3.1.5 \(\int \frac {\log ^{-1+q}(c x^n)}{x} \, dx\) [5]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {2339, 30} \begin {gather*} \frac {\log ^q\left (c x^n\right )}{n q} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*x^n]^(-1 + q)/x,x]

[Out]

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

Rule 30

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

Rule 2339

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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log ^q\left (c x^n\right )}{n q} \end {gather*}

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 = 0.20, size = 16, normalized size = 1.07

method	result	size
derivativedivides	\(\frac {\ln \left (c \,x^{n}\right )^{q}}{n q}\)	\(16\)
default	\(\frac {\ln \left (c \,x^{n}\right )^{q}}{n q}\)	\(16\)
risch	\(\frac {\left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i x^{n}\right )\right )}{2}\right )^{q}}{n q}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (c x^{n}\right )^{q}}{n q} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 25, normalized size = 1.67 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c x^{n} \right )}^{q - 1}}{x}\, dx \end {gather*}

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 = 7.35, size = 16, normalized size = 1.07 \begin {gather*} \frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q}}{n q} \end {gather*}

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)

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Mupad [B]
time = 0.28, size = 15, normalized size = 1.00 \begin {gather*} \frac {{\ln \left (c\,x^n\right )}^q}{n\,q} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*x^n)^(q - 1)/x,x)

[Out]

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

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