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

Optimal. Leaf size=34 \[ -\sqrt{x^2+1}+\sqrt{x^2+1} \log (x)+\tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.066878, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\sqrt{x^2+1}+\sqrt{x^2+1} \log (x)+\tanh ^{-1}\left (\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x*Log[x])/Sqrt[1 + x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.47028, size = 29, normalized size = 0.85 \[ \sqrt{x^{2} + 1} \log{\left (x \right )} - \sqrt{x^{2} + 1} + \operatorname{atanh}{\left (\sqrt{x^{2} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0207106, size = 40, normalized size = 1.18 \[ -\sqrt{x^2+1}+\sqrt{x^2+1} \log (x)+\log \left (\sqrt{x^2+1}+1\right )-\log (x) \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 + x^2] - Log[x] + Sqrt[1 + x^2]*Log[x] + Log[1 + Sqrt[1 + x^2]]

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Maple [A]  time = 0.038, size = 39, normalized size = 1.2 \[ 1-\sqrt{{x}^{2}+1}+{\frac{\ln \left ( x \right ) }{2} \left ( -2+2\,\sqrt{{x}^{2}+1} \right ) }+\ln \left ({\frac{1}{2}}+{\frac{1}{2}\sqrt{{x}^{2}+1}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*ln(x)/(x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.60265, size = 34, normalized size = 1. \[ \sqrt{x^{2} + 1} \log \left (x\right ) - \sqrt{x^{2} + 1} + \operatorname{arsinh}\left (\frac{1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.241642, size = 128, normalized size = 3.76 \[ \frac{x^{2} -{\left (x^{2} + 1\right )} \log \left (x\right ) +{\left (x - \sqrt{x^{2} + 1}\right )} \log \left (-x + \sqrt{x^{2} + 1} + 1\right ) -{\left (x - \sqrt{x^{2} + 1}\right )} \log \left (-x + \sqrt{x^{2} + 1} - 1\right ) + \sqrt{x^{2} + 1}{\left (x \log \left (x\right ) - x\right )} + 1}{x - \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.75978, size = 41, normalized size = 1.21 \[ - \frac{x}{\sqrt{1 + \frac{1}{x^{2}}}} + \sqrt{x^{2} + 1} \log{\left (x \right )} + \operatorname{asinh}{\left (\frac{1}{x} \right )} - \frac{1}{x \sqrt{1 + \frac{1}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.201371, size = 59, normalized size = 1.74 \[ \sqrt{x^{2} + 1}{\rm ln}\left (x\right ) - \sqrt{x^{2} + 1} + \frac{1}{2} \,{\rm ln}\left (\sqrt{x^{2} + 1} + 1\right ) - \frac{1}{2} \,{\rm ln}\left (\sqrt{x^{2} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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