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

Optimal result

Integrand size = 14, antiderivative size = 15 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {\log ^q\left (c x^n\right )}{n q} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 30} \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {\log ^q\left (c x^n\right )}{n q} \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {\log ^q\left (c x^n\right )}{n q} \]

[In]

Integrate[Log[c*x^n]^(-1 + q)/x,x]

[Out]

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

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 16, normalized size of antiderivative = 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	\(\frac {{\left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )}^{-1+q} \left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )}{n q}\)	\(118\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )} {\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q - 1}}{n q} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.27 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=- \begin {cases} - \log {\left (c \right )}^{q - 1} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\begin {cases} \frac {\log {\left (c x^{n} \right )}^{q}}{q} & \text {for}\: q \neq 0 \\\log {\left (\log {\left (c x^{n} \right )} \right )} & \text {otherwise} \end {cases}}{n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

-Piecewise((-log(c)**(q - 1)*log(x), Eq(n, 0)), (-Piecewise((log(c*x**n)**q/q, Ne(q, 0)), (log(log(c*x**n)), T
rue))/n, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {\log \left (c x^{n}\right )^{q}}{n q} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.53 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q}}{n q} \]

[In]

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

[Out]

(n*log(x) + log(c))^q/(n*q)

Mupad [B] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {{\ln \left (c\,x^n\right )}^q}{n\,q} \]

[In]

int(log(c*x^n)^(q - 1)/x,x)

[Out]

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