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

Optimal. Leaf size=33 \[ -\frac{\sqrt{x^2+1}}{x}-\frac{\sqrt{x^2+1} \log (x)}{x}+\sinh ^{-1}(x) \]

[Out]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

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