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

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

[Out]

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

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Rubi [A]  time = 0.00704155, 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 \[ -\left (a x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a*x^n)^n^(-1)),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.00404, size = 10, normalized size = 0.77 \[ - \left (a x^{n}\right )^{- \frac{1}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/((a*x**n)**(1/n)),x)

[Out]

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

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Mathematica [A]  time = 0.00242291, 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 = 0.003, size = 14, normalized size = 1.1 \[ - \left ( \sqrt [n]{a{x}^{n}} \right ) ^{-1} \]

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 = 1.45968, size = 24, normalized size = 1.85 \[ -a^{-\frac{1}{n}}{\left (x^{n}\right )}^{-\frac{1}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.225973, size = 16, normalized size = 1.23 \[ -\frac{1}{a^{\left (\frac{1}{n}\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/((a*x**n)**(1/n)),x)

[Out]

Exception raised: RecursionError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a x^{n}\right )^{\left (\frac{1}{n}\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((a*x^n)^(1/n)*x), x)