3.64 \(\int \frac{\log ^p(x)}{x} \, dx\)

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

[Out]

Log[x]^(1 + p)/(1 + p)

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Rubi [A]  time = 0.0249641, 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 \[ \frac{\log ^{p+1}(x)}{p+1} \]

Antiderivative was successfully verified.

[In]  Int[Log[x]^p/x,x]

[Out]

Log[x]^(1 + p)/(1 + p)

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Rubi in Sympy [A]  time = 1.63509, size = 8, normalized size = 0.67 \[ \frac{\log{\left (x \right )}^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(x)**p/x,x)

[Out]

log(x)**(p + 1)/(p + 1)

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Mathematica [A]  time = 0.00353357, size = 12, normalized size = 1. \[ \frac{\log ^{p+1}(x)}{p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[Log[x]^p/x,x]

[Out]

Log[x]^(1 + p)/(1 + p)

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Maple [A]  time = 0.001, size = 13, normalized size = 1.1 \[{\frac{ \left ( \ln \left ( x \right ) \right ) ^{1+p}}{1+p}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(x)^p/x,x)

[Out]

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

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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)^p/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218832, size = 16, normalized size = 1.33 \[ \frac{\log \left (x\right )^{p} \log \left (x\right )}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)^p*log(x)/(p + 1)

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Sympy [A]  time = 1.17423, size = 15, normalized size = 1.25 \[ \begin{cases} \frac{\log{\left (x \right )}^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (\log{\left (x \right )} \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(x)**p/x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.198559, size = 16, normalized size = 1.33 \[ \frac{{\rm ln}\left (x\right )^{p + 1}}{p + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(x)^(p + 1)/(p + 1)