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

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

[Out]

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

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Rubi [A]  time = 0.0016147, antiderivative size = 13, normalized size of antiderivative = 1., 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} \[ -\left (a x^n\right )^{-1/n} \]

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*}

Mathematica [A]  time = 0.0010883, size = 13, normalized size = 1. \[ -\left (a x^n\right )^{-1/n} \]

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 = 0.001, size = 14, normalized size = 1.1 \begin{align*} - \left ( \sqrt [n]{a{x}^{n}} \right ) ^{-1} \end{align*}

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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Maxima [A]  time = 1.02555, size = 24, normalized size = 1.85 \begin{align*} -\frac{1}{a^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}} \end{align*}

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 = 1.7453, size = 22, normalized size = 1.69 \begin{align*} -\frac{1}{a^{\left (\frac{1}{n}\right )} x} \end{align*}

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 = 1.36858, size = 32, normalized size = 2.46 \begin{align*} \begin{cases} - a^{- \frac{1}{n}} \left (x^{n}\right )^{- \frac{1}{n}} & \text{for}\: a \neq 0^{n} \\- \left (0^{n}\right )^{- \frac{1}{n}} \left (x^{n}\right )^{- \frac{1}{n}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**(-1/n)*(x**n)**(-1/n), Ne(a, 0**n)), (-(0**n)**(-1/n)*(x**n)**(-1/n), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a x^{n}\right )^{\left (\frac{1}{n}\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a*x^n)^(1/n)*x), x)