*a^2)/Sqrt[4 + a])*ArcTan[(1 - x)/Sq
rt[1 + Sqrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^2*Sqrt[1 + Sqrt[4 + a]]) + (3*ArcTan
h[(1 + (-1 + x)^2)/Sqrt[4 + a]])/(16*(4 + a)^(5/2))

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Rubi in Sympy [A]  time = 161.775, size = 292, normalized size = 0.84 \[ - \frac{3 \operatorname{atanh}{\left (\frac{- \left (x - 1\right )^{2} - 1}{\sqrt{a + 4}} \right )}}{16 \left (a + 4\right )^{\frac{5}{2}}} + \frac{\left (x - 1\right ) \left (2 a + \left (2 a + 10\right ) \left (x - 1\right ) + 2 \left (x - 1\right )^{3} + 2 \left (x - 1\right )^{2} + 10\right )}{16 \left (a + 3\right ) \left (a + 4\right ) \left (a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3\right )^{2}} + \frac{\left (x - 1\right ) \left (28 a^{2} + 268 a + \left (40 a + 136\right ) \left (x - 1\right )^{3} + \left (48 a + 168\right ) \left (x - 1\right )^{2} + \left (x - 1\right ) \left (24 a^{2} + 224 a + 488\right ) + 600\right )}{128 \left (a + 3\right )^{2} \left (a + 4\right )^{2} \left (a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3\right )} + \frac{3 \left (7 a^{2} + 47 a - 2 \sqrt{a + 4} \left (2 a + 7\right ) + 80\right ) \operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}}{64 \left (a + 3\right )^{2} \left (a + 4\right )^{\frac{5}{2}} \sqrt{\sqrt{a + 4} + 1}} - \frac{3 \left (7 a^{2} + 47 a + 2 \sqrt{a + 4} \left (2 a + 7\right ) + 80\right ) \operatorname{atan}{\left (\frac{x - 1}{\sqrt{- \sqrt{a + 4} + 1}} \right )}}{64 \left (a + 3\right )^{2} \left (a + 4\right )^{\frac{5}{2}} \sqrt{- \sqrt{a + 4} + 1}} \]

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)**3,x)

[Out]

-3*atanh((-(x - 1)**2 - 1)/sqrt(a + 4))/(16*(a + 4)**(5/2)) + (x - 1)*(2*a + (2*
a + 10)*(x - 1) + 2*(x - 1)**3 + 2*(x - 1)**2 + 10)/(16*(a + 3)*(a + 4)*(a - (x
- 1)**4 - 2*(x - 1)**2 + 3)**2) + (x - 1)*(28*a**2 + 268*a + (40*a + 136)*(x - 1
)**3 + (48*a + 168)*(x - 1)**2 + (x - 1)*(24*a**2 + 224*a + 488) + 600)/(128*(a
+ 3)**2*(a + 4)**2*(a - (x - 1)**4 - 2*(x - 1)**2 + 3)) + 3*(7*a**2 + 47*a - 2*s
qrt(a + 4)*(2*a + 7) + 80)*atan((x - 1)/sqrt(sqrt(a + 4) + 1))/(64*(a + 3)**2*(a
+ 4)**(5/2)*sqrt(sqrt(a + 4) + 1)) - 3*(7*a**2 + 47*a + 2*sqrt(a + 4)*(2*a + 7)
+ 80)*atan((x - 1)/sqrt(-sqrt(a + 4) + 1))/(64*(a + 3)**2*(a + 4)**(5/2)*sqrt(-
sqrt(a + 4) + 1))

