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

Optimal. Leaf size=23 \[ \frac{\log (x)}{a}-\frac{\log \left (a-b x^2\right )}{2 a} \]

[Out]

Log[x]/a - Log[a - b*x^2]/(2*a)

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Rubi [A]  time = 0.0367481, 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 \[ \frac{\log (x)}{a}-\frac{\log \left (a-b x^2\right )}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a - b*x^2)),x]

[Out]

Log[x]/a - Log[a - b*x^2]/(2*a)

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Rubi in Sympy [A]  time = 6.48221, size = 19, normalized size = 0.83 \[ \frac{\log{\left (x^{2} \right )}}{2 a} - \frac{\log{\left (a - b x^{2} \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0110714, size = 23, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a-b x^2\right )}{2 a} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a - b*x^2)),x]

[Out]

Log[x]/a - Log[a - b*x^2]/(2*a)

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Maple [A]  time = 0.006, size = 23, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{2}-a \right ) }{2\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.35458, size = 34, normalized size = 1.48 \[ -\frac{\log \left (b x^{2} - a\right )}{2 \, a} + \frac{\log \left (x^{2}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.210973, size = 27, normalized size = 1.17 \[ -\frac{\log \left (b x^{2} - a\right ) - 2 \, \log \left (x\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(log(b*x^2 - a) - 2*log(x))/a

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Sympy [A]  time = 0.530493, size = 15, normalized size = 0.65 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (- \frac{a}{b} + x^{2} \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-b*x**2+a),x)

[Out]

log(x)/a - log(-a/b + x**2)/(2*a)

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GIAC/XCAS [A]  time = 0.209211, size = 35, normalized size = 1.52 \[ \frac{{\rm ln}\left (x^{2}\right )}{2 \, a} - \frac{{\rm ln}\left ({\left | b x^{2} - a \right |}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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