3.254 \(\int \frac{1}{x \left (-1+b x^2\right )} \, dx\)

Optimal. Leaf size=18 \[ \frac{1}{2} \log \left (1-b x^2\right )-\log (x) \]

[Out]

-Log[x] + Log[1 - b*x^2]/2

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Rubi [A]  time = 0.0313369, antiderivative size = 18, 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 \[ \frac{1}{2} \log \left (1-b x^2\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(-1 + b*x^2)),x]

[Out]

-Log[x] + Log[1 - b*x^2]/2

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Rubi in Sympy [A]  time = 5.16496, size = 15, normalized size = 0.83 \[ - \frac{\log{\left (x^{2} \right )}}{2} + \frac{\log{\left (- b x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**2-1),x)

[Out]

-log(x**2)/2 + log(-b*x**2 + 1)/2

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Mathematica [A]  time = 0.00802293, size = 18, normalized size = 1. \[ \frac{1}{2} \log \left (1-b x^2\right )-\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(-1 + b*x^2)),x]

[Out]

-Log[x] + Log[1 - b*x^2]/2

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Maple [A]  time = 0.006, size = 16, normalized size = 0.9 \[ -\ln \left ( x \right ) +{\frac{\ln \left ( b{x}^{2}-1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^2-1),x)

[Out]

-ln(x)+1/2*ln(b*x^2-1)

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Maxima [A]  time = 1.33542, size = 23, normalized size = 1.28 \[ \frac{1}{2} \, \log \left (b x^{2} - 1\right ) - \frac{1}{2} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - 1)*x),x, algorithm="maxima")

[Out]

1/2*log(b*x^2 - 1) - 1/2*log(x^2)

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Fricas [A]  time = 0.227538, size = 20, normalized size = 1.11 \[ \frac{1}{2} \, \log \left (b x^{2} - 1\right ) - \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - 1)*x),x, algorithm="fricas")

[Out]

1/2*log(b*x^2 - 1) - log(x)

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Sympy [A]  time = 0.308755, size = 12, normalized size = 0.67 \[ - \log{\left (x \right )} + \frac{\log{\left (x^{2} - \frac{1}{b} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**2-1),x)

[Out]

-log(x) + log(x**2 - 1/b)/2

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GIAC/XCAS [A]  time = 0.219233, size = 24, normalized size = 1.33 \[ -\frac{1}{2} \,{\rm ln}\left (x^{2}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | b x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - 1)*x),x, algorithm="giac")

[Out]

-1/2*ln(x^2) + 1/2*ln(abs(b*x^2 - 1))