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Mathematica [C]  time = 0.208436, size = 284, normalized size = 0.81 \[ \frac{1}{128} \left (-\frac{3 \text{RootSum}\left [-\text{$\#$1}^4+4 \text{$\#$1}^3-8 \text{$\#$1}^2+8 \text{$\#$1}+a\&,\frac{4 \text{$\#$1}^2 a \log (x-\text{$\#$1})+14 \text{$\#$1}^2 \log (x-\text{$\#$1})+3 a^2 \log (x-\text{$\#$1})+4 \text{$\#$1} a^2 \log (x-\text{$\#$1})+31 a \log (x-\text{$\#$1})+16 \text{$\#$1} a \log (x-\text{$\#$1})+72 \log (x-\text{$\#$1})+8 \text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^3-3 \text{$\#$1}^2+4 \text{$\#$1}-2}\&\right ]}{\left (a^2+7 a+12\right )^2}+\frac{4 \left (a^2 \left (6 x^2-5 x+5\right )+a \left (12 x^3+31 x-7\right )+6 \left (7 x^3-12 x^2+28 x-14\right )\right )}{(a+3)^2 (a+4)^2 \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )}+\frac{16 \left (a x^2-a x+a+x^3+2 x\right )}{(a+3) (a+4) \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((16*(a + 2*x - a*x + a*x^2 + x^3))/((3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4*x^2 +
x^3))^2) + (4*(a^2*(5 - 5*x + 6*x^2) + 6*(-14 + 28*x - 12*x^2 + 7*x^3) + a*(-7 +
31*x + 12*x^3)))/((3 + a)^2*(4 + a)^2*(a - x*(-8 + 8*x - 4*x^2 + x^3))) - (3*Ro
otSum[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , (72*Log[x - #1] + 31*a*Log[x - #1] +
3*a^2*Log[x - #1] + 8*Log[x - #1]*#1 + 16*a*Log[x - #1]*#1 + 4*a^2*Log[x - #1]*
#1 + 14*Log[x - #1]*#1^2 + 4*a*Log[x - #1]*#1^2)/(-2 + 4*#1 - 3*#1^2 + #1^3) & ]
)/(12 + 7*a + a^2)^2)/128

