Optimal. Leaf size=26 \[ \frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \]
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Rubi [A]
time = 0.16, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {2626}
\begin {gather*} \frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2626
Rubi steps
\begin {align*} \int \frac {a d n x^m-a d m x^m \log \left (c x^n\right )-b d n (-1+q) \log ^q\left (c x^n\right )}{x \left (a x^m+b \log ^q\left (c x^n\right )\right )^2} \, dx &=\frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 26, normalized size = 1.00 \begin {gather*} \frac {d \log \left (c x^n\right )}{a x^m+b \log ^q\left (c x^n\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.09, size = 158, normalized size = 6.08
method | result | size |
risch | \(\frac {\left (2 \ln \left (c \right )+2 \ln \left (x^{n}\right )-i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}\right ) d}{2 a \,x^{m}+2 b \left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\mathrm {csgn}\left (i c \,x^{n}\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i c \right )\right ) \left (-\mathrm {csgn}\left (i c \,x^{n}\right )+\mathrm {csgn}\left (i x^{n}\right )\right )}{2}\right )^{q}}\) | \(158\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.45, size = 31, normalized size = 1.19 \begin {gather*} \frac {d \log \left (c\right ) + d \log \left (x^{n}\right )}{a x^{m} + b {\left (\log \left (c\right ) + \log \left (x^{n}\right )\right )}^{q}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 30, normalized size = 1.15 \begin {gather*} \frac {d n \log \left (x\right ) + d \log \left (c\right )}{{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 44.66, size = 22, normalized size = 0.85 \begin {gather*} \frac {d \log {\left (c x^{n} \right )}}{a x^{m} + b \log {\left (c x^{n} \right )}^{q}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 26, normalized size = 1.00 \begin {gather*} \frac {d\,\ln \left (c\,x^n\right )}{a\,x^m+b\,{\ln \left (c\,x^n\right )}^q} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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