Optimal. Leaf size=15 \[ -\frac{2}{n \sqrt{\log \left (a x^n\right )}} \]
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Rubi [A] time = 0.0140202, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2302, 30} \[ -\frac{2}{n \sqrt{\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{x \log ^{\frac{3}{2}}\left (a x^n\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=-\frac{2}{n \sqrt{\log \left (a x^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.0018364, size = 15, normalized size = 1. \[ -\frac{2}{n \sqrt{\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 14, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{n\sqrt{\ln \left ( a{x}^{n} \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13858, size = 18, normalized size = 1.2 \begin{align*} -\frac{2}{n \sqrt{\log \left (a x^{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.94426, size = 70, normalized size = 4.67 \begin{align*} -\frac{2 \, \sqrt{n \log \left (x\right ) + \log \left (a\right )}}{n^{2} \log \left (x\right ) + n \log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 165.01, size = 48, normalized size = 3.2 \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: \left (a = 1 \vee a = e^{- n \log{\left (x \right )}}\right ) \wedge \left (a = e^{- n \log{\left (x \right )}} \vee n = 0\right ) \\\frac{\log{\left (x \right )}}{\log{\left (a \right )}^{\frac{3}{2}}} & \text{for}\: n = 0 \\- \frac{2}{n \sqrt{n \log{\left (x \right )} + \log{\left (a \right )}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19797, size = 19, normalized size = 1.27 \begin{align*} -\frac{2}{\sqrt{n \log \left (x\right ) + \log \left (a\right )} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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