3.297 \(\int \frac{1}{x^3 \left (1-2 x^4+x^8\right )} \, dx\)

Optimal. Leaf size=32 \[ -\frac{3}{4 x^2}+\frac{3}{4} \tanh ^{-1}\left (x^2\right )+\frac{1}{4 x^2 \left (1-x^4\right )} \]

[Out]

-3/(4*x^2) + 1/(4*x^2*(1 - x^4)) + (3*ArcTanh[x^2])/4

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Rubi [A]  time = 0.0345444, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{3}{4 x^2}+\frac{3}{4} \tanh ^{-1}\left (x^2\right )+\frac{1}{4 x^2 \left (1-x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 - 2*x^4 + x^8)),x]

[Out]

-3/(4*x^2) + 1/(4*x^2*(1 - x^4)) + (3*ArcTanh[x^2])/4

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Rubi in Sympy [A]  time = 7.86682, size = 26, normalized size = 0.81 \[ \frac{3 \operatorname{atanh}{\left (x^{2} \right )}}{4} - \frac{3}{4 x^{2}} + \frac{1}{4 x^{2} \left (- x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(x**8-2*x**4+1),x)

[Out]

3*atanh(x**2)/4 - 3/(4*x**2) + 1/(4*x**2*(-x**4 + 1))

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Mathematica [A]  time = 0.0327269, size = 41, normalized size = 1.28 \[ \frac{1}{8} \left (-3 \log \left (1-x^2\right )+3 \log \left (x^2+1\right )+\frac{4-6 x^4}{x^2 \left (x^4-1\right )}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(1 - 2*x^4 + x^8)),x]

[Out]

((4 - 6*x^4)/(x^2*(-1 + x^4)) - 3*Log[1 - x^2] + 3*Log[1 + x^2])/8

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Maple [A]  time = 0.024, size = 50, normalized size = 1.6 \[ -{\frac{1}{-16+16\,x}}-{\frac{3\,\ln \left ( -1+x \right ) }{8}}+{\frac{1}{16+16\,x}}-{\frac{3\,\ln \left ( 1+x \right ) }{8}}-{\frac{1}{2\,{x}^{2}}}-{\frac{1}{8\,{x}^{2}+8}}+{\frac{3\,\ln \left ({x}^{2}+1 \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^8-2*x^4+1),x)

[Out]

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

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Maxima [A]  time = 0.770033, size = 50, normalized size = 1.56 \[ -\frac{3 \, x^{4} - 2}{4 \,{\left (x^{6} - x^{2}\right )}} + \frac{3}{8} \, \log \left (x^{2} + 1\right ) - \frac{3}{8} \, \log \left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 2*x^4 + 1)*x^3),x, algorithm="maxima")

[Out]

-1/4*(3*x^4 - 2)/(x^6 - x^2) + 3/8*log(x^2 + 1) - 3/8*log(x^2 - 1)

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Fricas [A]  time = 0.284432, size = 73, normalized size = 2.28 \[ -\frac{6 \, x^{4} - 3 \,{\left (x^{6} - x^{2}\right )} \log \left (x^{2} + 1\right ) + 3 \,{\left (x^{6} - x^{2}\right )} \log \left (x^{2} - 1\right ) - 4}{8 \,{\left (x^{6} - x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 2*x^4 + 1)*x^3),x, algorithm="fricas")

[Out]

-1/8*(6*x^4 - 3*(x^6 - x^2)*log(x^2 + 1) + 3*(x^6 - x^2)*log(x^2 - 1) - 4)/(x^6
- x^2)

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Sympy [A]  time = 0.399905, size = 36, normalized size = 1.12 \[ - \frac{3 x^{4} - 2}{4 x^{6} - 4 x^{2}} - \frac{3 \log{\left (x^{2} - 1 \right )}}{8} + \frac{3 \log{\left (x^{2} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(x**8-2*x**4+1),x)

[Out]

-(3*x**4 - 2)/(4*x**6 - 4*x**2) - 3*log(x**2 - 1)/8 + 3*log(x**2 + 1)/8

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GIAC/XCAS [A]  time = 0.287366, size = 51, normalized size = 1.59 \[ -\frac{3 \, x^{4} - 2}{4 \,{\left (x^{6} - x^{2}\right )}} + \frac{3}{8} \,{\rm ln}\left (x^{2} + 1\right ) - \frac{3}{8} \,{\rm ln}\left ({\left | x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^8 - 2*x^4 + 1)*x^3),x, algorithm="giac")

[Out]

-1/4*(3*x^4 - 2)/(x^6 - x^2) + 3/8*ln(x^2 + 1) - 3/8*ln(abs(x^2 - 1))