Optimal. Leaf size=33 \[ -\frac{\sqrt{x^2+1}}{x}-\frac{\sqrt{x^2+1} \log (x)}{x}+\sinh ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0744095, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sqrt{x^2+1}}{x}-\frac{\sqrt{x^2+1} \log (x)}{x}+\sinh ^{-1}(x) \]
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 = 6.08868, size = 26, normalized size = 0.79 \[ \operatorname{asinh}{\left (x \right )} - \frac{\sqrt{x^{2} + 1} \log{\left (x \right )}}{x} - \frac{\sqrt{x^{2} + 1}}{x} \]
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 = 0.027828, size = 21, normalized size = 0.64 \[ \sinh ^{-1}(x)-\frac{\sqrt{x^2+1} (\log (x)+1)}{x} \]
Antiderivative was successfully verified.
[In] Integrate[Log[x]/(x^2*Sqrt[1 + x^2]),x]
[Out]
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Maple [A] time = 0.05, size = 29, normalized size = 0.9 \[{\it Arcsinh} \left ( x \right ) +{\frac{1}{x} \left ( -\ln \left ( x \right ) \sqrt{{x}^{2}+1}-\sqrt{{x}^{2}+1} \right ) } \]
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 = 1.63262, size = 39, normalized size = 1.18 \[ -\frac{\sqrt{x^{2} + 1} \log \left (x\right )}{x} - \frac{\sqrt{x^{2} + 1}}{x} + \operatorname{arsinh}\left (x\right ) \]
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 = 0.252897, size = 90, normalized size = 2.73 \[ -\frac{\sqrt{x^{2} + 1} x \log \left (x\right ) -{\left (x^{2} + 1\right )} \log \left (x\right ) +{\left (x^{2} - \sqrt{x^{2} + 1} x\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) - 1}{x^{2} - \sqrt{x^{2} + 1} x} \]
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 = 11.5661, size = 26, normalized size = 0.79 \[ \operatorname{asinh}{\left (x \right )} - \frac{\sqrt{x^{2} + 1} \log{\left (x \right )}}{x} - \frac{\sqrt{x^{2} + 1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(ln(x)/x**2/(x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.203719, size = 78, normalized size = 2.36 \[ \frac{2 \,{\rm ln}\left (x\right )}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} + \frac{2}{{\left (x - \sqrt{x^{2} + 1}\right )}^{2} - 1} -{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(log(x)/(sqrt(x^2 + 1)*x^2),x, algorithm="giac")
[Out]