Optimal. Leaf size=74 \[ \frac{b n \text{PolyLog}(2,-e x)}{e}+\frac{(e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{e}-x \left (a+b \log \left (c x^n\right )\right )-\frac{b n (e x+1) \log (e x+1)}{e}+2 b n x \]
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Rubi [A] time = 0.0886038, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {2389, 2295, 2370, 2411, 43, 2351, 2315} \[ \frac{b n \text{PolyLog}(2,-e x)}{e}+\frac{(e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{e}-x \left (a+b \log \left (c x^n\right )\right )-\frac{b n (e x+1) \log (e x+1)}{e}+2 b n x \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2295
Rule 2370
Rule 2411
Rule 43
Rule 2351
Rule 2315
Rubi steps
\begin{align*} \int \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx &=-x \left (a+b \log \left (c x^n\right )\right )+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-(b n) \int \left (-1+\frac{(1+e x) \log (1+e x)}{e x}\right ) \, dx\\ &=b n x-x \left (a+b \log \left (c x^n\right )\right )+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{(b n) \int \frac{(1+e x) \log (1+e x)}{x} \, dx}{e}\\ &=b n x-x \left (a+b \log \left (c x^n\right )\right )+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{(b n) \operatorname{Subst}\left (\int \frac{x \log (x)}{-\frac{1}{e}+\frac{x}{e}} \, dx,x,1+e x\right )}{e^2}\\ &=b n x-x \left (a+b \log \left (c x^n\right )\right )+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{(b n) \operatorname{Subst}\left (\int \left (e \log (x)+\frac{e \log (x)}{-1+x}\right ) \, dx,x,1+e x\right )}{e^2}\\ &=b n x-x \left (a+b \log \left (c x^n\right )\right )+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}-\frac{(b n) \operatorname{Subst}(\int \log (x) \, dx,x,1+e x)}{e}-\frac{(b n) \operatorname{Subst}\left (\int \frac{\log (x)}{-1+x} \, dx,x,1+e x\right )}{e}\\ &=2 b n x-x \left (a+b \log \left (c x^n\right )\right )-\frac{b n (1+e x) \log (1+e x)}{e}+\frac{(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}+\frac{b n \text{Li}_2(-e x)}{e}\\ \end{align*}
Mathematica [A] time = 0.0311021, size = 90, normalized size = 1.22 \[ \frac{b n \text{PolyLog}(2,-e x)-a e x+a e x \log (e x+1)+a \log (e x+1)+b ((e x+1) \log (e x+1)-e x) \log \left (c x^n\right )+2 b e n x-b e n x \log (e x+1)-b n \log (e x+1)}{e} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.064, size = 557, normalized size = 7.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31946, size = 170, normalized size = 2.3 \begin{align*} \frac{{\left (\log \left (e x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-e x\right )\right )} b n}{e} - \frac{{\left (b{\left (n - \log \left (c\right )\right )} - a\right )} \log \left (e x + 1\right )}{e} + \frac{{\left ({\left (2 \, e n - e \log \left (c\right )\right )} b - a e\right )} x -{\left (b n \log \left (x\right ) +{\left ({\left (e n - e \log \left (c\right )\right )} b - a e\right )} x\right )} \log \left (e x + 1\right ) -{\left (b e x -{\left (b e x + b\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a \log \left (e x + 1\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (e x + 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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