Optimal. Leaf size=20 \[ -\frac{1}{b n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.024049, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2302, 30} \[ -\frac{1}{b n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2302
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2} \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}\\ &=-\frac{1}{b n \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.0045306, size = 20, normalized size = 1. \[ -\frac{1}{b n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 21, normalized size = 1.1 \begin{align*} -{\frac{1}{bn \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13194, size = 27, normalized size = 1.35 \begin{align*} -\frac{1}{{\left (b \log \left (c x^{n}\right ) + a\right )} b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.804778, size = 59, normalized size = 2.95 \begin{align*} -\frac{1}{b^{2} n^{2} \log \left (x\right ) + b^{2} n \log \left (c\right ) + a b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 49.4998, size = 70, normalized size = 3.5 \begin{align*} \begin{cases} \frac{\tilde{\infty } \log{\left (x \right )}}{\log{\left (c \right )}^{2}} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\tilde{\infty } n \log{\left (x \right )} & \text{for}\: a = - b \left (n \log{\left (x \right )} + \log{\left (c \right )}\right ) \\\frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{\left (a + b \log{\left (c \right )}\right )^{2}} & \text{for}\: n = 0 \\- \frac{1}{a b n + b^{2} n^{2} \log{\left (x \right )} + b^{2} n \log{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18794, size = 28, normalized size = 1.4 \begin{align*} -\frac{1}{{\left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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