Optimal. Leaf size=18 \[ -\frac {\left (a x^n\right )^{-1/n}}{2 x} \]
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Rubi [A] time = 0.00, antiderivative size = 18, 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} \[ -\frac {\left (a x^n\right )^{-1/n}}{2 x} \]
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*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \[ -\frac {\left (a x^n\right )^{-1/n}}{2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 12, normalized size = 0.67 \[ -\frac {1}{2 \, a^{\left (\frac {1}{n}\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a x^{n}\right )^{\left (\frac {1}{n}\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.94 \[ -\frac {\left (a \,x^{n}\right )^{-\frac {1}{n}}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a x^{n}\right )^{\left (\frac {1}{n}\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{x^2\,{\left (a\,x^n\right )}^{1/n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.03, size = 60, normalized size = 3.33 \[ \begin {cases} - \frac {a^{- \frac {1}{n}} \left (x^{n}\right )^{- \frac {1}{n}}}{2 x} & \text {for}\: a \neq 0^{n} \\- \frac {1}{0^{n} \tilde {\infty }^{n} x \left (0^{n}\right )^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} + x \left (0^{n}\right )^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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