3.122 \(\int \frac{1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^3} \, dx\)
Optimal. Leaf size=252 \[ \frac{(x-1) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^2}-\frac{3 \left (7 a^2+\left (4 \sqrt{a+4}+47\right ) a+14 \sqrt{a+4}+80\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{64 (a+3)^2 (a+4)^{5/2} \sqrt{1-\sqrt{a+4}}}-\frac{3 \left (-\frac{7 a^2+47 a+80}{\sqrt{a+4}}+4 a+14\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{64 (a+3)^2 (a+4)^2 \sqrt{\sqrt{a+4}+1}}+\frac{(x-1) \left (6 (2 a+7) (x-1)^2+(a+6) (7 a+25)\right )}{32 (a+3)^2 (a+4)^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )} \]
[Out]
((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)/S
qrt[4 + a])*ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^2*Sqrt
[1 + Sqrt[4 + a]])
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Rubi [A] time = 1.41356, antiderivative size = 252, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227
\[ -\frac{(1-x) \left (a+(x-1)^2+5\right )}{8 \left (a^2+7 a+12\right ) \left (a-(1-x)^4-2 (1-x)^2+3\right )^2}+\frac{3 \left (7 a^2+\left (4 \sqrt{a+4}+47\right ) a+14 \sqrt{a+4}+80\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{a+4}}}\right )}{64 (a+3)^2 (a+4)^{5/2} \sqrt{1-\sqrt{a+4}}}+\frac{3 \left (-\frac{7 a^2+47 a+80}{\sqrt{a+4}}+4 a+14\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )}{64 (a+3)^2 (a+4)^2 \sqrt{\sqrt{a+4}+1}}-\frac{(1-x) \left (6 (2 a+7) (1-x)^2+(a+6) (7 a+25)\right )}{32 (a+3)^2 (a+4)^2 \left (a-(1-x)^4-2 (1-x)^2+3\right )} \]
Antiderivative was successfully verified.
[In] Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-3),x]
[Out]
-(((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 - Sqrt[4 + a]]) + (3*(14 + 4*a - (80 + 47*a + 7*a^2)/Sqrt[4 +
a])*ArcTan[(1 - x)/Sqrt[1 + Sqrt[4 + a]]])/(64*(3 + a)^2*(4 + a)^2*Sqrt[1 + Sqr
t[4 + a]])
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Rubi in Sympy [A] time = 120.835, size = 226, normalized size = 0.9 \[ \frac{\left (x - 1\right ) \left (2 a + 2 \left (x - 1\right )^{2} + 10\right )}{16 \left (a + 3\right ) \left (a + 4\right ) \left (a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3\right )^{2}} + \frac{\left (x - 1\right ) \left (28 a^{2} + 268 a + \left (48 a + 168\right ) \left (x - 1\right )^{2} + 600\right )}{128 \left (a + 3\right )^{2} \left (a + 4\right )^{2} \left (a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3\right )} + \frac{3 \left (7 a^{2} + 47 a - 2 \sqrt{a + 4} \left (2 a + 7\right ) + 80\right ) \operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}}{64 \left (a + 3\right )^{2} \left (a + 4\right )^{\frac{5}{2}} \sqrt{\sqrt{a + 4} + 1}} - \frac{3 \left (7 a^{2} + 47 a + 2 \sqrt{a + 4} \left (2 a + 7\right ) + 80\right ) \operatorname{atan}{\left (\frac{x - 1}{\sqrt{- \sqrt{a + 4} + 1}} \right )}}{64 \left (a + 3\right )^{2} \left (a + 4\right )^{\frac{5}{2}} \sqrt{- \sqrt{a + 4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**4+4*x**3-8*x**2+a+8*x)**3,x)
[Out]
(x - 1)*(2*a + 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 + (48*a + 168)*(x - 1)**2 + 600)/(128*(
