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

Optimal. Leaf size=23 \[ \frac{2}{3} (\log (x)+1)^{3/2}-2 \sqrt{\log (x)+1} \]

[Out]

-2*Sqrt[1 + Log[x]] + (2*(1 + Log[x])^(3/2))/3

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Rubi [A]  time = 0.065073, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2}{3} (\log (x)+1)^{3/2}-2 \sqrt{\log (x)+1} \]

Antiderivative was successfully verified.

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

[Out]

-2*Sqrt[1 + Log[x]] + (2*(1 + Log[x])^(3/2))/3

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Rubi in Sympy [A]  time = 5.06574, size = 24, normalized size = 1.04 \[ - \frac{4 \left (\log{\left (x \right )} + 1\right )^{\frac{3}{2}}}{3} + 2 \sqrt{\log{\left (x \right )} + 1} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*(log(x) + 1)**(3/2)/3 + 2*sqrt(log(x) + 1)*log(x)

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Mathematica [A]  time = 0.00760408, size = 16, normalized size = 0.7 \[ \frac{2}{3} (\log (x)-2) \sqrt{\log (x)+1} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-2 + Log[x])*Sqrt[1 + Log[x]])/3

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Maple [A]  time = 0.009, size = 18, normalized size = 0.8 \[{\frac{2}{3} \left ( 1+\ln \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-2\,\sqrt{1+\ln \left ( x \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x)/x/(1+ln(x))^(1/2),x)

[Out]

2/3*(1+ln(x))^(3/2)-2*(1+ln(x))^(1/2)

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Maxima [A]  time = 1.50231, size = 23, normalized size = 1. \[ \frac{2}{3} \,{\left (\log \left (x\right ) + 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{\log \left (x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(log(x) + 1)^(3/2) - 2*sqrt(log(x) + 1)

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Fricas [A]  time = 0.203332, size = 16, normalized size = 0.7 \[ \frac{2}{3} \, \sqrt{\log \left (x\right ) + 1}{\left (\log \left (x\right ) - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(log(x) + 1)*(log(x) - 2)

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Sympy [A]  time = 1.43454, size = 20, normalized size = 0.87 \[ \frac{2 \left (\log{\left (x \right )} + 1\right )^{\frac{3}{2}}}{3} - 2 \sqrt{\log{\left (x \right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(log(x) + 1)**(3/2)/3 - 2*sqrt(log(x) + 1)

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GIAC/XCAS [A]  time = 0.2355, size = 23, normalized size = 1. \[ \frac{2}{3} \,{\left ({\rm ln}\left (x\right ) + 1\right )}^{\frac{3}{2}} - 2 \, \sqrt{{\rm ln}\left (x\right ) + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(ln(x) + 1)^(3/2) - 2*sqrt(ln(x) + 1)