Optimal. Leaf size=15 \[ \log \left (a x+b \log ^2\left (c x^n\right )\right ) \]
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Rubi [A] time = 0.0820021, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2561, 2541} \[ \log \left (a x+b \log ^2\left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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Rule 2561
Rule 2541
Rubi steps
\begin{align*} \int \frac{a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx &=\int \frac{a x+2 b n \log \left (c x^n\right )}{x \left (a x+b \log ^2\left (c x^n\right )\right )} \, dx\\ &=\log \left (a x+b \log ^2\left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0953352, size = 15, normalized size = 1. \[ \log \left (a x+b \log ^2\left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.066, size = 428, normalized size = 28.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18479, size = 43, normalized size = 2.87 \begin{align*} \log \left (\frac{b \log \left (c\right )^{2} + 2 \, b \log \left (c\right ) \log \left (x^{n}\right ) + b \log \left (x^{n}\right )^{2} + a x}{b}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81171, size = 84, normalized size = 5.6 \begin{align*} \log \left (b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} + a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.39266, size = 48, normalized size = 3.2 \begin{align*} \begin{cases} \log{\left (x + \frac{b n^{2} \log{\left (x \right )}^{2}}{a} + \frac{2 b n \log{\left (c \right )} \log{\left (x \right )}}{a} + \frac{b \log{\left (c \right )}^{2}}{a} \right )} & \text{for}\: a \neq 0 \\2 \log{\left (n \log{\left (x \right )} + \log{\left (c \right )} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14868, size = 38, normalized size = 2.53 \begin{align*} \log \left (b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} + a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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