3.18 \(\int \frac{\log (x)}{x^2 \sqrt{-1+x^2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{\sqrt{x^2-1}}{x}+\frac{\sqrt{x^2-1} \log (x)}{x}-\tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0802764, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\sqrt{x^2-1}}{x}+\frac{\sqrt{x^2-1} \log (x)}{x}-\tanh ^{-1}\left (\frac{x}{\sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Log[x]/(x^2*Sqrt[-1 + x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.19531, size = 34, normalized size = 0.79 \[ - \operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} - 1}} \right )} + \frac{\sqrt{x^{2} - 1} \log{\left (x \right )}}{x} + \frac{\sqrt{x^{2} - 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(x/sqrt(x**2 - 1)) + sqrt(x**2 - 1)*log(x)/x + sqrt(x**2 - 1)/x

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Mathematica [A]  time = 0.0204133, size = 43, normalized size = 1. \[ \frac{\sqrt{x^2-1}}{x}+\frac{\sqrt{x^2-1} \log (x)}{x}-\log \left (\sqrt{x^2-1}+x\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.109, size = 89, normalized size = 2.1 \[ -{\arcsin \left ( x \right ) \sqrt{-{\it signum} \left ({x}^{2}-1 \right ) }{\frac{1}{\sqrt{{\it signum} \left ({x}^{2}-1 \right ) }}}}+{\frac{1}{x} \left ( -{1\sqrt{-{\it signum} \left ({x}^{2}-1 \right ) }\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{\it signum} \left ({x}^{2}-1 \right ) }}}}-{\ln \left ( x \right ) \sqrt{-{\it signum} \left ({x}^{2}-1 \right ) }\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{\it signum} \left ({x}^{2}-1 \right ) }}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x)/x^2/(x^2-1)^(1/2),x)

[Out]

-1/signum(x^2-1)^(1/2)*(-signum(x^2-1))^(1/2)*arcsin(x)+(-1/signum(x^2-1)^(1/2)*
(-signum(x^2-1))^(1/2)*(-x^2+1)^(1/2)-1/signum(x^2-1)^(1/2)*(-signum(x^2-1))^(1/
2)*ln(x)*(-x^2+1)^(1/2))/x

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Maxima [A]  time = 1.59195, size = 55, normalized size = 1.28 \[ \frac{\sqrt{x^{2} - 1} \log \left (x\right )}{x} + \frac{\sqrt{x^{2} - 1}}{x} - \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227597, size = 89, normalized size = 2.07 \[ \frac{\sqrt{x^{2} - 1} x \log \left (x\right ) -{\left (x^{2} - 1\right )} \log \left (x\right ) +{\left (x^{2} - \sqrt{x^{2} - 1} x\right )} \log \left (-x + \sqrt{x^{2} - 1}\right ) + 1}{x^{2} - \sqrt{x^{2} - 1} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(sqrt(x^2 - 1)*x*log(x) - (x^2 - 1)*log(x) + (x^2 - sqrt(x^2 - 1)*x)*log(-x + sq
rt(x^2 - 1)) + 1)/(x^2 - sqrt(x^2 - 1)*x)

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Sympy [A]  time = 150.189, size = 37, normalized size = 0.86 \[ \left (\begin{cases} \frac{\sqrt{x^{2} - 1}}{x} & \text{for}\: x > -1 \wedge x < 1 \end{cases}\right ) \log{\left (x \right )} - \begin{cases} \mathrm{NaN} & \text{for}\: x < -1 \\\operatorname{acosh}{\left (x \right )} - i \pi - \frac{\sqrt{x^{2} - 1}}{x} & \text{for}\: x < 1 \\\mathrm{NaN} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(x)/x**2/(x**2-1)**(1/2),x)

[Out]

Piecewise((sqrt(x**2 - 1)/x, (x > -1) & (x < 1)))*log(x) - Piecewise((nan, x < -
1), (acosh(x) - I*pi - sqrt(x**2 - 1)/x, x < 1), (nan, True))

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GIAC/XCAS [A]  time = 0.203329, size = 84, normalized size = 1.95 \[ \frac{2 \,{\rm ln}\left (x\right )}{{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 1} + \frac{2}{{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 1} + \frac{1}{2} \,{\rm ln}\left ({\left (x - \sqrt{x^{2} - 1}\right )}^{2}\right ) -{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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