Optimal. Leaf size=15 \[ \frac {\log ^q\left (c x^n\right )}{n q} \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 30}
\begin {gather*} \frac {\log ^q\left (c x^n\right )}{n q} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2339
Rubi steps
\begin {align*} \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx &=\frac {\text {Subst}\left (\int x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\log ^q\left (c x^n\right )}{n q}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log ^q\left (c x^n\right )}{n q} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 16, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {\ln \left (c \,x^{n}\right )^{q}}{n q}\) | \(16\) |
default | \(\frac {\ln \left (c \,x^{n}\right )^{q}}{n q}\) | \(16\) |
risch | \(\frac {\left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i x^{n}\right )\right )}{2}\right )^{q}}{n q}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (c x^{n}\right )^{q}}{n q} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 25, normalized size = 1.67 \begin {gather*} \frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )} {\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q - 1}}{n q} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (c x^{n} \right )}^{q - 1}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.35, size = 16, normalized size = 1.07 \begin {gather*} \frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q}}{n q} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 15, normalized size = 1.00 \begin {gather*} \frac {{\ln \left (c\,x^n\right )}^q}{n\,q} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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