Optimal. Leaf size=39 \[ \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d}+\frac {b n \text {Li}_2\left (-\frac {d x}{e}\right )}{d} \]
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Rubi [A]
time = 0.05, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2370, 2354,
2438} \begin {gather*} \frac {b n \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{d}+\frac {\log \left (\frac {d x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2354
Rule 2370
Rule 2438
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{\left (d+\frac {e}{x}\right ) x} \, dx &=\int \frac {a+b \log \left (c x^n\right )}{e+d x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d}-\frac {(b n) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{d}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )}{d}+\frac {b n \text {Li}_2\left (-\frac {d x}{e}\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 0.95 \begin {gather*} \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x}{e}\right )+b n \text {Li}_2\left (-\frac {d x}{e}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.06, size = 195, normalized size = 5.00
method | result | size |
risch | \(\frac {b \ln \left (d x +e \right ) \ln \left (x^{n}\right )}{d}-\frac {b n \ln \left (d x +e \right ) \ln \left (-\frac {d x}{e}\right )}{d}-\frac {b n \dilog \left (-\frac {d x}{e}\right )}{d}-\frac {i \ln \left (d x +e \right ) b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 d}+\frac {i \ln \left (d x +e \right ) b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 d}+\frac {i \ln \left (d x +e \right ) b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 d}-\frac {i \ln \left (d x +e \right ) b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 d}+\frac {\ln \left (d x +e \right ) b \ln \left (c \right )}{d}+\frac {a \ln \left (d x +e \right )}{d}\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c x^{n} \right )}}{d x + e}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\left (d+\frac {e}{x}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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