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 = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 2304
Rule 2305
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.62, size = 87, normalized size = 1.67 \[ -\frac {2 \, n^{2} p^{2} \log \relax (x)^{2} + n^{2} p^{2} + 2 \, n p^{2} \log \relax (b) + 2 \, p^{2} \log \relax (b)^{2} + 2 \, {\left (n p + 2 \, p \log \relax (b)\right )} \log \relax (c) + 2 \, \log \relax (c)^{2} + 2 \, {\left (n^{2} p^{2} + 2 \, n p^{2} \log \relax (b) + 2 \, n p \log \relax (c)\right )} \log \relax (x)}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 94, normalized size = 1.81 \[ -\frac {n^{2} p^{2} \log \relax (x)^{2}}{2 \, x^{2}} - \frac {{\left (n^{2} p^{2} + 2 \, n p^{2} \log \relax (b) + 2 \, n p \log \relax (c)\right )} \log \relax (x)}{2 \, x^{2}} - \frac {n^{2} p^{2} + 2 \, n p^{2} \log \relax (b) + 2 \, p^{2} \log \relax (b)^{2} + 2 \, n p \log \relax (c) + 4 \, p \log \relax (b) \log \relax (c) + 2 \, \log \relax (c)^{2}}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 46, normalized size = 0.88 \[ -\frac {n^{2} p^{2}}{4 \, x^{2}} - \frac {n p \log \left (\left (b x^{n}\right )^{p} c\right )}{2 \, x^{2}} - \frac {\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.79, size = 46, normalized size = 0.88 \[ -\frac {{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{2\,x^2}-\frac {n^2\,p^2}{4\,x^2}-\frac {n\,p\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.44, size = 134, normalized size = 2.58 \[ - \frac {n^{2} p^{2} \log {\relax (x )}^{2}}{2 x^{2}} - \frac {n^{2} p^{2} \log {\relax (x )}}{2 x^{2}} - \frac {n^{2} p^{2}}{4 x^{2}} - \frac {n p^{2} \log {\relax (b )} \log {\relax (x )}}{x^{2}} - \frac {n p^{2} \log {\relax (b )}}{2 x^{2}} - \frac {n p \log {\relax (c )} \log {\relax (x )}}{x^{2}} - \frac {n p \log {\relax (c )}}{2 x^{2}} - \frac {p^{2} \log {\relax (b )}^{2}}{2 x^{2}} - \frac {p \log {\relax (b )} \log {\relax (c )}}{x^{2}} - \frac {\log {\relax (c )}^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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