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

Optimal. Leaf size=13 \[ -\left (a x^n\right )^{-1/n} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 30} \begin {gather*} -\left (a x^n\right )^{-1/n} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\left (a x^n\right )^{-1/n}}{x} \, dx &=\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*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\left (a x^n\right )^{-1/n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 14, normalized size = 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	\(-\left (a \,x^{n}\right )^{-\frac {1}{n}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.31, size = 18, normalized size = 1.38 \begin {gather*} -\frac {1}{a^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 12, normalized size = 0.92 \begin {gather*} -\frac {1}{a^{\left (\frac {1}{n}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.30, size = 10, normalized size = 0.77 \begin {gather*} - \left (a x^{n}\right )^{- \frac {1}{n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.23, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{\left (a x^{n}\right )^{\left (\frac {1}{n}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.02, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{{\left (a\,x^n\right )}^{1/n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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