3.121 \(\int \frac{1}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^2} \, dx\)
Optimal. Leaf size=169 \[ \frac{(x-1) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )}-\frac{\left (3 a+\sqrt{a+4}+10\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{1-\sqrt{a+4}}}\right )}{8 (a+3) (a+4)^{3/2} \sqrt{1-\sqrt{a+4}}}+\frac{\left (3 a-\sqrt{a+4}+10\right ) \tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )}{8 (a+3) (a+4)^{3/2} \sqrt{\sqrt{a+4}+1}} \]
[Out]
((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]])
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Rubi [A] time = 0.627456, antiderivative size = 169, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182
\[ -\frac{(1-x) \left (a+(x-1)^2+5\right )}{4 \left (a^2+7 a+12\right ) \left (a-(1-x)^4-2 (1-x)^2+3\right )}+\frac{\left (3 a+\sqrt{a+4}+10\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{1-\sqrt{a+4}}}\right )}{8 (a+3) (a+4)^{3/2} \sqrt{1-\sqrt{a+4}}}-\frac{\left (3 a-\sqrt{a+4}+10\right ) \tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )}{8 (a+3) (a+4)^{3/2} \sqrt{\sqrt{a+4}+1}} \]
Antiderivative was successfully verified.
[In] Int[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-2),x]
[Out]
-((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]])
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Rubi in Sympy [A] time = 58.9326, size = 143, normalized size = 0.85 \[ \frac{\left (x - 1\right ) \left (2 a + 2 \left (x - 1\right )^{2} + 10\right )}{8 \left (a + 3\right ) \left (a + 4\right ) \left (a - \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3\right )} + \frac{\left (3 a - \sqrt{a + 4} + 10\right ) \operatorname{atan}{\left (\frac{x - 1}{\sqrt{\sqrt{a + 4} + 1}} \right )}}{8 \left (a + 3\right ) \left (a + 4\right )^{\frac{3}{2}} \sqrt{\sqrt{a + 4} + 1}} - \frac{\left (3 a + \sqrt{a + 4} + 10\right ) \operatorname{atan}{\left (\frac{x - 1}{\sqrt{- \sqrt{a + 4} + 1}} \right )}}{8 \left (a + 3\right ) \left (a + 4\right )^{\frac{3}{2}} \sqrt{- \sqrt{a + 4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**4+4*x**3-8*x**2+a+8*x)**2,x)
[Out]
(x - 1)*(2*a + 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 = 0.0804479, size = 150, normalized size = 0.89 \[ \frac{(x-1) \left (a+x^2-2 x+6\right )}{4 (a+3) (a+4) \left (a-x \left (x^3-4 x^2+8 x-8\right )\right )}-\frac{\text{RootSum}\left [-\text{$\#$1}^4+4 \text{$\#$1}^3-8 \text{$\#$1}^2+8 \text{$\#$1}+a\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})+3 a \log (x-\text{$\#$1})-2 \text{$\#$1} \log (x-\text{$\#$1})+12 \log (x-\text{$\#$1})}{\text{$\#$1}^3-3 \text{$\#$1}^2+4 \text{$\#$1}-2}\&\right ]}{16 \left (a^2+7 a+12\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(-2),x]
[Out]
((-1 + x)*(6 + a - 2*x + x^2))/(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 & , (12*Log[x - #1] + 3*a*Log[x
- #1] - 2*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 = 0.029, size = 158, normalized size = 0.9 \[{\frac{1}{{x}^{4}-4\,{x}^{3}+8\,{x}^{2}-a-8\,x} \left ( -{\frac{{x}^{3}}{4\,{a}^{2}+28\,a+48}}+{\frac{3\,{x}^{2}}{4\,{a}^{2}+28\,a+48}}-{\frac{ \left ( 8+a \right ) x}{4\,{a}^{2}+28\,a+48}}+{\frac{6+a}{4\,{a}^{2}+28\,a+48}} \right ) }+{\frac{1}{16}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}-8\,{\it \_Z}-a \right ) }{\frac{ \left ( -{{\it \_R}}^{2}+2\,{\it \_R}-3\,a-12 \right ) \ln \left ( x-{\it \_R} \right ) }{ \left ({{\it \_R}}^{3}-3\,{{\it \_R}}^{2}+4\,{\it \_R}-2 \right ) \left ( 4+a \right ) \left ( 3+a \right ) }}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^4+4*x^3-8*x^2+a+8*x)^2,x)
