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

Optimal. Leaf size=23 \[ \frac{\log (x)}{a}-\frac{\log \left (a-b x^3\right )}{3 a} \]

[Out]

Log[x]/a - Log[a - b*x^3]/(3*a)

_______________________________________________________________________________________

Rubi [A]  time = 0.0339012, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\log (x)}{a}-\frac{\log \left (a-b x^3\right )}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a - b*x^3)),x]

[Out]

Log[x]/a - Log[a - b*x^3]/(3*a)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.21322, size = 19, normalized size = 0.83 \[ \frac{\log{\left (x^{3} \right )}}{3 a} - \frac{\log{\left (a - b x^{3} \right )}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0112752, size = 23, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a-b x^3\right )}{3 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a - b*x^3)),x]

[Out]

Log[x]/a - Log[a - b*x^3]/(3*a)

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 23, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{3}-a \right ) }{3\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-b*x^3+a),x)

[Out]

ln(x)/a-1/3/a*ln(b*x^3-a)

_______________________________________________________________________________________

Maxima [A]  time = 1.43427, size = 34, normalized size = 1.48 \[ -\frac{\log \left (b x^{3} - a\right )}{3 \, a} + \frac{\log \left (x^{3}\right )}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.241599, size = 27, normalized size = 1.17 \[ -\frac{\log \left (b x^{3} - a\right ) - 3 \, \log \left (x\right )}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(log(b*x^3 - a) - 3*log(x))/a

_______________________________________________________________________________________

Sympy [A]  time = 0.667913, size = 15, normalized size = 0.65 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (- \frac{a}{b} + x^{3} \right )}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-b*x**3+a),x)

[Out]

log(x)/a - log(-a/b + x**3)/(3*a)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.242507, size = 32, normalized size = 1.39 \[ -\frac{{\rm ln}\left ({\left | b x^{3} - a \right |}\right )}{3 \, a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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