3.141 \(\int \frac{1}{x \left (a^5+x^5\right )} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5} \]

[Out]

Log[x]/a^5 - Log[a^5 + x^5]/(5*a^5)

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Rubi [A]  time = 0.0236483, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a^5 + x^5)),x]

[Out]

Log[x]/a^5 - Log[a^5 + x^5]/(5*a^5)

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Rubi in Sympy [A]  time = 2.3961, size = 22, normalized size = 1. \[ \frac{\log{\left (x^{5} \right )}}{5 a^{5}} - \frac{\log{\left (a^{5} + x^{5} \right )}}{5 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a**5+x**5),x)

[Out]

log(x**5)/(5*a**5) - log(a**5 + x**5)/(5*a**5)

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Mathematica [A]  time = 0.00616191, size = 22, normalized size = 1. \[ \frac{\log (x)}{a^5}-\frac{\log \left (a^5+x^5\right )}{5 a^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a^5 + x^5)),x]

[Out]

Log[x]/a^5 - Log[a^5 + x^5]/(5*a^5)

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Maple [B]  time = 0.01, size = 49, normalized size = 2.2 \[ -{\frac{\ln \left ( a+x \right ) }{5\,{a}^{5}}}+{\frac{\ln \left ( x \right ) }{{a}^{5}}}-{\frac{\ln \left ({a}^{4}-{a}^{3}x+{a}^{2}{x}^{2}-a{x}^{3}+{x}^{4} \right ) }{5\,{a}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a^5+x^5),x)

[Out]

-1/5*ln(a+x)/a^5+ln(x)/a^5-1/5/a^5*ln(a^4-a^3*x+a^2*x^2-a*x^3+x^4)

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Maxima [A]  time = 1.33773, size = 31, normalized size = 1.41 \[ -\frac{\log \left (a^{5} + x^{5}\right )}{5 \, a^{5}} + \frac{\log \left (x^{5}\right )}{5 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^5 + x^5)*x),x, algorithm="maxima")

[Out]

-1/5*log(a^5 + x^5)/a^5 + 1/5*log(x^5)/a^5

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Fricas [A]  time = 0.206647, size = 24, normalized size = 1.09 \[ -\frac{\log \left (a^{5} + x^{5}\right ) - 5 \, \log \left (x\right )}{5 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^5 + x^5)*x),x, algorithm="fricas")

[Out]

-1/5*(log(a^5 + x^5) - 5*log(x))/a^5

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Sympy [A]  time = 0.322804, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x \right )}}{a^{5}} - \frac{\log{\left (a^{5} + x^{5} \right )}}{5 a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a**5+x**5),x)

[Out]

log(x)/a**5 - log(a**5 + x**5)/(5*a**5)

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GIAC/XCAS [A]  time = 0.207787, size = 30, normalized size = 1.36 \[ -\frac{{\rm ln}\left ({\left | a^{5} + x^{5} \right |}\right )}{5 \, a^{5}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a^5 + x^5)*x),x, algorithm="giac")

[Out]

-1/5*ln(abs(a^5 + x^5))/a^5 + ln(abs(x))/a^5