3.90 \(\int \frac{1}{\sqrt{b x^2}} \, dx\)

Optimal. Leaf size=13 \[ \frac{x \log (x)}{\sqrt{b x^2}} \]

[Out]

(x*Log[x])/Sqrt[b*x^2]

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Rubi [A]  time = 0.00678204, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{x \log (x)}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[b*x^2],x]

[Out]

(x*Log[x])/Sqrt[b*x^2]

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Rubi in Sympy [A]  time = 1.48988, size = 14, normalized size = 1.08 \[ \frac{\sqrt{b x^{2}} \log{\left (x \right )}}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x**2)*log(x)/(b*x)

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Mathematica [A]  time = 0.0026933, size = 13, normalized size = 1. \[ \frac{x \log (x)}{\sqrt{b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[b*x^2],x]

[Out]

(x*Log[x])/Sqrt[b*x^2]

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Maple [A]  time = 0., size = 12, normalized size = 0.9 \[{x\ln \left ( x \right ){\frac{1}{\sqrt{b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43424, size = 8, normalized size = 0.62 \[ \frac{\log \left (x\right )}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/sqrt(b)

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Fricas [A]  time = 0.243999, size = 22, normalized size = 1.69 \[ \frac{\sqrt{b x^{2}} \log \left (x\right )}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b*x^2)*log(x)/(b*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2)**(1/2),x)

[Out]

Integral(1/sqrt(b*x**2), x)

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GIAC/XCAS [A]  time = 0.247026, size = 24, normalized size = 1.85 \[ \frac{{\rm ln}\left (\sqrt{b}{\left | x \right |}{\left |{\rm sign}\left (x\right ) \right |}\right )}{\sqrt{b}{\rm sign}\left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(sqrt(b)*abs(x)*abs(sign(x)))/(sqrt(b)*sign(x))