Optimal. Leaf size=52 \[ -\frac {2 n^2 p^2}{27 x^3}-\frac {2 n p \log \left (c \left (b x^n\right )^p\right )}{9 x^3}-\frac {\log ^2\left (c \left (b x^n\right )^p\right )}{3 x^3} \]
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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 )}{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} \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^4} \, dx &=\text {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}+\text {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*}
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Mathematica [A]
time = 0.00, size = 52, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )^{2}}{x^{4}}\, dx\]
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 {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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 88, normalized size = 1.69 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.89, size = 51, normalized size = 0.98 \begin {gather*} - \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 {\left (c \left (b x^{n}\right )^{p} \right )}^{2}}{3 x^{3}} \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. 95 vs.
\(2 (46) = 92\).
time = 4.67, size = 95, normalized size = 1.83 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.88, size = 46, normalized size = 0.88 \begin {gather*} -\frac {{\ln \left (c\,{\left (b\,x^n\right )}^p\right )}^2}{3\,x^3}-\frac {2\,n^2\,p^2}{27\,x^3}-\frac {2\,n\,p\,\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{9\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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