3.5 \(\int \frac{\log ^{-1+q}(c x^n)}{x} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log ^q\left (c x^n\right )}{n q} \]

[Out]

Log[c*x^n]^q/(n*q)

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Rubi [A]  time = 0.0217537, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2302, 30} \[ \frac{\log ^q\left (c x^n\right )}{n q} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*x^n]^(-1 + q)/x,x]

[Out]

Log[c*x^n]^q/(n*q)

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\log ^{-1+q}\left (c x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\log ^q\left (c x^n\right )}{n q}\\ \end{align*}

Mathematica [A]  time = 0.0020717, size = 15, normalized size = 1. \[ \frac{\log ^q\left (c x^n\right )}{n q} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*x^n]^(-1 + q)/x,x]

[Out]

Log[c*x^n]^q/(n*q)

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Maple [A]  time = 0.003, size = 16, normalized size = 1.1 \begin{align*}{\frac{ \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q}}{nq}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*x^n)^(-1+q)/x,x)

[Out]

ln(c*x^n)^q/n/q

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)^(-1+q)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.94719, size = 74, normalized size = 4.93 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (c\right )\right )}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q - 1}}{n q} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)^(-1+q)/x,x, algorithm="fricas")

[Out]

(n*log(x) + log(c))*(n*log(x) + log(c))^(q - 1)/(n*q)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c x^{n} \right )}^{q - 1}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*x**n)**(-1+q)/x,x)

[Out]

Integral(log(c*x**n)**(q - 1)/x, x)

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Giac [A]  time = 1.30102, size = 22, normalized size = 1.47 \begin{align*} \frac{{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q}}{n q} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*x^n)^(-1+q)/x,x, algorithm="giac")

[Out]

(n*log(x) + log(c))^q/(n*q)