Optimal. Leaf size=18 \[ x \log \left (c \left (b x^n\right )^p\right )-n p x \]
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Rubi [A] time = 0.0070388, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2295, 2445} \[ x \log \left (c \left (b x^n\right )^p\right )-n p x \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2445
Rubi steps
\begin{align*} \int \log \left (c \left (b x^n\right )^p\right ) \, dx &=\operatorname{Subst}\left (\int \log \left (b^p c x^{n p}\right ) \, dx,b^p c x^{n p},c \left (b x^n\right )^p\right )\\ &=-n p x+x \log \left (c \left (b x^n\right )^p\right )\\ \end{align*}
Mathematica [A] time = 0.0007363, size = 18, normalized size = 1. \[ x \log \left (c \left (b x^n\right )^p\right )-n p x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 19, normalized size = 1.1 \begin{align*} -npx+x\ln \left ( c \left ( b{x}^{n} \right ) ^{p} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14816, size = 24, normalized size = 1.33 \begin{align*} -n p x + x \log \left (\left (b x^{n}\right )^{p} c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.871845, size = 62, normalized size = 3.44 \begin{align*} n p x \log \left (x\right ) - n p x + p x \log \left (b\right ) + x \log \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.430165, size = 24, normalized size = 1.33 \begin{align*} n p x \log{\left (x \right )} - n p x + p x \log{\left (b \right )} + x \log{\left (c \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27637, size = 28, normalized size = 1.56 \begin{align*} n p x \log \left (x\right ) - n p x + p x \log \left (b\right ) + x \log \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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