[Out]
(-1/4/(a^2+7*a+12)*x^3+3/4/(a^2+7*a+12)*x^2-1/4*(8+a)/(a^2+7*a+12)*x+1/4*(6+a)/(
a^2+7*a+12))/(x^4-4*x^3+8*x^2-a-8*x)+1/16*sum((-_R^2+2*_R-3*a-12)/(_R^3-3*_R^2+4
*_R-2)/(4+a)/(3+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. \[ -\frac{x^{3} +{\left (a + 8\right )} x - 3 \, x^{2} - a - 6}{4 \,{\left ({\left (a^{2} + 7 \, a + 12\right )} x^{4} - 4 \,{\left (a^{2} + 7 \, a + 12\right )} x^{3} - a^{3} + 8 \,{\left (a^{2} + 7 \, a + 12\right )} x^{2} - 7 \, a^{2} - 8 \,{\left (a^{2} + 7 \, a + 12\right )} x - 12 \, a\right )}} - \frac{\int \frac{x^{2} + 3 \, a - 2 \, x + 12}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x}\,{d x}}{4 \,{\left (a^{2} + 7 \, a + 12\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 - 4*x^3 + 8*x^2 - a - 8*x)^(-2),x, algorithm="maxima")
[Out]
-1/4*(x^3 + (a + 8)*x - 3*x^2 - a - 6)/((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((x^2 + 3*a - 2*x + 12)/(x^4 - 4*x^3 + 8*x^2 - a - 8*x), x)/(a^2 + 7*
a + 12)
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Fricas [A] time = 0.286674, size = 2630, normalized size = 15.56 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 - 4*x^3 + 8*x^2 - a - 8*x)^(-2),x, algorithm="fricas")
[Out]
-1/16*(4*x^3 - ((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)*sqrt((15*a^2 + (a^6 + 21*a^5
+ 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728)*sqrt((81*a^2 + 558*a + 961)/(a^
9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^4 + 111105*a^3 + 156492*a^
2 + 128304*a + 46656)) + 105*a + 184)/(a^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a
^2 + 3024*a + 1728))*log(-81*a^2 + (81*a^2 + 567*a + 992)*x + (27*a^4 + 408*a^3
+ 2309*a^2 - 2*(2*a^7 + 49*a^6 + 513*a^5 + 2975*a^4 + 10321*a^3 + 21420*a^2 + 24
624*a + 12096)*sqrt((81*a^2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 +
15327*a^5 + 50598*a^4 + 111105*a^3 + 156492*a^2 + 128304*a + 46656)) + 5800*a +
5456)*sqrt((15*a^2 + (a^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 172
8)*sqrt((81*a^2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 +
50598*a^4 + 111105*a^3 + 156492*a^2 + 128304*a + 46656)) + 105*a + 184)/(a^6 + 2
1*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728)) - 567*a - 992) + ((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)*sqrt((15*a^2 + (a^6 + 21*a^5 + 183*a^4 + 847*a^3 +
2196*a^2 + 3024*a + 1728)*sqrt((81*a^2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 +
3088*a^6 + 15327*a^5 + 50598*a^4 + 111105*a^3 + 156492*a^2 + 128304*a + 46656))
+ 105*a + 184)/(a^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728))*lo
g(-81*a^2 + (81*a^2 + 567*a + 992)*x - (27*a^4 + 408*a^3 + 2309*a^2 - 2*(2*a^7 +
49*a^6 + 513*a^5 + 2975*a^4 + 10321*a^3 + 21420*a^2 + 24624*a + 12096)*sqrt((81
*a^2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^4 +
111105*a^3 + 156492*a^2 + 128304*a + 46656)) + 5800*a + 5456)*sqrt((15*a^2 + (a
^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728)*sqrt((81*a^2 + 558*a
+ 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^4 + 111105*a^3
+ 156492*a^2 + 128304*a + 46656)) + 105*a + 184)/(a^6 + 21*a^5 + 183*a^4 + 847*a
^3 + 2196*a^2 + 3024*a + 1728)) - 567*a - 992) - ((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)*sqrt((15*a^2 - (a^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 172
8)*sqrt((81*a^2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 +
50598*a^4 + 111105*a^3 + 156492*a^2 + 128304*a + 46656)) + 105*a + 184)/(a^6 + 2
1*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728))*log(-81*a^2 + (81*a^2 + 5
