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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]