Optimal. Leaf size=22 \[ \frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
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Rubi [A] time = 0.0122896, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2301} \[ \frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
Antiderivative was successfully verified.
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Rule 2301
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n}\\ \end{align*}
Mathematica [A] time = 0.0010606, size = 21, normalized size = 0.95 \[ a \log (x)+\frac{b \log ^2\left (c x^n\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 27, normalized size = 1.2 \begin{align*}{\frac{ \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}b}{2\,n}}+{\frac{a\ln \left ( c{x}^{n} \right ) }{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11712, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{2 \, b n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.87814, size = 57, normalized size = 2.59 \begin{align*} \frac{1}{2} \, b n \log \left (x\right )^{2} +{\left (b \log \left (c\right ) + a\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.94959, size = 36, normalized size = 1.64 \begin{align*} - \begin{cases} - a \log{\left (x \right )} & \text{for}\: b = 0 \\- \left (a + b \log{\left (c \right )}\right ) \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{\left (a + b \log{\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13865, size = 26, normalized size = 1.18 \begin{align*} \frac{1}{2} \, b n \log \left (x\right )^{2} + b \log \left (c\right ) \log \left (x\right ) + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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