3.121 \(\int \frac{1}{x \left (a^3+x^3\right )} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a^3}-\frac{\log \left (a^3+x^3\right )}{3 a^3} \]

[Out]

Log[x]/a^3 - Log[a^3 + x^3]/(3*a^3)

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Rubi [A]  time = 0.0236263, 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 \[ \frac{\log (x)}{a^3}-\frac{\log \left (a^3+x^3\right )}{3 a^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a^3 + x^3)),x]

[Out]

Log[x]/a^3 - Log[a^3 + x^3]/(3*a^3)

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Rubi in Sympy [A]  time = 2.39399, size = 22, normalized size = 1. \[ \frac{\log{\left (x^{3} \right )}}{3 a^{3}} - \frac{\log{\left (a^{3} + x^{3} \right )}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**3)/(3*a**3) - log(a**3 + x**3)/(3*a**3)

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Mathematica [A]  time = 0.00625151, size = 22, normalized size = 1. \[ \frac{\log (x)}{a^3}-\frac{\log \left (a^3+x^3\right )}{3 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a^3 + x^3)),x]

[Out]

Log[x]/a^3 - Log[a^3 + x^3]/(3*a^3)

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Maple [A]  time = 0.01, size = 34, normalized size = 1.6 \[ -{\frac{\ln \left ( a+x \right ) }{3\,{a}^{3}}}-{\frac{\ln \left ({a}^{2}-ax+{x}^{2} \right ) }{3\,{a}^{3}}}+{\frac{\ln \left ( x \right ) }{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a^3+x^3),x)

[Out]

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

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Maxima [A]  time = 1.37183, size = 31, normalized size = 1.41 \[ -\frac{\log \left (a^{3} + x^{3}\right )}{3 \, a^{3}} + \frac{\log \left (x^{3}\right )}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(a^3 + x^3)/a^3 + 1/3*log(x^3)/a^3

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Fricas [A]  time = 0.2078, size = 24, normalized size = 1.09 \[ -\frac{\log \left (a^{3} + x^{3}\right ) - 3 \, \log \left (x\right )}{3 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(log(a^3 + x^3) - 3*log(x))/a^3

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Sympy [A]  time = 0.254326, size = 19, normalized size = 0.86 \[ \frac{\log{\left (x \right )}}{a^{3}} - \frac{\log{\left (a^{3} + x^{3} \right )}}{3 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a**3+x**3),x)

[Out]

log(x)/a**3 - log(a**3 + x**3)/(3*a**3)

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GIAC/XCAS [A]  time = 0.207743, size = 30, normalized size = 1.36 \[ -\frac{{\rm ln}\left ({\left | a^{3} + x^{3} \right |}\right )}{3 \, a^{3}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*ln(abs(a^3 + x^3))/a^3 + ln(abs(x))/a^3