Optimal. Leaf size=52 \[ -\frac {n^2 p^2}{4 x^2}-\frac {n p \log \left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{2 x^2} \]
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Rubi [A]
time = 0.05, 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 = {2342, 2341,
2495} \begin {gather*} -\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} \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^3} \, dx &=\text {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}+\text {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 = 0.00, size = 43, normalized size = 0.83 \begin {gather*} -\frac {n^2 p^2+2 n p \log \left (c \left (b x^n\right )^p\right )+2 \log ^2\left (c \left (b x^n\right )^p\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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 = 0.28, size = 46, normalized size = 0.88 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 87, normalized size = 1.67 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.49, size = 48, normalized size = 0.92 \begin {gather*} - \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 {\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (46) = 92\).
time = 5.07, size = 94, normalized size = 1.81 \begin {gather*} -\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 {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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