a + 3)**2*(a + 4)**2*(a - (x - 1)**4 - 2*(x - 1)**2 + 3)) + 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)) - 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 = 0.187523, size = 254, normalized size = 1.01 \[ \frac{1}{128} \left (-\frac{3 \text{RootSum}\left [-\text{$\#$1}^4+4 \text{$\#$1}^3-8 \text{$\#$1}^2+8 \text{$\#$1}+a\&,\frac{4 \text{$\#$1}^2 a \log (x-\text{$\#$1})+14 \text{$\#$1}^2 \log (x-\text{$\#$1})+7 a^2 \log (x-\text{$\#$1})+55 a \log (x-\text{$\#$1})-8 \text{$\#$1} a \log (x-\text{$\#$1})+108 \log (x-\text{$\#$1})-28 \text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^3-3 \text{$\#$1}^2+4 \text{$\#$1}-2}\&\right ]}{\left (a^2+7 a+12\right )^2}+\frac{4 (x-1) \left (7 a^2+a \left (12 x^2-24 x+79\right )+6 \left (7 x^2-14 x+32\right )\right )}{(a+3)^2 (a+4)^2 \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )}+\frac{16 (x-1) \left (a+x^2-2 x+6\right )}{(a+3) (a+4) \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-3),x]
[Out]
((16*(-1 + x)*(6 + a - 2*x + x^2))/((3 + a)*(4 + a)*(a - x*(-8 + 8*x - 4*x^2 + x
^3))^2) + (4*(-1 + x)*(7*a^2 + 6*(32 - 14*x + 7*x^2) + a*(79 - 24*x + 12*x^2)))/
((3 + a)^2*(4 + a)^2*(a - x*(-8 + 8*x - 4*x^2 + x^3))) - (3*RootSum[a + 8*#1 - 8
*#1^2 + 4*#1^3 - #1^4 & , (108*Log[x - #1] + 55*a*Log[x - #1] + 7*a^2*Log[x - #1
] - 28*Log[x - #1]*#1 - 8*a*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 = 0.045, size = 398, normalized size = 1.6 \[ -{\frac{1}{ \left ({x}^{4}-4\,{x}^{3}+8\,{x}^{2}-a-8\,x \right ) ^{2}} \left ({\frac{ \left ( 6\,a+21 \right ){x}^{7}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}-{\frac{ \left ( 147+42\,a \right ){x}^{6}}{ \left ( 16\,{a}^{2}+128\,a+256 \right ) \left ({a}^{2}+6\,a+9 \right ) }}+{\frac{ \left ( 7\,{a}^{2}+343\,a+1116 \right ){x}^{5}}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}-{\frac{ \left ( 35\,{a}^{2}+875\,a+2640 \right ){x}^{4}}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}+{\frac{ \left ( 34\,{a}^{2}+679\,a+1968 \right ){x}^{3}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}-{\frac{ \left ( 32\,{a}^{2}+623\,a+1800 \right ){x}^{2}}{16\,{a}^{4}+224\,{a}^{3}+1168\,{a}^{2}+2688\,a+2304}}-{\frac{ \left ( 11\,{a}^{3}+107\,{a}^{2}-84\,a-1152 \right ) x}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}}+{\frac{11\,{a}^{3}+131\,{a}^{2}+408\,a+288}{32\,{a}^{4}+448\,{a}^{3}+2336\,{a}^{2}+5376\,a+4608}} \right ) }-{\frac{3}{128}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-8\,{\it \_Z}-a \right ) }{\frac{ \left ( 108+2\, \left ( 7+2\,a \right ){{\it \_R}}^{2}+4\, \left ( -2\,a-7 \right ){\it \_R}+7\,{a}^{2}+55\,a \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ({{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+4\,{\it \_R}-2 \right ) \left ({a}^{3}+10\,{a}^{2}+33\,a+36 \right ) \left ( 4+a \right ) }}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-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^7-21/16*(7+2*a)/(a^2+8*a+16)/(a^2
+6*a+9)*x^6+1/32*(7*a^2+343*a+1116)/(a^4+14*a^3+73*a^2+168*a+144)*x^5-5/32*(7*a^
2+175*a+528)/(a^4+14*a^3+73*a^2+168*a+144)*x^4+1/16*(34*a^2+679*a+1968)/(a^4+14*
a^3+73*a^2+168*a+144)*x^3-1/16*(32*a^2+623*a+1800)/(a^4+14*a^3+73*a^2+168*a+144)
*x^2-1/32*(11*a^3+107*a^2-84*a-1152)/(a^4+14*a^3+73*a^2+168*a+144)*x+1/32*(11*a^
3+131*a^2+408*a+288)/(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((108+2*(7+2*a)*_R^2+4*(-2*a-7)*_R+7*a^2+55*a)/(_R^3-3*_R^2+4*_R-2)/(a^3+
10*a^2+33*a+36)/(4+a)*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. \[ -\frac{6 \,{\left (2 \, a + 7\right )} x^{7} - 42 \,{\left (2 \, a + 7\right )} x^{6} +{\left (7 \, a^{2} + 343 \, a + 1116\right )} x^{5} - 5 \,{\left (7 \, a^{2} + 175 \, a + 528\right )} x^{4} + 2 \,{\left (34 \, a^{2} + 679 \, a + 1968\right )} x^{3} + 11 \, a^{3} - 2 \,{\left (32 \, a^{2} + 623 \, a + 1800\right )} x^{2} + 131 \, a^{2} -{\left (11 \, a^{3} + 107 \, a^{2} - 84 \, a - 1152\right )} x + 408 \, a + 288}{32 \,{\left ({\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )} x^{8} - 8 \,{\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )} x^{7} + 32 \,{\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )} x^{6} + a^{6} - 80 \,{\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )} x^{5} + 14 \, a^{5} - 2 \,{\left (a^{5} - 50 \, a^{4} - 823 \, a^{3} - 4504 \, a^{2} - 10608 \, a - 9216\right )} x^{4} + 73 \, a^{4} + 8 \,{\left (a^{5} - 2 \, a^{4} - 151 \, a^{3} - 1000 \, a^{2} - 2544 \, a - 2304\right )} x^{3} + 168 \, a^{3} - 16 \,{\left (a^{5} + 10 \, a^{4} + 17 \, a^{3} - 124 \, a^{2} - 528 \, a - 576\right )} x^{2} + 144 \, a^{2} + 16 \,{\left (a^{5} + 14 \, a^{4} + 73 \, a^{3} + 168 \, a^{2} + 144 \, a\right )} x\right )}} - \frac{3 \, \int \frac{2 \,{\left (2 \, a + 7\right )} x^{2} + 7 \, a^{2} - 4 \,{\left (2 \, a + 7\right )} x + 55 \, a + 108}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x}}{32 \,{\left (a^{4} + 14 \, a^{3} + 73 \, a^{2} + 168 \, a + 144\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x)^3,x, algorithm="maxima")
[Out]
-1/32*(6*(2*a + 7)*x^7 - 42*(2*a + 7)*x^6 + (7*a^2 + 343*a + 1116)*x^5 - 5*(7*a^
2 + 175*a + 528)*x^4 + 2*(34*a^2 + 679*a + 1968)*x^3 + 11*a^3 - 2*(32*a^2 + 623*
a + 1800)*x^2 + 131*a^2 - (11*a^3 + 107*a^2 - 84*a - 1152)*x + 408*a + 288)/((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 + 7
3*a^2 + 168*a + 144)*x^5 + 14*a^5 - 2*(a^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 10608
*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 2544*a - 2304)*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 + 7*a^2 - 4*(2*a + 7)*x + 55*a + 108)/(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 [A] time = 0.286467, size = 5361, normalized size = 21.27 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x)^3,x, algorithm="fricas")
[Out]
-1/128*(24*(2*a + 7)*x^7 - 168*(2*a + 7)*x^6 + 4*(7*a^2 + 343*a + 1116)*x^5 - 20
*(7*a^2 + 175*a + 528)*x^4 + 8*(34*a^2 + 679*a + 1968)*x^3 + 44*a^3 - 8*(32*a^2
+ 623*a + 1800)*x^2 - 3*((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 - 10608*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 -
1000*a^2 - 2544*a - 2304)*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)*sq
rt((105*a^4 + 1470*a^3 + 7749*a^2 + (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*
a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)*sqr
t((2401*a^4 + 33124*a^3 + 171966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165
*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 9
4320045*a^7 + 241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 59
2064640*a^2 + 277136640*a + 60466176)) + 18228*a + 16144)/(a^10 + 35*a^9 + 550*a
^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 +
725760*a + 248832))*log(-64827*a^4 - 907578*a^3 - 4780647*a^2 + 27*(2401*a^4 + 3
3614*a^3 + 177061*a^2 + 415884*a + 367536)*x + 27*(343*a^7 + 8981*a^6 + 100811*a
^5 + 628887*a^4 + 2354874*a^3 + 5293208*a^2 - (11*a^12 + 462*a^11 + 8881*a^10 +
103320*a^9 + 810205*a^8 + 4511542*a^7 + 18292039*a^6 + 54410692*a^5 + 117844800*
a^4 + 181238400*a^3 + 187875072*a^2 + 117863424*a + 33841152)*sqrt((2401*a^4 + 3
3124*a^3 + 171966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a
^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 2
41870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 2
77136640*a + 60466176)) + 6613472*a + 3543424)*sqrt((105*a^4 + 1470*a^3 + 7749*a
^2 + (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 +
735840*a^3 + 950400*a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a^3 + 17196
