3.131 \(\int \frac{1}{x^3 \left (a^4-x^4\right )} \, dx\)

Optimal. Leaf size=26 \[ \frac{\tanh ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^6}-\frac{1}{2 a^4 x^2} \]

[Out]

-1/(2*a^4*x^2) + ArcTanh[x^2/a^2]/(2*a^6)

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Rubi [A]  time = 0.03274, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tanh ^{-1}\left (\frac{x^2}{a^2}\right )}{2 a^6}-\frac{1}{2 a^4 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a^4 - x^4)),x]

[Out]

-1/(2*a^4*x^2) + ArcTanh[x^2/a^2]/(2*a^6)

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Rubi in Sympy [A]  time = 3.71084, size = 22, normalized size = 0.85 \[ - \frac{1}{2 a^{4} x^{2}} + \frac{\operatorname{atanh}{\left (\frac{x^{2}}{a^{2}} \right )}}{2 a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a**4-x**4),x)

[Out]

-1/(2*a**4*x**2) + atanh(x**2/a**2)/(2*a**6)

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Mathematica [A]  time = 0.0108749, size = 50, normalized size = 1.92 \[ -\frac{\log (a-x)}{4 a^6}-\frac{\log (a+x)}{4 a^6}-\frac{1}{2 a^4 x^2}+\frac{\log \left (a^2+x^2\right )}{4 a^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a^4 - x^4)),x]

[Out]

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

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Maple [A]  time = 0.012, size = 43, normalized size = 1.7 \[ -{\frac{\ln \left ( a+x \right ) }{4\,{a}^{6}}}-{\frac{1}{2\,{a}^{4}{x}^{2}}}+{\frac{\ln \left ({a}^{2}+{x}^{2} \right ) }{4\,{a}^{6}}}-{\frac{\ln \left ( -a+x \right ) }{4\,{a}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a^4-x^4),x)

[Out]

-1/4*ln(a+x)/a^6-1/2/a^4/x^2+1/4/a^6*ln(a^2+x^2)-1/4/a^6*ln(-a+x)

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Maxima [A]  time = 1.34828, size = 50, normalized size = 1.92 \[ \frac{\log \left (a^{2} + x^{2}\right )}{4 \, a^{6}} - \frac{\log \left (-a^{2} + x^{2}\right )}{4 \, a^{6}} - \frac{1}{2 \, a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.211081, size = 55, normalized size = 2.12 \[ \frac{x^{2} \log \left (a^{2} + x^{2}\right ) - x^{2} \log \left (-a^{2} + x^{2}\right ) - 2 \, a^{2}}{4 \, a^{6} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.693625, size = 34, normalized size = 1.31 \[ - \frac{1}{2 a^{4} x^{2}} - \frac{\frac{\log{\left (- a^{2} + x^{2} \right )}}{4} - \frac{\log{\left (a^{2} + x^{2} \right )}}{4}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a**4-x**4),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.204585, size = 51, normalized size = 1.96 \[ \frac{{\rm ln}\left (a^{2} + x^{2}\right )}{4 \, a^{6}} - \frac{{\rm ln}\left ({\left | -a^{2} + x^{2} \right |}\right )}{4 \, a^{6}} - \frac{1}{2 \, a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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