3.618 \(\int \frac{\log ^n(x)}{x} \, dx\)

Optimal. Leaf size=12 \[ \frac{\log ^{n+1}(x)}{n+1} \]

[Out]

Log[x]^(1 + n)/(1 + n)

_______________________________________________________________________________________

Rubi [A]  time = 0.027003, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\log ^{n+1}(x)}{n+1} \]

Antiderivative was successfully verified.

[In]  Int[Log[x]^n/x,x]

[Out]

Log[x]^(1 + n)/(1 + n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.67372, size = 8, normalized size = 0.67 \[ \frac{\log{\left (x \right )}^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(x)**n/x,x)

[Out]

log(x)**(n + 1)/(n + 1)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00357997, size = 12, normalized size = 1. \[ \frac{\log ^{n+1}(x)}{n+1} \]

Antiderivative was successfully verified.

[In]  Integrate[Log[x]^n/x,x]

[Out]

Log[x]^(1 + n)/(1 + n)

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 13, normalized size = 1.1 \[{\frac{ \left ( \ln \left ( x \right ) \right ) ^{1+n}}{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x)^n/x,x)

[Out]

ln(x)^(1+n)/(1+n)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x)^n/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.218214, size = 16, normalized size = 1.33 \[ \frac{\log \left (x\right )^{n} \log \left (x\right )}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x)^n/x,x, algorithm="fricas")

[Out]

log(x)^n*log(x)/(n + 1)

_______________________________________________________________________________________

Sympy [A]  time = 1.2244, size = 15, normalized size = 1.25 \[ \begin{cases} \frac{\log{\left (x \right )}^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left (\log{\left (x \right )} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(x)**n/x,x)

[Out]

Piecewise((log(x)**(n + 1)/(n + 1), Ne(n, -1)), (log(log(x)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.198872, size = 16, normalized size = 1.33 \[ \frac{{\rm ln}\left (x\right )^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x)^n/x,x, algorithm="giac")

[Out]

ln(x)^(n + 1)/(n + 1)