Optimal. Leaf size=52 \[ -\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac{n p \log \left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac{n^2 p^2}{4 x^2} \]
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Rubi [A] time = 0.0696167, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2305, 2304, 2445} \[ -\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac{n p \log \left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac{n^2 p^2}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rule 2445
Rubi steps
\begin{align*} \int \frac{\log ^2\left (c \left (b x^n\right )^p\right )}{x^3} \, dx &=\operatorname{Subst}\left (\int \frac{\log ^2\left (b^p c x^{n p}\right )}{x^3} \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{2 x^2}+\operatorname{Subst}\left ((n p) \int \frac{\log \left (b^p c x^{n p}\right )}{x^3} \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-\frac{n^2 p^2}{4 x^2}-\frac{n p \log \left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0043444, size = 43, normalized size = 0.83 \[ -\frac{2 \log ^2\left (c \left (b x^n\right )^p\right )+2 n p \log \left (c \left (b x^n\right )^p\right )+n^2 p^2}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15784, size = 62, normalized size = 1.19 \begin{align*} -\frac{n^{2} p^{2}}{4 \, x^{2}} - \frac{n p \log \left (\left (b x^{n}\right )^{p} c\right )}{2 \, x^{2}} - \frac{\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.836053, size = 231, normalized size = 4.44 \begin{align*} -\frac{2 \, n^{2} p^{2} \log \left (x\right )^{2} + n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + 2 \, p^{2} \log \left (b\right )^{2} + 2 \,{\left (n p + 2 \, p \log \left (b\right )\right )} \log \left (c\right ) + 2 \, \log \left (c\right )^{2} + 2 \,{\left (n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + 2 \, n p \log \left (c\right )\right )} \log \left (x\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.1379, size = 134, normalized size = 2.58 \begin{align*} - \frac{n^{2} p^{2} \log{\left (x \right )}^{2}}{2 x^{2}} - \frac{n^{2} p^{2} \log{\left (x \right )}}{2 x^{2}} - \frac{n^{2} p^{2}}{4 x^{2}} - \frac{n p^{2} \log{\left (b \right )} \log{\left (x \right )}}{x^{2}} - \frac{n p^{2} \log{\left (b \right )}}{2 x^{2}} - \frac{n p \log{\left (c \right )} \log{\left (x \right )}}{x^{2}} - \frac{n p \log{\left (c \right )}}{2 x^{2}} - \frac{p^{2} \log{\left (b \right )}^{2}}{2 x^{2}} - \frac{p \log{\left (b \right )} \log{\left (c \right )}}{x^{2}} - \frac{\log{\left (c \right )}^{2}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32222, size = 127, normalized size = 2.44 \begin{align*} -\frac{n^{2} p^{2} \log \left (x\right )^{2}}{2 \, x^{2}} - \frac{{\left (n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + 2 \, n p \log \left (c\right )\right )} \log \left (x\right )}{2 \, x^{2}} - \frac{n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + 2 \, p^{2} \log \left (b\right )^{2} + 2 \, n p \log \left (c\right ) + 4 \, p \log \left (b\right ) \log \left (c\right ) + 2 \, \log \left (c\right )^{2}}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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