6*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^1
1 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 241870050*a^6 + 4
77857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 604
66176)) + 18228*a + 16144)/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129
367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)) - 11228868*
a - 9923472) + 3*((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 - 10608*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a
^2 - 2544*a - 2304)*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)*sqrt((105
*a^4 + 1470*a^3 + 7749*a^2 + (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 1
29367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)*sqrt((2401
*a^4 + 33124*a^3 + 171966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 +
16780*a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045
*a^7 + 241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640
*a^2 + 277136640*a + 60466176)) + 18228*a + 16144)/(a^10 + 35*a^9 + 550*a^8 + 51
10*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*
a + 248832))*log(-64827*a^4 - 907578*a^3 - 4780647*a^2 + 27*(2401*a^4 + 33614*a^
3 + 177061*a^2 + 415884*a + 367536)*x - 27*(343*a^7 + 8981*a^6 + 100811*a^5 + 62
8887*a^4 + 2354874*a^3 + 5293208*a^2 - (11*a^12 + 462*a^11 + 8881*a^10 + 103320*
a^9 + 810205*a^8 + 4511542*a^7 + 18292039*a^6 + 54410692*a^5 + 117844800*a^4 + 1
81238400*a^3 + 187875072*a^2 + 117863424*a + 33841152)*sqrt((2401*a^4 + 33124*a^
3 + 171966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 1
67090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 24187005
0*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 27713664
0*a + 60466176)) + 6613472*a + 3543424)*sqrt((105*a^4 + 1470*a^3 + 7749*a^2 + (a
^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840
*a^3 + 950400*a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 +
398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^11 + 121
8460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 241870050*a^6 + 47785731
3*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176))
+ 18228*a + 16144)/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5
+ 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)) - 11228868*a - 992
3472) - 3*((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 - 45
04*a^2 - 10608*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 25
44*a - 2304)*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)*sqrt((105*a^4 +
1470*a^3 + 7749*a^2 - (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a
^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)*sqrt((2401*a^4 +
33124*a^3 + 171966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*
a^12 + 167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 +
241870050*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 +
277136640*a + 60466176)) + 18228*a + 16144)/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7
+ 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248
832))*log(-64827*a^4 - 907578*a^3 - 4780647*a^2 + 27*(2401*a^4 + 33614*a^3 + 177
061*a^2 + 415884*a + 367536)*x + 27*(343*a^7 + 8981*a^6 + 100811*a^5 + 628887*a^
4 + 2354874*a^3 + 5293208*a^2 + (11*a^12 + 462*a^11 + 8881*a^10 + 103320*a^9 + 8
10205*a^8 + 4511542*a^7 + 18292039*a^6 + 54410692*a^5 + 117844800*a^4 + 18123840
0*a^3 + 187875072*a^2 + 117863424*a + 33841152)*sqrt((2401*a^4 + 33124*a^3 + 171
966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a
^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 241870050*a^6 +
