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

Optimal. Leaf size=36 \[ -\frac{7}{12 x^3}+\frac{1}{4 x^3 \left (1-x^4\right )}+\frac{7}{8} \tan ^{-1}(x)+\frac{7}{8} \tanh ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0259804, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{7}{12 x^3}+\frac{1}{4 x^3 \left (1-x^4\right )}+\frac{7}{8} \tan ^{-1}(x)+\frac{7}{8} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.30421, size = 31, normalized size = 0.86 \[ \frac{7 \operatorname{atan}{\left (x \right )}}{8} + \frac{7 \operatorname{atanh}{\left (x \right )}}{8} - \frac{7}{12 x^{3}} + \frac{1}{4 x^{3} \left (- x^{4} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0310668, size = 38, normalized size = 1.06 \[ \frac{1}{48} \left (-\frac{12 x}{x^4-1}-\frac{16}{x^3}-21 \log (1-x)+21 \log (x+1)+42 \tan ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-16/x^3 - (12*x)/(-1 + x^4) + 42*ArcTan[x] - 21*Log[1 - x] + 21*Log[1 + x])/48

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Maple [A]  time = 0.021, size = 47, normalized size = 1.3 \[ -{\frac{1}{-16+16\,x}}-{\frac{7\,\ln \left ( -1+x \right ) }{16}}-{\frac{1}{16+16\,x}}+{\frac{7\,\ln \left ( 1+x \right ) }{16}}-{\frac{1}{3\,{x}^{3}}}+{\frac{x}{8\,{x}^{2}+8}}+{\frac{7\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(x^8-2*x^4+1),x)

[Out]

-1/16/(-1+x)-7/16*ln(-1+x)-1/16/(1+x)+7/16*ln(1+x)-1/3/x^3+1/8*x/(x^2+1)+7/8*arc
tan(x)

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Maxima [A]  time = 0.86121, size = 50, normalized size = 1.39 \[ -\frac{7 \, x^{4} - 4}{12 \,{\left (x^{7} - x^{3}\right )}} + \frac{7}{8} \, \arctan \left (x\right ) + \frac{7}{16} \, \log \left (x + 1\right ) - \frac{7}{16} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(7*x^4 - 4)/(x^7 - x^3) + 7/8*arctan(x) + 7/16*log(x + 1) - 7/16*log(x - 1
)

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Fricas [A]  time = 0.269199, size = 85, normalized size = 2.36 \[ -\frac{28 \, x^{4} - 42 \,{\left (x^{7} - x^{3}\right )} \arctan \left (x\right ) - 21 \,{\left (x^{7} - x^{3}\right )} \log \left (x + 1\right ) + 21 \,{\left (x^{7} - x^{3}\right )} \log \left (x - 1\right ) - 16}{48 \,{\left (x^{7} - x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(28*x^4 - 42*(x^7 - x^3)*arctan(x) - 21*(x^7 - x^3)*log(x + 1) + 21*(x^7 -
x^3)*log(x - 1) - 16)/(x^7 - x^3)

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Sympy [A]  time = 0.583853, size = 39, normalized size = 1.08 \[ - \frac{7 x^{4} - 4}{12 x^{7} - 12 x^{3}} - \frac{7 \log{\left (x - 1 \right )}}{16} + \frac{7 \log{\left (x + 1 \right )}}{16} + \frac{7 \operatorname{atan}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(7*x**4 - 4)/(12*x**7 - 12*x**3) - 7*log(x - 1)/16 + 7*log(x + 1)/16 + 7*atan(x
)/8

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GIAC/XCAS [A]  time = 0.28787, size = 46, normalized size = 1.28 \[ -\frac{x}{4 \,{\left (x^{4} - 1\right )}} - \frac{1}{3 \, x^{3}} + \frac{7}{8} \, \arctan \left (x\right ) + \frac{7}{16} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{7}{16} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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