Optimal. Leaf size=32 \[ \frac {(d x)^n \log \left (c x^n\right )}{d n}-\frac {(d x)^n}{d n} \]
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Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2304} \[ \frac {(d x)^n \log \left (c x^n\right )}{d n}-\frac {(d x)^n}{d n} \]
Antiderivative was successfully verified.
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Rule 2304
Rubi steps
\begin {align*} \int (d x)^{-1+n} \log \left (c x^n\right ) \, dx &=-\frac {(d x)^n}{d n}+\frac {(d x)^n \log \left (c x^n\right )}{d n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 0.62 \[ \frac {(d x)^n \left (\log \left (c x^n\right )-1\right )}{d n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 20, normalized size = 0.62 \[ \frac {{\left (n \log \relax (x) + \log \relax (c) - 1\right )} d^{n - 1} x^{n}}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 50, normalized size = 1.56 \[ \frac {\frac {1}{d}^{n} x^{n} {\left | d \right |}^{2 \, n} \log \relax (c)}{d n} + \frac {d^{n} x^{n} \log \relax (x)}{d} - \frac {d^{n} x^{n}}{d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 263, normalized size = 8.22 \[ \frac {x \,{\mathrm e}^{\frac {\left (n -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right )}{2}} \ln \left (x^{n}\right )}{n}+\frac {\left (-i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \relax (c )-2\right ) x \,{\mathrm e}^{\frac {\left (n -1\right ) \left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right )}{2}}}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 32, normalized size = 1.00 \[ -\frac {d^{n - 1} x^{n}}{n} + \frac {\left (d x\right )^{n} \log \left (c x^{n}\right )}{d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \ln \left (c\,x^n\right )\,{\left (d\,x\right )}^{n-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.72, size = 68, normalized size = 2.12 \[ \begin {cases} \tilde {\infty } x \log {\relax (c )} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\log {\relax (c )} \log {\relax (x )}}{d} & \text {for}\: n = 0 \\0^{n - 1} \left (n x \log {\relax (x )} - n x + x \log {\relax (c )}\right ) & \text {for}\: d = 0 \\\frac {d^{n} x^{n} \log {\relax (x )}}{d} + \frac {d^{n} x^{n} \log {\relax (c )}}{d n} - \frac {d^{n} x^{n}}{d n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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