Optimal. Leaf size=14 \[ a x+b \log ^2\left (c x^n\right ) \]
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Rubi [A] time = 0.0092727, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2301} \[ a x+b \log ^2\left (c x^n\right ) \]
Antiderivative was successfully verified.
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Rule 2301
Rubi steps
\begin{align*} \int \left (a+\frac{2 b n \log \left (c x^n\right )}{x}\right ) \, dx &=a x+(2 b n) \int \frac{\log \left (c x^n\right )}{x} \, dx\\ &=a x+b \log ^2\left (c x^n\right )\\ \end{align*}
Mathematica [A] time = 0.0018801, size = 14, normalized size = 1. \[ a x+b \log ^2\left (c x^n\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 15, normalized size = 1.1 \begin{align*} ax+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01534, size = 19, normalized size = 1.36 \begin{align*} b \log \left (c x^{n}\right )^{2} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88183, size = 59, normalized size = 4.21 \begin{align*} b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.67615, size = 60, normalized size = 4.29 \begin{align*} a x + 2 b n \left (\begin{cases} \frac{\log{\left (c x^{n} \right )}^{2}}{2 n} & \text{for}\: \left |{c x^{n}}\right | < 1 \\\frac{\log{\left (\frac{x^{- n}}{c} \right )}^{2}}{2 n} & \text{for}\: \frac{1}{\left |{c x^{n}}\right |} < 1 \\\frac{{G_{3, 3}^{3, 0}\left (\begin{matrix} & 1, 1, 1 \\0, 0, 0 & \end{matrix} \middle |{c x^{n}} \right )}}{n} + \frac{{G_{3, 3}^{0, 3}\left (\begin{matrix} 1, 1, 1 & \\ & 0, 0, 0 \end{matrix} \middle |{c x^{n}} \right )}}{n} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24135, size = 27, normalized size = 1.93 \begin{align*}{\left (n \log \left (x\right )^{2} + 2 \, \log \left (c\right ) \log \left (x\right )\right )} b n + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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