477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 6
0466176)) + 6613472*a + 3543424)*sqrt((105*a^4 + 1470*a^3 + 7749*a^2 - (a^10 + 3
5*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 +
950400*a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 + 398164
*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^
10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 241870050*a^6 + 477857313*a^5 +
714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176)) + 1822
8*a + 16144)/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 3730
20*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)) - 11228868*a - 9923472) +
3*((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
- 10608*a - 9216)*x^4 + 73*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 2544*a -
2304)*x^3 + 168*a^3 - 16*(a^5 + 10*a^4 + 17*a^3 - 124*a^2 - 528*a - 576)*x^2 + 1
44*a^2 + 16*(a^5 + 14*a^4 + 73*a^3 + 168*a^2 + 144*a)*x)*sqrt((105*a^4 + 1470*a^
3 + 7749*a^2 - (a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 37
3020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a
^3 + 171966*a^2 + 398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 +
167090*a^11 + 1218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 2418700
50*a^6 + 477857313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 2771366
40*a + 60466176)) + 18228*a + 16144)/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085
*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 950400*a^2 + 725760*a + 248832))*l
og(-64827*a^4 - 907578*a^3 - 4780647*a^2 + 27*(2401*a^4 + 33614*a^3 + 177061*a^2
+ 415884*a + 367536)*x - 27*(343*a^7 + 8981*a^6 + 100811*a^5 + 628887*a^4 + 235
4874*a^3 + 5293208*a^2 + (11*a^12 + 462*a^11 + 8881*a^10 + 103320*a^9 + 810205*a
^8 + 4511542*a^7 + 18292039*a^6 + 54410692*a^5 + 117844800*a^4 + 181238400*a^3 +
187875072*a^2 + 117863424*a + 33841152)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2
+ 398164*a + 346921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^11 + 1
218460*a^10 + 6722130*a^9 + 28570320*a^8 + 94320045*a^7 + 241870050*a^6 + 477857
313*a^5 + 714317940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176
)) + 6613472*a + 3543424)*sqrt((105*a^4 + 1470*a^3 + 7749*a^2 - (a^10 + 35*a^9 +
550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4 + 735840*a^3 + 950400*
a^2 + 725760*a + 248832)*sqrt((2401*a^4 + 33124*a^3 + 171966*a^2 + 398164*a + 34
6921)/(a^15 + 50*a^14 + 1165*a^13 + 16780*a^12 + 167090*a^11 + 1218460*a^10 + 67
22130*a^9 + 28570320*a^8 + 94320045*a^7 + 241870050*a^6 + 477857313*a^5 + 714317
940*a^4 + 782071200*a^3 + 592064640*a^2 + 277136640*a + 60466176)) + 18228*a + 1
6144)/(a^10 + 35*a^9 + 550*a^8 + 5110*a^7 + 31085*a^6 + 129367*a^5 + 373020*a^4
+ 735840*a^3 + 950400*a^2 + 725760*a + 248832)) - 11228868*a - 9923472) + 524*a^
2 - 4*(11*a^3 + 107*a^2 - 84*a - 1152)*x + 1632*a + 1152)/((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 + 1
4*a^3 + 73*a^2 + 168*a + 144)*x^6 + a^6 - 80*(a^4 + 14*a^3 + 73*a^2 + 168*a + 14
4)*x^5 + 14*a^5 - 2*(a^5 - 50*a^4 - 823*a^3 - 4504*a^2 - 10608*a - 9216)*x^4 + 7
3*a^4 + 8*(a^5 - 2*a^4 - 151*a^3 - 1000*a^2 - 2544*a - 2304)*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)
_______________________________________________________________________________________
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(1/(-x**4+4*x**3-8*x**2+a+8*x)**3,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x)^3,x, algorithm="giac")
[Out]
integrate(-1/(x^4 - 4*x^3 + 8*x^2 - a - 8*x)^3, x)