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

Optimal. Leaf size=58 \[ \frac{1}{128 \left (4-x^2\right )}+\frac{1}{64 \left (4-x^2\right )^2}+\frac{1}{24 \left (4-x^2\right )^3}-\frac{1}{512} \log \left (4-x^2\right )+\frac{\log (x)}{256} \]

[Out]

1/(24*(4 - x^2)^3) + 1/(64*(4 - x^2)^2) + 1/(128*(4 - x^2)) + Log[x]/256 - Log[4
- x^2]/512

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Rubi [A]  time = 0.0616172, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{128 \left (4-x^2\right )}+\frac{1}{64 \left (4-x^2\right )^2}+\frac{1}{24 \left (4-x^2\right )^3}-\frac{1}{512} \log \left (4-x^2\right )+\frac{\log (x)}{256} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(-4 + x^2)^4),x]

[Out]

1/(24*(4 - x^2)^3) + 1/(64*(4 - x^2)^2) + 1/(128*(4 - x^2)) + Log[x]/256 - Log[4
- x^2]/512

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Rubi in Sympy [A]  time = 3.24212, size = 42, normalized size = 0.72 \[ \frac{\log{\left (x^{2} \right )}}{512} - \frac{\log{\left (- x^{2} + 4 \right )}}{512} + \frac{1}{128 \left (- x^{2} + 4\right )} + \frac{1}{64 \left (- x^{2} + 4\right )^{2}} + \frac{1}{24 \left (- x^{2} + 4\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2)/512 - log(-x**2 + 4)/512 + 1/(128*(-x**2 + 4)) + 1/(64*(-x**2 + 4)**2)
+ 1/(24*(-x**2 + 4)**3)

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Mathematica [A]  time = 0.0264754, size = 40, normalized size = 0.69 \[ \frac{-3 \log \left (4-x^2\right )-\frac{4 \left (3 x^4-30 x^2+88\right )}{\left (x^2-4\right )^3}+6 \log (x)}{1536} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(-4 + x^2)^4),x]

[Out]

((-4*(88 - 30*x^2 + 3*x^4))/(-4 + x^2)^3 + 6*Log[x] - 3*Log[4 - x^2])/1536

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Maple [A]  time = 0.02, size = 60, normalized size = 1. \[{\frac{1}{1536\, \left ( 2+x \right ) ^{3}}}+{\frac{3}{2048\, \left ( 2+x \right ) ^{2}}}+{\frac{11}{8192+4096\,x}}-{\frac{\ln \left ( 2+x \right ) }{512}}+{\frac{\ln \left ( x \right ) }{256}}-{\frac{1}{1536\, \left ( -2+x \right ) ^{3}}}+{\frac{3}{2048\, \left ( -2+x \right ) ^{2}}}-{\frac{11}{-8192+4096\,x}}-{\frac{\ln \left ( -2+x \right ) }{512}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^2-4)^4,x)

[Out]

1/1536/(2+x)^3+3/2048/(2+x)^2+11/4096/(2+x)-1/512*ln(2+x)+1/256*ln(x)-1/1536/(-2
+x)^3+3/2048/(-2+x)^2-11/4096/(-2+x)-1/512*ln(-2+x)

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Maxima [A]  time = 1.33462, size = 62, normalized size = 1.07 \[ -\frac{3 \, x^{4} - 30 \, x^{2} + 88}{384 \,{\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )}} - \frac{1}{512} \, \log \left (x^{2} - 4\right ) + \frac{1}{512} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/384*(3*x^4 - 30*x^2 + 88)/(x^6 - 12*x^4 + 48*x^2 - 64) - 1/512*log(x^2 - 4) +
1/512*log(x^2)

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Fricas [A]  time = 0.197571, size = 99, normalized size = 1.71 \[ -\frac{12 \, x^{4} - 120 \, x^{2} + 3 \,{\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )} \log \left (x^{2} - 4\right ) - 6 \,{\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )} \log \left (x\right ) + 352}{1536 \,{\left (x^{6} - 12 \, x^{4} + 48 \, x^{2} - 64\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1536*(12*x^4 - 120*x^2 + 3*(x^6 - 12*x^4 + 48*x^2 - 64)*log(x^2 - 4) - 6*(x^6
- 12*x^4 + 48*x^2 - 64)*log(x) + 352)/(x^6 - 12*x^4 + 48*x^2 - 64)

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Sympy [A]  time = 0.198873, size = 41, normalized size = 0.71 \[ - \frac{3 x^{4} - 30 x^{2} + 88}{384 x^{6} - 4608 x^{4} + 18432 x^{2} - 24576} + \frac{\log{\left (x \right )}}{256} - \frac{\log{\left (x^{2} - 4 \right )}}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(x**2-4)**4,x)

[Out]

-(3*x**4 - 30*x**2 + 88)/(384*x**6 - 4608*x**4 + 18432*x**2 - 24576) + log(x)/25
6 - log(x**2 - 4)/512

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GIAC/XCAS [A]  time = 0.20528, size = 57, normalized size = 0.98 \[ \frac{11 \, x^{6} - 156 \, x^{4} + 768 \, x^{2} - 1408}{3072 \,{\left (x^{2} - 4\right )}^{3}} + \frac{1}{512} \,{\rm ln}\left (x^{2}\right ) - \frac{1}{512} \,{\rm ln}\left ({\left | x^{2} - 4 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3072*(11*x^6 - 156*x^4 + 768*x^2 - 1408)/(x^2 - 4)^3 + 1/512*ln(x^2) - 1/512*l
n(abs(x^2 - 4))