\(\int \frac {(a x^n)^{-1/n}}{x} \, dx\) [185]

Optimal result

Integrand size = 15, antiderivative size = 13 \[ \int \frac {\left (a x^n\right )^{-1/n}}{x} \, dx=-\left (a x^n\right )^{-1/n} \]

[Out]

-1/((a*x^n)^(1/n))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 30} \[ \int \frac {\left (a x^n\right )^{-1/n}}{x} \, dx=-\left (a x^n\right )^{-1/n} \]

[In]

Int[1/(x*(a*x^n)^n^(-1)),x]

[Out]

-(a*x^n)^(-n^(-1))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

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

Rubi steps \begin{align*} \text {integral}& = \left (x \left (a x^n\right )^{-1/n}\right ) \int \frac {1}{x^2} \, dx \\ & = -\left (a x^n\right )^{-1/n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a x^n\right )^{-1/n}}{x} \, dx=-\left (a x^n\right )^{-1/n} \]

[In]

Integrate[1/(x*(a*x^n)^n^(-1)),x]

[Out]

-(a*x^n)^(-n^(-1))

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
gosper	\(-\left (a \,x^{n}\right )^{-\frac {1}{n}}\)	\(14\)
derivativedivides	\(-\left (a \,x^{n}\right )^{-\frac {1}{n}}\)	\(14\)
default	\(-\left (a \,x^{n}\right )^{-\frac {1}{n}}\)	\(14\)
parallelrisch	\(-\left (a \,x^{n}\right )^{-\frac {1}{n}}\)	\(14\)

[In]

int(1/x/((a*x^n)^(1/n)),x,method=_RETURNVERBOSE)

[Out]

-1/((a*x^n)^(1/n))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a x^n\right )^{-1/n}}{x} \, dx=-\frac {1}{a^{\left (\frac {1}{n}\right )} x} \]

[In]

integrate(1/x/((a*x^n)^(1/n)),x, algorithm="fricas")

[Out]

-1/(a^(1/n)*x)

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a x^n\right )^{-1/n}}{x} \, dx=- \left (a x^{n}\right )^{- \frac {1}{n}} \]

[In]

integrate(1/x/((a*x**n)**(1/n)),x)

[Out]

-1/(a*x**n)**(1/n)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a x^n\right )^{-1/n}}{x} \, dx=-\frac {1}{a^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}} \]

[In]

integrate(1/x/((a*x^n)^(1/n)),x, algorithm="maxima")

[Out]

-1/(a^(1/n)*(x^n)^(1/n))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a x^n\right )^{-1/n}}{x} \, dx=-\frac {1}{\left (a x^{n}\right )^{\left (\frac {1}{n}\right )}} \]

[In]

integrate(1/x/((a*x^n)^(1/n)),x, algorithm="giac")

[Out]

-1/(a*x^n)^(1/n)

Mupad [B] (verification not implemented)

Time = 5.60 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a x^n\right )^{-1/n}}{x} \, dx=-\frac {1}{{\left (a\,x^n\right )}^{1/n}} \]

[In]

int(1/(x*(a*x^n)^(1/n)),x)

[Out]

-1/(a*x^n)^(1/n)