Optimal. Leaf size=23 \[ -\frac{\log \left (c \left (b x^n\right )^p\right )}{x}-\frac{n p}{x} \]
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Rubi [A] time = 0.0311087, antiderivative size = 23, 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 = {2304, 2445} \[ -\frac{\log \left (c \left (b x^n\right )^p\right )}{x}-\frac{n p}{x} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2445
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (b x^n\right )^p\right )}{x^2} \, dx &=\operatorname{Subst}\left (\int \frac{\log \left (b^p c x^{n p}\right )}{x^2} \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-\frac{n p}{x}-\frac{\log \left (c \left (b x^n\right )^p\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.001183, size = 23, normalized size = 1. \[ -\frac{\log \left (c \left (b x^n\right )^p\right )}{x}-\frac{n p}{x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17727, size = 31, normalized size = 1.35 \begin{align*} -\frac{n p}{x} - \frac{\log \left (\left (b x^{n}\right )^{p} c\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.889436, size = 58, normalized size = 2.52 \begin{align*} -\frac{n p \log \left (x\right ) + n p + p \log \left (b\right ) + \log \left (c\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.19603, size = 26, normalized size = 1.13 \begin{align*} - \frac{n p \log{\left (x \right )}}{x} - \frac{n p}{x} - \frac{p \log{\left (b \right )}}{x} - \frac{\log{\left (c \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32019, size = 34, normalized size = 1.48 \begin{align*} -\frac{n p \log \left (x\right )}{x} - \frac{n p + p \log \left (b\right ) + \log \left (c\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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