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

Optimal. Leaf size=30 \[ \frac{1}{2 \left (1-b x^2\right )}-\frac{1}{2} \log \left (1-b x^2\right )+\log (x) \]

[Out]

1/(2*(1 - b*x^2)) + Log[x] - Log[1 - b*x^2]/2

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Rubi [A]  time = 0.0523323, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{2 \left (1-b x^2\right )}-\frac{1}{2} \log \left (1-b x^2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(-1 + b*x^2)^2),x]

[Out]

1/(2*(1 - b*x^2)) + Log[x] - Log[1 - b*x^2]/2

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Rubi in Sympy [A]  time = 6.94858, size = 26, normalized size = 0.87 \[ \frac{\log{\left (x^{2} \right )}}{2} - \frac{\log{\left (- b x^{2} + 1 \right )}}{2} + \frac{1}{2 \left (- b x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2)/2 - log(-b*x**2 + 1)/2 + 1/(2*(-b*x**2 + 1))

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Mathematica [A]  time = 0.0240394, size = 26, normalized size = 0.87 \[ \frac{1}{2-2 b x^2}-\frac{1}{2} \log \left (1-b x^2\right )+\log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(-1 + b*x^2)^2),x]

[Out]

(2 - 2*b*x^2)^(-1) + Log[x] - Log[1 - b*x^2]/2

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Maple [A]  time = 0.015, size = 25, normalized size = 0.8 \[ \ln \left ( x \right ) -{\frac{1}{2\,b{x}^{2}-2}}-{\frac{\ln \left ( b{x}^{2}-1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^2-1)^2,x)

[Out]

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

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Maxima [A]  time = 1.35406, size = 38, normalized size = 1.27 \[ -\frac{1}{2 \,{\left (b x^{2} - 1\right )}} - \frac{1}{2} \, \log \left (b x^{2} - 1\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221936, size = 54, normalized size = 1.8 \[ -\frac{{\left (b x^{2} - 1\right )} \log \left (b x^{2} - 1\right ) - 2 \,{\left (b x^{2} - 1\right )} \log \left (x\right ) + 1}{2 \,{\left (b x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.37295, size = 22, normalized size = 0.73 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} - \frac{1}{b} \right )}}{2} - \frac{1}{2 b x^{2} - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**2-1)**2,x)

[Out]

log(x) - log(x**2 - 1/b)/2 - 1/(2*b*x**2 - 2)

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GIAC/XCAS [A]  time = 0.209082, size = 49, normalized size = 1.63 \[ \frac{b x^{2} - 2}{2 \,{\left (b x^{2} - 1\right )}} + \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | b x^{2} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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