Optimal. Leaf size=52 \[ -\frac{b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{b^2 n^2}{4 x^2} \]
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Rubi [A] time = 0.0352318, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2305, 2304} \[ -\frac{b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{b^2 n^2}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+(b n) \int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac{b^2 n^2}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0129237, size = 41, normalized size = 0.79 \[ -\frac{2 \left (a+b \log \left (c x^n\right )\right )^2+b n \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.122, size = 703, normalized size = 13.5 \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 [A] time = 1.15863, size = 96, normalized size = 1.85 \begin{align*} -\frac{1}{4} \, b^{2}{\left (\frac{n^{2}}{x^{2}} + \frac{2 \, n \log \left (c x^{n}\right )}{x^{2}}\right )} - \frac{b^{2} \log \left (c x^{n}\right )^{2}}{2 \, x^{2}} - \frac{a b n}{2 \, x^{2}} - \frac{a b \log \left (c x^{n}\right )}{x^{2}} - \frac{a^{2}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.863724, size = 204, normalized size = 3.92 \begin{align*} -\frac{2 \, b^{2} n^{2} \log \left (x\right )^{2} + b^{2} n^{2} + 2 \, b^{2} \log \left (c\right )^{2} + 2 \, a b n + 2 \, a^{2} + 2 \,{\left (b^{2} n + 2 \, a b\right )} \log \left (c\right ) + 2 \,{\left (b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, a b n\right )} \log \left (x\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.74155, size = 128, normalized size = 2.46 \begin{align*} - \frac{a^{2}}{2 x^{2}} - \frac{a b n \log{\left (x \right )}}{x^{2}} - \frac{a b n}{2 x^{2}} - \frac{a b \log{\left (c \right )}}{x^{2}} - \frac{b^{2} n^{2} \log{\left (x \right )}^{2}}{2 x^{2}} - \frac{b^{2} n^{2} \log{\left (x \right )}}{2 x^{2}} - \frac{b^{2} n^{2}}{4 x^{2}} - \frac{b^{2} n \log{\left (c \right )} \log{\left (x \right )}}{x^{2}} - \frac{b^{2} n \log{\left (c \right )}}{2 x^{2}} - \frac{b^{2} \log{\left (c \right )}^{2}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23047, size = 122, normalized size = 2.35 \begin{align*} -\frac{b^{2} n^{2} \log \left (x\right )^{2}}{2 \, x^{2}} - \frac{{\left (b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, a b n\right )} \log \left (x\right )}{2 \, x^{2}} - \frac{b^{2} n^{2} + 2 \, b^{2} n \log \left (c\right ) + 2 \, b^{2} \log \left (c\right )^{2} + 2 \, a b n + 4 \, a b \log \left (c\right ) + 2 \, a^{2}}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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