Optimal. Leaf size=46 \[ -\frac {2 n^2 p^2}{x}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{x}-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{x} \]
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Rubi [A]
time = 0.05, antiderivative size = 46, 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 = {2342, 2341,
2495} \begin {gather*} -\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{x}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{x}-\frac {2 n^2 p^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2342
Rule 2495
Rubi steps
\begin {align*} \int \frac {\log ^2\left (c \left (b x^n\right )^p\right )}{x^2} \, dx &=\text {Subst}\left (\int \frac {\log ^2\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 {\log ^2\left (c \left (b x^n\right )^p\right )}{x}+\text {Subst}\left ((2 n p) \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 {2 n^2 p^2}{x}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{x}-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 40, normalized size = 0.87 \begin {gather*} -\frac {2 n^2 p^2+2 n p \log \left (c \left (b x^n\right )^p\right )+\log ^2\left (c \left (b x^n\right )^p\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 46, normalized size = 1.00 \begin {gather*} -\frac {2 \, n^{2} p^{2}}{x} - \frac {2 \, n p \log \left (\left (b x^{n}\right )^{p} c\right )}{x} - \frac {\log \left (\left (b x^{n}\right )^{p} c\right )^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 81, normalized size = 1.76 \begin {gather*} -\frac {n^{2} p^{2} \log \left (x\right )^{2} + 2 \, n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + p^{2} \log \left (b\right )^{2} + 2 \, {\left (n p + p \log \left (b\right )\right )} \log \left (c\right ) + \log \left (c\right )^{2} + 2 \, {\left (n^{2} p^{2} + n p^{2} \log \left (b\right ) + n p \log \left (c\right )\right )} \log \left (x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 41, normalized size = 0.89 \begin {gather*} - \frac {2 n^{2} p^{2}}{x} - \frac {2 n p \log {\left (c \left (b x^{n}\right )^{p} \right )}}{x} - \frac {\log {\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.84, size = 90, normalized size = 1.96 \begin {gather*} -\frac {n^{2} p^{2} \log \left (x\right )^{2}}{x} - \frac {2 \, {\left (n^{2} p^{2} + n p^{2} \log \left (b\right ) + n p \log \left (c\right )\right )} \log \left (x\right )}{x} - \frac {2 \, n^{2} p^{2} + 2 \, n p^{2} \log \left (b\right ) + p^{2} \log \left (b\right )^{2} + 2 \, n p \log \left (c\right ) + 2 \, p \log \left (b\right ) \log \left (c\right ) + \log \left (c\right )^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.85, size = 40, normalized size = 0.87 \begin {gather*} -\frac {2\,n^2\,p^2+2\,n\,p\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )+{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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