67*a + 992)*x + (27*a^4 + 408*a^3 + 2309*a^2 + 2*(2*a^7 + 49*a^6 + 513*a^5 + 297
5*a^4 + 10321*a^3 + 21420*a^2 + 24624*a + 12096)*sqrt((81*a^2 + 558*a + 961)/(a^
9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^4 + 111105*a^3 + 156492*a^
2 + 128304*a + 46656)) + 5800*a + 5456)*sqrt((15*a^2 - (a^6 + 21*a^5 + 183*a^4 +
847*a^3 + 2196*a^2 + 3024*a + 1728)*sqrt((81*a^2 + 558*a + 961)/(a^9 + 30*a^8 +
399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^4 + 111105*a^3 + 156492*a^2 + 128304*a
+ 46656)) + 105*a + 184)/(a^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a
+ 1728)) - 567*a - 992) + ((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)*sqrt((15*a^2 - (a
^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728)*sqrt((81*a^2 + 558*a
+ 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 15327*a^5 + 50598*a^4 + 111105*a^3
+ 156492*a^2 + 128304*a + 46656)) + 105*a + 184)/(a^6 + 21*a^5 + 183*a^4 + 847*a
^3 + 2196*a^2 + 3024*a + 1728))*log(-81*a^2 + (81*a^2 + 567*a + 992)*x - (27*a^4
+ 408*a^3 + 2309*a^2 + 2*(2*a^7 + 49*a^6 + 513*a^5 + 2975*a^4 + 10321*a^3 + 214
20*a^2 + 24624*a + 12096)*sqrt((81*a^2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 +
3088*a^6 + 15327*a^5 + 50598*a^4 + 111105*a^3 + 156492*a^2 + 128304*a + 46656))
+ 5800*a + 5456)*sqrt((15*a^2 - (a^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3
024*a + 1728)*sqrt((81*a^2 + 558*a + 961)/(a^9 + 30*a^8 + 399*a^7 + 3088*a^6 + 1
5327*a^5 + 50598*a^4 + 111105*a^3 + 156492*a^2 + 128304*a + 46656)) + 105*a + 18
4)/(a^6 + 21*a^5 + 183*a^4 + 847*a^3 + 2196*a^2 + 3024*a + 1728)) - 567*a - 992)
+ 4*(a + 8)*x - 12*x^2 - 4*a - 24)/((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)
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Sympy [A] time = 14.9847, size = 292, normalized size = 1.73 \[ - \frac{- a + x^{3} - 3 x^{2} + x \left (a + 8\right ) - 6}{- 4 a^{3} - 28 a^{2} - 48 a + x^{4} \left (4 a^{2} + 28 a + 48\right ) + x^{3} \left (- 16 a^{2} - 112 a - 192\right ) + x^{2} \left (32 a^{2} + 224 a + 384\right ) + x \left (- 32 a^{2} - 224 a - 384\right )} + \operatorname{RootSum}{\left (t^{4} \left (65536 a^{9} + 2162688 a^{8} + 31653888 a^{7} + 269680640 a^{6} + 1473773568 a^{5} + 5357174784 a^{4} + 12952010752 a^{3} + 20082327552 a^{2} + 18119393280 a + 7247757312\right ) + t^{2} \left (- 7680 a^{5} - 145920 a^{4} - 1107968 a^{3} - 4202496 a^{2} - 7962624 a - 6029312\right ) - 81 a^{2} - 576 a - 1024, \left ( t \mapsto t \log{\left (x + \frac{- 16384 t^{3} a^{7} - 401408 t^{3} a^{6} - 4202496 t^{3} a^{5} - 24371200 t^{3} a^{4} - 84549632 t^{3} a^{3} - 175472640 t^{3} a^{2} - 201719808 t^{3} a - 99090432 t^{3} + 432 t a^{4} + 7488 t a^{3} + 47024 t a^{2} + 128096 t a + 128512 t - 81 a^{2} - 567 a - 992}{81 a^{2} + 567 a + 992} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**4+4*x**3-8*x**2+a+8*x)**2,x)
[Out]
-(-a + x**3 - 3*x**2 + x*(a + 8) - 6)/(-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 + 20
082327552*a**2 + 18119393280*a + 7247757312) + _t**2*(-7680*a**5 - 145920*a**4 -
1107968*a**3 - 4202496*a**2 - 7962624*a - 6029312) - 81*a**2 - 576*a - 1024, La
mbda(_t, _t*log(x + (-16384*_t**3*a**7 - 401408*_t**3*a**6 - 4202496*_t**3*a**5
- 24371200*_t**3*a**4 - 84549632*_t**3*a**3 - 175472640*_t**3*a**2 - 201719808*_
t**3*a - 99090432*_t**3 + 432*_t*a**4 + 7488*_t*a**3 + 47024*_t*a**2 + 128096*_t
*a + 128512*_t - 81*a**2 - 567*a - 992)/(81*a**2 + 567*a + 992))))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 - 4*x^3 + 8*x^2 - a - 8*x)^(-2),x, algorithm="giac")
[Out]
integrate((x^4 - 4*x^3 + 8*x^2 - a - 8*x)^(-2), x)