Optimal. Leaf size=71 \[ \frac {b n \log (x)}{e m}-\frac {b n \left (d+e \log \left (f x^m\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m} \]
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Rubi [A]
time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2339, 29, 2413,
12, 2436, 2332} \begin {gather*} \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-\frac {b n \left (d+e \log \left (f x^m\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2}+\frac {b n \log (x)}{e m} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 29
Rule 2332
Rule 2339
Rule 2413
Rule 2436
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e \log \left (f x^m\right )\right )} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-(b n) \int \frac {\log \left (d+e \log \left (f x^m\right )\right )}{e m x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-\frac {(b n) \int \frac {\log \left (d+e \log \left (f x^m\right )\right )}{x} \, dx}{e m}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-\frac {(b n) \text {Subst}\left (\int \log (d+e x) \, dx,x,\log \left (f x^m\right )\right )}{e m^2}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}-\frac {(b n) \text {Subst}\left (\int \log (x) \, dx,x,d+e \log \left (f x^m\right )\right )}{e^2 m^2}\\ &=\frac {b n \log (x)}{e m}-\frac {b n \left (d+e \log \left (f x^m\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2}+\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e m}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 58, normalized size = 0.82 \begin {gather*} \frac {b e m n \log (x)+\left (a e m-b d n-b e n \log \left (f x^m\right )+b e m \log \left (c x^n\right )\right ) \log \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2} \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 = 0.15, size = 1744, normalized size = 24.56
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1744\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 120, normalized size = 1.69 \begin {gather*} \frac {b e^{\left (-1\right )} \log \left (c x^{n}\right ) \log \left ({\left (e \log \left (f\right ) + e \log \left (x^{m}\right ) + d\right )} e^{\left (-1\right )}\right )}{m} - \frac {{\left ({\left (e \log \left (f\right ) + e \log \left (x^{m}\right ) + d\right )} e^{\left (-1\right )} \log \left ({\left (e \log \left (f\right ) + e \log \left (x^{m}\right ) + d\right )} e^{\left (-1\right )}\right ) - {\left (e \log \left (f\right ) + e \log \left (x^{m}\right ) + d\right )} e^{\left (-1\right )}\right )} b n e^{\left (-1\right )}}{m^{2}} + \frac {a e^{\left (-1\right )} \log \left ({\left (e \log \left (f\right ) + e \log \left (x^{m}\right ) + d\right )} e^{\left (-1\right )}\right )}{m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 56, normalized size = 0.79 \begin {gather*} \frac {{\left (b m n e \log \left (x\right ) + {\left (b m e \log \left (c\right ) - b n e \log \left (f\right ) - b d n + a m e\right )} \log \left (m e \log \left (x\right ) + e \log \left (f\right ) + d\right )\right )} e^{\left (-2\right )}}{m^{2}} \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 {a + b \log {\left (c x^{n} \right )}}{x \left (d + e \log {\left (f x^{m} \right )}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.49, size = 85, normalized size = 1.20 \begin {gather*} \frac {b n e^{\left (-1\right )} \log \left (x\right )}{m} + \frac {{\left (b m e \log \left (c\right ) - b n e \log \left (f\right ) - b d n + a m e\right )} e^{\left (-2\right )} \log \left (\frac {1}{4} \, {\left (\pi m {\left (\mathrm {sgn}\left (x\right ) - 1\right )} e + \pi {\left (\mathrm {sgn}\left (f\right ) - 1\right )} e\right )}^{2} + {\left (m e \log \left ({\left | x \right |}\right ) + e \log \left ({\left | f \right |}\right ) + d\right )}^{2}\right )}{2 \, m^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (d+e\,\ln \left (f\,x^m\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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