Optimal. Leaf size=27 \[ -\frac{\log \left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac{n p}{9 x^3} \]
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Rubi [A] time = 0.0296407, antiderivative size = 27, normalized size of antiderivative = 1., 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 = {2304, 2445} \[ -\frac{\log \left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac{n p}{9 x^3} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2445
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (b x^n\right )^p\right )}{x^4} \, dx &=\operatorname{Subst}\left (\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{n p}{9 x^3}-\frac{\log \left (c \left (b x^n\right )^p\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0012108, size = 27, normalized size = 1. \[ -\frac{\log \left (c \left (b x^n\right )^p\right )}{3 x^3}-\frac{n p}{9 x^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) }{{x}^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15947, size = 31, normalized size = 1.15 \begin{align*} -\frac{n p}{9 \, x^{3}} - \frac{\log \left (\left (b x^{n}\right )^{p} c\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.879031, size = 74, normalized size = 2.74 \begin{align*} -\frac{3 \, n p \log \left (x\right ) + n p + 3 \, p \log \left (b\right ) + 3 \, \log \left (c\right )}{9 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.92444, size = 39, normalized size = 1.44 \begin{align*} - \frac{n p \log{\left (x \right )}}{3 x^{3}} - \frac{n p}{9 x^{3}} - \frac{p \log{\left (b \right )}}{3 x^{3}} - \frac{\log{\left (c \right )}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31124, size = 38, normalized size = 1.41 \begin{align*} -\frac{n p \log \left (x\right )}{3 \, x^{3}} - \frac{n p + 3 \, p \log \left (b\right ) + 3 \, \log \left (c\right )}{9 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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