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 = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 2304
Rule 2305
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.45, size = 42, normalized size = 0.79 \[ \frac {{\left (n^{2} \log \relax (x)^{2} + \log \relax (c)^{2} + 2 \, {\left (n \log \relax (c) - n\right )} \log \relax (x) - 2 \, \log \relax (c) + 2\right )} d^{n - 1} x^{n}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 99, normalized size = 1.87 \[ \frac {d^{n} n x^{n} \log \relax (x)^{2}}{d} + \frac {\frac {1}{d}^{n} x^{n} {\left | d \right |}^{2 \, n} \log \relax (c)^{2}}{d n} + \frac {2 \, d^{n} x^{n} \log \relax (c) \log \relax (x)}{d} - \frac {2 \, d^{n} x^{n} \log \relax (x)}{d} - \frac {2 \, d^{n} x^{n} \log \relax (c)}{d n} + \frac {2 \, d^{n} x^{n}}{d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 750, normalized size = 14.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 53, normalized size = 1.00 \[ -\frac {2 \, d^{n - 1} x^{n} \log \left (c x^{n}\right )}{n} + \frac {2 \, d^{n - 1} x^{n}}{n} + \frac {\left (d x\right )^{n} \log \left (c x^{n}\right )^{2}}{d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\ln \left (c\,x^n\right )}^2\,{\left (d\,x\right )}^{n-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 43.86, size = 163, normalized size = 3.08 \[ \begin {cases} \tilde {\infty } x \log {\relax (c )}^{2} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\log {\relax (c )}^{2} \log {\relax (x )}}{d} & \text {for}\: n = 0 \\0^{n - 1} \left (n^{2} x \log {\relax (x )}^{2} - 2 n^{2} x \log {\relax (x )} + 2 n^{2} x + 2 n x \log {\relax (c )} \log {\relax (x )} - 2 n x \log {\relax (c )} + x \log {\relax (c )}^{2}\right ) & \text {for}\: d = 0 \\\frac {d^{n} n x^{n} \log {\relax (x )}^{2}}{d} + \frac {2 d^{n} x^{n} \log {\relax (c )} \log {\relax (x )}}{d} - \frac {2 d^{n} x^{n} \log {\relax (x )}}{d} + \frac {d^{n} x^{n} \log {\relax (c )}^{2}}{d n} - \frac {2 d^{n} x^{n} \log {\relax (c )}}{d n} + \frac {2 d^{n} x^{n}}{d n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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