3.308 \(\int \frac{1}{x^8 \left (1-2 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=43 \[ -\frac{11}{28 x^7}-\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1-x^4\right )}+\frac{11}{8} \tan ^{-1}(x)+\frac{11}{8} \tanh ^{-1}(x) \]

[Out]

-11/(28*x^7) - 11/(12*x^3) + 1/(4*x^7*(1 - x^4)) + (11*ArcTan[x])/8 + (11*ArcTan
h[x])/8

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Rubi [A]  time = 0.0332619, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{11}{28 x^7}-\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1-x^4\right )}+\frac{11}{8} \tan ^{-1}(x)+\frac{11}{8} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(1 - 2*x^4 + x^8)),x]

[Out]

-11/(28*x^7) - 11/(12*x^3) + 1/(4*x^7*(1 - x^4)) + (11*ArcTan[x])/8 + (11*ArcTan
h[x])/8

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Rubi in Sympy [A]  time = 7.28232, size = 37, normalized size = 0.86 \[ \frac{11 \operatorname{atan}{\left (x \right )}}{8} + \frac{11 \operatorname{atanh}{\left (x \right )}}{8} - \frac{11}{12 x^{3}} - \frac{11}{28 x^{7}} + \frac{1}{4 x^{7} \left (- x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(x**8-2*x**4+1),x)

[Out]

11*atan(x)/8 + 11*atanh(x)/8 - 11/(12*x**3) - 11/(28*x**7) + 1/(4*x**7*(-x**4 +
1))

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Mathematica [A]  time = 0.0342513, size = 43, normalized size = 1. \[ \frac{1}{336} \left (-\frac{48}{x^7}-\frac{84 x}{x^4-1}-\frac{224}{x^3}-231 \log (1-x)+231 \log (x+1)+462 \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(1 - 2*x^4 + x^8)),x]

[Out]

(-48/x^7 - 224/x^3 - (84*x)/(-1 + x^4) + 462*ArcTan[x] - 231*Log[1 - x] + 231*Lo
g[1 + x])/336

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Maple [A]  time = 0.024, size = 52, normalized size = 1.2 \[ -{\frac{1}{-16+16\,x}}-{\frac{11\,\ln \left ( -1+x \right ) }{16}}-{\frac{1}{16+16\,x}}+{\frac{11\,\ln \left ( 1+x \right ) }{16}}-{\frac{1}{7\,{x}^{7}}}-{\frac{2}{3\,{x}^{3}}}+{\frac{x}{8\,{x}^{2}+8}}+{\frac{11\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(x^8-2*x^4+1),x)

[Out]

-1/16/(-1+x)-11/16*ln(-1+x)-1/16/(1+x)+11/16*ln(1+x)-1/7/x^7-2/3/x^3+1/8*x/(x^2+
1)+11/8*arctan(x)

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Maxima [A]  time = 0.854868, size = 57, normalized size = 1.33 \[ -\frac{77 \, x^{8} - 44 \, x^{4} - 12}{84 \,{\left (x^{11} - x^{7}\right )}} + \frac{11}{8} \, \arctan \left (x\right ) + \frac{11}{16} \, \log \left (x + 1\right ) - \frac{11}{16} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 2*x^4 + 1)*x^8),x, algorithm="maxima")

[Out]

-1/84*(77*x^8 - 44*x^4 - 12)/(x^11 - x^7) + 11/8*arctan(x) + 11/16*log(x + 1) -
11/16*log(x - 1)

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Fricas [A]  time = 0.280728, size = 92, normalized size = 2.14 \[ -\frac{308 \, x^{8} - 176 \, x^{4} - 462 \,{\left (x^{11} - x^{7}\right )} \arctan \left (x\right ) - 231 \,{\left (x^{11} - x^{7}\right )} \log \left (x + 1\right ) + 231 \,{\left (x^{11} - x^{7}\right )} \log \left (x - 1\right ) - 48}{336 \,{\left (x^{11} - x^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 2*x^4 + 1)*x^8),x, algorithm="fricas")

[Out]

-1/336*(308*x^8 - 176*x^4 - 462*(x^11 - x^7)*arctan(x) - 231*(x^11 - x^7)*log(x
+ 1) + 231*(x^11 - x^7)*log(x - 1) - 48)/(x^11 - x^7)

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Sympy [A]  time = 0.730099, size = 44, normalized size = 1.02 \[ - \frac{11 \log{\left (x - 1 \right )}}{16} + \frac{11 \log{\left (x + 1 \right )}}{16} + \frac{11 \operatorname{atan}{\left (x \right )}}{8} - \frac{77 x^{8} - 44 x^{4} - 12}{84 x^{11} - 84 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(x**8-2*x**4+1),x)

[Out]

-11*log(x - 1)/16 + 11*log(x + 1)/16 + 11*atan(x)/8 - (77*x**8 - 44*x**4 - 12)/(
84*x**11 - 84*x**7)

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GIAC/XCAS [A]  time = 0.280882, size = 55, normalized size = 1.28 \[ -\frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{14 \, x^{4} + 3}{21 \, x^{7}} + \frac{11}{8} \, \arctan \left (x\right ) + \frac{11}{16} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{11}{16} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 2*x^4 + 1)*x^8),x, algorithm="giac")

[Out]

-1/4*x/(x^4 - 1) - 1/21*(14*x^4 + 3)/x^7 + 11/8*arctan(x) + 11/16*ln(abs(x + 1))
- 11/16*ln(abs(x - 1))