3.269 \(\int \frac{\sqrt{1+\log (x)}}{x \log (x)} \, dx\)

Optimal. Leaf size=22 \[ 2 \sqrt{\log (x)+1}-2 \tanh ^{-1}\left (\sqrt{\log (x)+1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0912611, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ 2 \sqrt{\log (x)+1}-2 \tanh ^{-1}\left (\sqrt{\log (x)+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.2017, size = 20, normalized size = 0.91 \[ 2 \sqrt{\log{\left (x \right )} + 1} - 2 \operatorname{atanh}{\left (\sqrt{\log{\left (x \right )} + 1} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0155022, size = 37, normalized size = 1.68 \[ \log \left (1-\sqrt{\log (x)+1}\right )-\log \left (\sqrt{\log (x)+1}+1\right )+2 \sqrt{\log (x)+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 30, normalized size = 1.4 \[ 2\,\sqrt{1+\ln \left ( x \right ) }+\ln \left ( -1+\sqrt{1+\ln \left ( x \right ) } \right ) -\ln \left ( 1+\sqrt{1+\ln \left ( x \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+ln(x))^(1/2)/x/ln(x),x)

[Out]

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

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Maxima [A]  time = 1.3562, size = 39, normalized size = 1.77 \[ 2 \, \sqrt{\log \left (x\right ) + 1} - \log \left (\sqrt{\log \left (x\right ) + 1} + 1\right ) + \log \left (\sqrt{\log \left (x\right ) + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23841, size = 39, normalized size = 1.77 \[ 2 \, \sqrt{\log \left (x\right ) + 1} - \log \left (\sqrt{\log \left (x\right ) + 1} + 1\right ) + \log \left (\sqrt{\log \left (x\right ) + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.950221, size = 32, normalized size = 1.45 \[ 2 \sqrt{\log{\left (x \right )} + 1} + \log{\left (\sqrt{\log{\left (x \right )} + 1} - 1 \right )} - \log{\left (\sqrt{\log{\left (x \right )} + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out