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

Optimal. Leaf size=24 \[ \frac{\log (x)}{a^4}-\frac{\log \left (a^4-x^4\right )}{4 a^4} \]

[Out]

Log[x]/a^4 - Log[a^4 - x^4]/(4*a^4)

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Rubi [A]  time = 0.0272014, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\log (x)}{a^4}-\frac{\log \left (a^4-x^4\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a^4 - x^4)),x]

[Out]

Log[x]/a^4 - Log[a^4 - x^4]/(4*a^4)

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Rubi in Sympy [A]  time = 2.59572, size = 22, normalized size = 0.92 \[ \frac{\log{\left (x^{4} \right )}}{4 a^{4}} - \frac{\log{\left (a^{4} - x^{4} \right )}}{4 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**4)/(4*a**4) - log(a**4 - x**4)/(4*a**4)

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Mathematica [A]  time = 0.00884881, size = 24, normalized size = 1. \[ \frac{\log (x)}{a^4}-\frac{\log \left (x^4-a^4\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a^4 - x^4)),x]

[Out]

Log[x]/a^4 - Log[-a^4 + x^4]/(4*a^4)

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Maple [A]  time = 0.01, size = 41, normalized size = 1.7 \[ -{\frac{\ln \left ( a+x \right ) }{4\,{a}^{4}}}+{\frac{\ln \left ( x \right ) }{{a}^{4}}}-{\frac{\ln \left ({a}^{2}+{x}^{2} \right ) }{4\,{a}^{4}}}-{\frac{\ln \left ( -a+x \right ) }{4\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a^4-x^4),x)

[Out]

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

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Maxima [A]  time = 1.35116, size = 34, normalized size = 1.42 \[ -\frac{\log \left (-a^{4} + x^{4}\right )}{4 \, a^{4}} + \frac{\log \left (x^{4}\right )}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.198396, size = 27, normalized size = 1.12 \[ -\frac{\log \left (-a^{4} + x^{4}\right ) - 4 \, \log \left (x\right )}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(log(-a^4 + x^4) - 4*log(x))/a^4

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Sympy [A]  time = 0.311318, size = 19, normalized size = 0.79 \[ \frac{\log{\left (x \right )}}{a^{4}} - \frac{\log{\left (- a^{4} + x^{4} \right )}}{4 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a**4-x**4),x)

[Out]

log(x)/a**4 - log(-a**4 + x**4)/(4*a**4)

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GIAC/XCAS [A]  time = 0.202234, size = 35, normalized size = 1.46 \[ \frac{{\rm ln}\left (x^{4}\right )}{4 \, a^{4}} - \frac{{\rm ln}\left ({\left | -a^{4} + x^{4} \right |}\right )}{4 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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