Optimal. Leaf size=18 \[ -\frac{1}{a x+b \log ^2\left (c x^n\right )} \]
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Rubi [A] time = 0.133133, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {2561, 2544} \[ -\frac{1}{a x+b \log ^2\left (c x^n\right )} \]
Antiderivative was successfully verified.
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Rule 2561
Rule 2544
Rubi steps
\begin{align*} \int \frac{a x^2+2 b n x \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, dx &=\int \frac{x \left (a x+2 b n \log \left (c x^n\right )\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^2} \, 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 )^2} \, dx\\ &=-\frac{1}{a x+b \log ^2\left (c x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.0133318, size = 18, normalized size = 1. \[ -\frac{1}{a x+b \log ^2\left (c x^n\right )} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.072, size = 451, normalized size = 25.1 \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 [A] time = 1.21717, size = 42, normalized size = 2.33 \begin{align*} -\frac{1}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86091, size = 84, normalized size = 4.67 \begin{align*} -\frac{1}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.45101, size = 42, normalized size = 2.33 \begin{align*} -\frac{1}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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