Optimal. Leaf size=53 \[ \frac{(d x)^n \log ^2\left (c x^n\right )}{d n}-\frac{2 (d x)^n \log \left (c x^n\right )}{d n}+\frac{2 (d x)^n}{d n} \]
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Rubi [A] time = 0.031621, antiderivative size = 53, 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} \[ \frac{(d x)^n \log ^2\left (c x^n\right )}{d n}-\frac{2 (d x)^n \log \left (c x^n\right )}{d n}+\frac{2 (d x)^n}{d n} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int (d x)^{-1+n} \log ^2\left (c x^n\right ) \, dx &=\frac{(d x)^n \log ^2\left (c x^n\right )}{d n}-2 \int (d x)^{-1+n} \log \left (c x^n\right ) \, dx\\ &=\frac{2 (d x)^n}{d n}-\frac{2 (d x)^n \log \left (c x^n\right )}{d n}+\frac{(d x)^n \log ^2\left (c x^n\right )}{d n}\\ \end{align*}
Mathematica [A] time = 0.0056123, size = 30, normalized size = 0.57 \[ \frac{(d x)^n \left (\log ^2\left (c x^n\right )-2 \log \left (c x^n\right )+2\right )}{d n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.121, size = 750, normalized size = 14.2 \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02732, size = 113, normalized size = 2.13 \begin{align*} \frac{{\left (n^{2} \log \left (x\right )^{2} + \log \left (c\right )^{2} + 2 \,{\left (n \log \left (c\right ) - n\right )} \log \left (x\right ) - 2 \, \log \left (c\right ) + 2\right )} d^{n - 1} x^{n}}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 113.492, size = 163, normalized size = 3.08 \begin{align*} \begin{cases} \tilde{\infty } x \log{\left (c \right )}^{2} & \text{for}\: d = 0 \wedge n = 0 \\\frac{\log{\left (c \right )}^{2} \log{\left (x \right )}}{d} & \text{for}\: n = 0 \\0^{n - 1} \left (n^{2} x \log{\left (x \right )}^{2} - 2 n^{2} x \log{\left (x \right )} + 2 n^{2} x + 2 n x \log{\left (c \right )} \log{\left (x \right )} - 2 n x \log{\left (c \right )} + x \log{\left (c \right )}^{2}\right ) & \text{for}\: d = 0 \\\frac{d^{n} n x^{n} \log{\left (x \right )}^{2}}{d} + \frac{2 d^{n} x^{n} \log{\left (c \right )} \log{\left (x \right )}}{d} - \frac{2 d^{n} x^{n} \log{\left (x \right )}}{d} + \frac{d^{n} x^{n} \log{\left (c \right )}^{2}}{d n} - \frac{2 d^{n} x^{n} \log{\left (c \right )}}{d n} + \frac{2 d^{n} x^{n}}{d n} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2535, size = 123, normalized size = 2.32 \begin{align*} \frac{d^{n} n x^{n} \log \left (x\right )^{2}}{d} + \frac{2 \, d^{n} x^{n} \log \left (c\right ) \log \left (x\right )}{d} + \frac{d^{n} x^{n} \log \left (c\right )^{2}}{d n} - \frac{2 \, d^{n} x^{n} \log \left (x\right )}{d} - \frac{2 \, d^{n} x^{n} \log \left (c\right )}{d n} + \frac{2 \, d^{n} x^{n}}{d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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