Optimal. Leaf size=27 \[ -\frac {n p}{4 x^2}-\frac {\log \left (c \left (b x^n\right )^p\right )}{2 x^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2341, 2495}
\begin {gather*} -\frac {\log \left (c \left (b x^n\right )^p\right )}{2 x^2}-\frac {n p}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2495
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (b x^n\right )^p\right )}{x^3} \, dx &=\text {Subst}\left (\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 p}{4 x^2}-\frac {\log \left (c \left (b x^n\right )^p\right )}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {n p}{4 x^2}-\frac {\log \left (c \left (b x^n\right )^p\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (b \,x^{n}\right )^{p}\right )}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 23, normalized size = 0.85 \begin {gather*} -\frac {n p}{4 \, x^{2}} - \frac {\log \left (\left (b x^{n}\right )^{p} c\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2 \, n p \log \left (x\right ) + n p + 2 \, p \log \left (b\right ) + 2 \, \log \left (c\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 = 24, normalized size = 0.89 \begin {gather*} - \frac {n p}{4 x^{2}} - \frac {\log {\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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Giac [A]
time = 5.65, size = 28, normalized size = 1.04 \begin {gather*} -\frac {n p \log \left (x\right )}{2 \, x^{2}} - \frac {n p + 2 \, p \log \left (b\right ) + 2 \, \log \left (c\right )}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.84, size = 23, normalized size = 0.85 \begin {gather*} -\frac {\ln \left (c\,{\left (b\,x^n\right )}^p\right )}{2\,x^2}-\frac {n\,p}{4\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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