Optimal. Leaf size=52 \[ -\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac{2 n p \log \left (c \left (b x^n\right )^p\right )}{9 x^3}-\frac{2 n^2 p^2}{27 x^3} \]
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Rubi [A] time = 0.0702828, 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} \[ -\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac{2 n p \log \left (c \left (b x^n\right )^p\right )}{9 x^3}-\frac{2 n^2 p^2}{27 x^3} \]
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^4} \, dx &=\operatorname{Subst}\left (\int \frac{\log ^2\left (b^p c x^{n p}\right )}{x^4} \, 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 )}{3 x^3}+\operatorname{Subst}\left (\frac{1}{3} (2 n p) \int \frac{\log \left (b^p c x^{n p}\right )}{x^4} \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-\frac{2 n^2 p^2}{27 x^3}-\frac{2 n p \log \left (c \left (b x^n\right )^p\right )}{9 x^3}-\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0017769, size = 52, normalized size = 1. \[ -\frac{\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac{2 n p \log \left (c \left (b x^n\right )^p\right )}{9 x^3}-\frac{2 n^2 p^2}{27 x^3} \]
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}^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10404, size = 62, normalized size = 1.19 \begin{align*} -\frac{2 \, n^{2} p^{2}}{27 \, x^{3}} - \frac{2 \, n p \log \left (\left (b x^{n}\right )^{p} c\right )}{9 \, x^{3}} - \frac{\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.773725, size = 235, normalized size = 4.52 \begin{align*} -\frac{9 \, n^{2} p^{2} \log \left (x\right )^{2} + 2 \, n^{2} p^{2} + 6 \, n p^{2} \log \left (b\right ) + 9 \, p^{2} \log \left (b\right )^{2} + 6 \,{\left (n p + 3 \, p \log \left (b\right )\right )} \log \left (c\right ) + 9 \, \log \left (c\right )^{2} + 6 \,{\left (n^{2} p^{2} + 3 \, n p^{2} \log \left (b\right ) + 3 \, n p \log \left (c\right )\right )} \log \left (x\right )}{27 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.73749, size = 151, normalized size = 2.9 \begin{align*} - \frac{n^{2} p^{2} \log{\left (x \right )}^{2}}{3 x^{3}} - \frac{2 n^{2} p^{2} \log{\left (x \right )}}{9 x^{3}} - \frac{2 n^{2} p^{2}}{27 x^{3}} - \frac{2 n p^{2} \log{\left (b \right )} \log{\left (x \right )}}{3 x^{3}} - \frac{2 n p^{2} \log{\left (b \right )}}{9 x^{3}} - \frac{2 n p \log{\left (c \right )} \log{\left (x \right )}}{3 x^{3}} - \frac{2 n p \log{\left (c \right )}}{9 x^{3}} - \frac{p^{2} \log{\left (b \right )}^{2}}{3 x^{3}} - \frac{2 p \log{\left (b \right )} \log{\left (c \right )}}{3 x^{3}} - \frac{\log{\left (c \right )}^{2}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31286, size = 128, normalized size = 2.46 \begin{align*} -\frac{n^{2} p^{2} \log \left (x\right )^{2}}{3 \, x^{3}} - \frac{2 \,{\left (n^{2} p^{2} + 3 \, n p^{2} \log \left (b\right ) + 3 \, n p \log \left (c\right )\right )} \log \left (x\right )}{9 \, x^{3}} - \frac{2 \, n^{2} p^{2} + 6 \, n p^{2} \log \left (b\right ) + 9 \, p^{2} \log \left (b\right )^{2} + 6 \, n p \log \left (c\right ) + 18 \, p \log \left (b\right ) \log \left (c\right ) + 9 \, \log \left (c\right )^{2}}{27 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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