Optimal. Leaf size=18 \[ -\frac{\left (a x^n\right )^{-1/n}}{2 x} \]
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Rubi [A] time = 0.0017376, antiderivative size = 18, 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} \[ -\frac{\left (a x^n\right )^{-1/n}}{2 x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{\left (a x^n\right )^{-1/n}}{x^2} \, dx &=\left (x \left (a x^n\right )^{-1/n}\right ) \int \frac{1}{x^3} \, dx\\ &=-\frac{\left (a x^n\right )^{-1/n}}{2 x}\\ \end{align*}
Mathematica [A] time = 0.0016529, size = 18, normalized size = 1. \[ -\frac{\left (a x^n\right )^{-1/n}}{2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 17, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,x\sqrt [n]{a{x}^{n}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a x^{n}\right )^{\left (\frac{1}{n}\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72035, size = 27, normalized size = 1.5 \begin{align*} -\frac{1}{2 \, a^{\left (\frac{1}{n}\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.15301, size = 60, normalized size = 3.33 \begin{align*} \begin{cases} - \frac{a^{- \frac{1}{n}} \left (x^{n}\right )^{- \frac{1}{n}}}{2 x} & \text{for}\: a \neq 0^{n} \\- \frac{1}{0^{n} \tilde{\infty }^{n} x \left (0^{n}\right )^{\frac{1}{n}} \left (x^{n}\right )^{\frac{1}{n}} + x \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.
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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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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