Optimal. Leaf size=158 \[ \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {\log (x) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+\frac {\log \left (-\frac {e x}{d}\right ) \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right )^2}{4 b g}+n \left (b f+a g+2 b g \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )-2 b g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right ) \]
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Rubi [A]
time = 0.16, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2482, 2481,
2422, 2354, 2421, 6724} \begin {gather*} n \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )-2 b g n^2 \text {PolyLog}\left (3,\frac {e x}{d}+1\right )-\frac {\log (x) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a g+2 b g \log \left (c (d+e x)^n\right )+b f\right )^2}{4 b g}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (g \log \left (c (d+e x)^n\right )+f\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2354
Rule 2421
Rule 2422
Rule 2481
Rule 2482
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )}{x} \, dx &=\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-(b e n) \int \frac {\log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx-(e g n) \int \frac {\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx\\ &=\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-(b n) \text {Subst}\left (\int \frac {\left (f+g \log \left (c x^n\right )\right ) \log \left (-\frac {d}{e}+\frac {x}{e}\right )}{x} \, dx,x,d+e x\right )-(g n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (-\frac {d}{e}+\frac {x}{e}\right )}{x} \, dx,x,d+e x\right )\\ &=-\frac {g \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {b \log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+\frac {b \text {Subst}\left (\int \frac {\left (f+g \log \left (c x^n\right )\right )^2}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{2 e g}+\frac {g \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{2 b e}\\ &=-\frac {g \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {b \log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+\frac {b \log \left (-\frac {e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}-(b n) \text {Subst}\left (\int \frac {\left (f+g \log \left (c x^n\right )\right ) \log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )-(g n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )\\ &=-\frac {g \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {b \log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+\frac {b \log \left (-\frac {e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )+b n \left (f+g \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )-2 \left (\left (b g n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )\right )\\ &=-\frac {g \log (x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 b}+\log (x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )-\frac {b \log (x) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+\frac {b \log \left (-\frac {e x}{d}\right ) \left (f+g \log \left (c (d+e x)^n\right )\right )^2}{2 g}+g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )+b n \left (f+g \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )-2 b g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 227, normalized size = 1.44 \begin {gather*} a f \log (x)+b f \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+a g \log \left (-\frac {e x}{d}\right ) \log \left (c (d+e x)^n\right )+b g \log (x) \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right )^2+2 b g n \left (-n \log (d+e x)+\log \left (c (d+e x)^n\right )\right ) \left (\log (x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\text {Li}_2\left (-\frac {e x}{d}\right )\right )+b f n \text {Li}_2\left (\frac {d+e x}{d}\right )+a g n \text {Li}_2\left (\frac {d+e x}{d}\right )+2 b g n^2 \left (\frac {1}{2} \log ^2(d+e x) \log \left (1-\frac {d+e x}{d}\right )+\log (d+e x) \text {Li}_2\left (\frac {d+e x}{d}\right )-\text {Li}_3\left (\frac {d+e x}{d}\right )\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
4.
time = 0.73, size = 1534, normalized size = 9.71
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1534\) |
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 {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \left (f + g \log {\left (c \left (d + e x\right )^{n} \right )}\right )}{x}\, 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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )\,\left (f+g\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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