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Maple [C]  time = 0.036, size = 405, normalized size = 1.2 \[ -{\frac{1}{ \left ({x}^{4}-4\,{x}^{3}+8\,{x}^{2}-a-8\,x \right ) ^{2}} \left ({\frac{ \left ( 6\,a+21 \right ){x}^{7}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}+{\frac{ \left ( 3\,{a}^{2}-24\,a-120 \right ){x}^{6}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}-{\frac{ \left ( 29\,{a}^{2}-127\,a-792 \right ){x}^{5}}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}+{\frac{ \left ( 73\,{a}^{2}-227\,a-1668 \right ){x}^{4}}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}-{\frac{ \left ( 62\,{a}^{2}-103\,a-1104 \right ){x}^{3}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}-{\frac{ \left ( 5\,{a}^{3}-26\,{a}^{2}+140\,a+1008 \right ){x}^{2}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}+{\frac{ \left ( 9\,{a}^{3}-51\,{a}^{2}-120\,a+576 \right ) x}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}-{\frac{3\,a \left ( 3\,{a}^{2}+7\,a-12 \right ) }{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}} \right ) }-{\frac{3}{128}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-8\,{\it \_Z}-a \right ) }{\frac{ \left ( 72+2\, \left ( 7+2\,a \right ){{\it \_R}}^{2}+4\, \left ({a}^{2}+4\,a+2 \right ){\it \_R}+3\,{a}^{2}+31\,a \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ({a}^{4}+14\,{a}^{3}+73\,{a}^{2}+168\,a+144 \right ) \left ({{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+4\,{\it \_R}-2 \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(3/16*(7+2*a)/(a^4+14*a^3+73*a^2+168*a+144)*x^7+3/16*(a^2-8*a-40)/(a^4+14*a^3+7
3*a^2+168*a+144)*x^6-1/32*(29*a^2-127*a-792)/(a^4+14*a^3+73*a^2+168*a+144)*x^5+1
/32*(73*a^2-227*a-1668)/(a^4+14*a^3+73*a^2+168*a+144)*x^4-1/16*(62*a^2-103*a-110
4)/(a^4+14*a^3+73*a^2+168*a+144)*x^3-1/16*(5*a^3-26*a^2+140*a+1008)/(a^4+14*a^3+
73*a^2+168*a+144)*x^2+3/32*(3*a^3-17*a^2-40*a+192)/(a^4+14*a^3+73*a^2+168*a+144)
*x-3/32*a*(3*a^2+7*a-12)/(a^4+14*a^3+73*a^2+168*a+144))/(x^4-4*x^3+8*x^2-a-8*x)^
2-3/128*sum((72+2*(7+2*a)*_R^2+4*(a^2+4*a+2)*_R+3*a^2+31*a)/(a^4+14*a^3+73*a^2+1
68*a+144)/(_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. \[ -\frac{6 \,{\left (2 \, a + 7\right )} x^{7} + 6 \,{\left (a^{2} - 8 \, a - 40\right )} x^{6} -{\left (29 \, a^{2} - 127 \, a - 792\right )} x^{5} +{\left (73 \, a^{2} - 227 \, a - 1668\right )} x^{4} - 2 \,{\left (62 \, a^{2} - 103 \, a - 1104\right )} x^{3} - 9 \, a^{3} - 2 \,{\left (5 \, a^{3} - 26 \, a^{2} + 140 \, a + 1008\right )} x^{2} - 21 \, a^{2} + 3 \,{\left (3 \, a^{3} - 17 \, a^{2} - 40 \, a + 192\right )} x + 36 \, a}{32 \,{\left ({\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )} x^{8} - 8 \,{\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )} x^{7} + 32 \,{\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )} x^{6} + a^{6} - 80 \,{\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )} x^{5} + 14 \, a^{5} - 2 \,{\left (a^{5} - 50 \, a^{4} - 823 \, a^{3} - 4504 \, a^{2} - 10608 \, a - 9216\right )} x^{4} + 73 \, a^{4} + 8 \,{\left (a^{5} - 2 \, a^{4} - 151 \, a^{3} - 1000 \, a^{2} - 2544 \, a - 2304\right )} x^{3} + 168 \, a^{3} - 16 \,{\left (a^{5} + 10 \, a^{4} + 17 \, a^{3} - 124 \, a^{2} - 528 \, a - 576\right )} x^{2} + 144 \, a^{2} + 16 \,{\left (a^{5} + 14 \, a^{4} + 73 \, a^{3} + 168 \, a^{2} + 144 \, a\right )} x\right )}} - \frac{3 \, \int \frac{2 \,{\left (2 \, a + 7\right )} x^{2} + 3 \, a^{2} + 4 \,{\left (a^{2} + 4 \, a + 2\right )} x + 31 \, a + 72}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x}}{32 \,{\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(6*(2*a + 7)*x^7 + 6*(a^2 - 8*a - 40)*x^6 - (29*a^2 - 127*a - 792)*x^5 + (
73*a^2 - 227*a - 1668)*x^4 - 2*(62*a^2 - 103*a - 1104)*x^3 - 9*a^3 - 2*(5*a^3 -
26*a^2 + 140*a + 1008)*x^2 - 21*a^2 + 3*(3*a^3 - 17*a^2 - 40*a + 192)*x + 36*a)/
((a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^8 - 8*(a^4 + 14*a^3 + 73*a^2 + 168*a +
144)*x^7 + 32*(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)*x^6 + a^6 - 80*(a^4 + 14*a^3
+ 73*a^2 + 168*a + 144)*x^5 + 14*a^5 - 2*(a^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 1
0608*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 2544*a - 230
4)*x^3 + 168*a^3 - 16*(a^5 + 10*a^4 + 17*a^3 - 124*a^2 - 528*a - 576)*x^2 + 144*
a^2 + 16*(a^5 + 14*a^4 + 73*a^3 + 168*a^2 + 144*a)*x) - 3/32*integrate((2*(2*a +
7)*x^2 + 3*a^2 + 4*(a^2 + 4*a + 2)*x + 31*a + 72)/(x^4 - 4*x^3 + 8*x^2 - a - 8*
x), x)/(a^4 + 14*a^3 + 73*a^2 + 168*a + 144)

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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)^3,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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