Optimal. Leaf size=193 \[ 2 a b n x-6 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )+2 b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 n^2 (1+e x) \log (1+e x)}{e}-\frac {2 b n (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{e} \]
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Rubi [A]
time = 0.24, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 12, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {2436, 2332,
2417, 2388, 2338, 6874, 2458, 45, 2393, 2352, 2421, 6724} \begin {gather*} \frac {2 b n \text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 b^2 n^2 \text {PolyLog}(2,-e x)}{e}-\frac {2 b^2 n^2 \text {PolyLog}(3,-e x)}{e}-\frac {2 b n (e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {(e x+1) \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{e}+2 b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2+2 a b n x+2 b^2 n x \log \left (c x^n\right )+\frac {2 b^2 n^2 (e x+1) \log (e x+1)}{e}-6 b^2 n^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2338
Rule 2352
Rule 2388
Rule 2393
Rule 2417
Rule 2421
Rule 2436
Rule 2458
Rule 6724
Rule 6874
Rubi steps
\begin {align*} \int \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x) \, dx &=-x \left (a+b \log \left (c x^n\right )\right )^2+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}-(2 b n) \int \left (-a-b \log \left (c x^n\right )+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e x}\right ) \, dx\\ &=2 a b n x-x \left (a+b \log \left (c x^n\right )\right )^2+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx-\frac {(2 b n) \int \frac {(1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{e}\\ &=2 a b n x-2 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )-x \left (a+b \log \left (c x^n\right )\right )^2+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}-\frac {(2 b n) \int \left (e \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)+\frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x}\right ) \, dx}{e}\\ &=2 a b n x-2 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )-x \left (a+b \log \left (c x^n\right )\right )^2+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}-(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx-\frac {(2 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx}{e}\\ &=2 a b n x-2 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )+2 b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e}+\left (2 b^2 n^2\right ) \int \left (-1+\frac {(1+e x) \log (1+e x)}{e x}\right ) \, dx-\frac {\left (2 b^2 n^2\right ) \int \frac {\text {Li}_2(-e x)}{x} \, dx}{e}\\ &=2 a b n x-4 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )+2 b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{e}+\frac {\left (2 b^2 n^2\right ) \int \frac {(1+e x) \log (1+e x)}{x} \, dx}{e}\\ &=2 a b n x-4 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )+2 b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{e}+\frac {\left (2 b^2 n^2\right ) \text {Subst}\left (\int \frac {x \log (x)}{-\frac {1}{e}+\frac {x}{e}} \, dx,x,1+e x\right )}{e^2}\\ &=2 a b n x-4 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )+2 b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{e}+\frac {\left (2 b^2 n^2\right ) \text {Subst}\left (\int \left (e \log (x)+\frac {e \log (x)}{-1+x}\right ) \, dx,x,1+e x\right )}{e^2}\\ &=2 a b n x-4 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )+2 b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2-\frac {2 b n (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{e}+\frac {\left (2 b^2 n^2\right ) \text {Subst}(\int \log (x) \, dx,x,1+e x)}{e}+\frac {\left (2 b^2 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{-1+x} \, dx,x,1+e x\right )}{e}\\ &=2 a b n x-6 b^2 n^2 x+2 b^2 n x \log \left (c x^n\right )+2 b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 b^2 n^2 (1+e x) \log (1+e x)}{e}-\frac {2 b n (1+e x) \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{e}+\frac {(1+e x) \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{e}-\frac {2 b^2 n^2 \text {Li}_2(-e x)}{e}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)}{e}-\frac {2 b^2 n^2 \text {Li}_3(-e x)}{e}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 294, normalized size = 1.52 \begin {gather*} \frac {-a^2 e x+4 a b e n x-6 b^2 e n^2 x-2 a b e x \log \left (c x^n\right )+4 b^2 e n x \log \left (c x^n\right )-b^2 e x \log ^2\left (c x^n\right )+a^2 \log (1+e x)-2 a b n \log (1+e x)+2 b^2 n^2 \log (1+e x)+a^2 e x \log (1+e x)-2 a b e n x \log (1+e x)+2 b^2 e n^2 x \log (1+e x)+2 a b \log \left (c x^n\right ) \log (1+e x)-2 b^2 n \log \left (c x^n\right ) \log (1+e x)+2 a b e x \log \left (c x^n\right ) \log (1+e x)-2 b^2 e n x \log \left (c x^n\right ) \log (1+e x)+b^2 \log ^2\left (c x^n\right ) \log (1+e x)+b^2 e x \log ^2\left (c x^n\right ) \log (1+e x)+2 b n \left (a-b n+b \log \left (c x^n\right )\right ) \text {Li}_2(-e x)-2 b^2 n^2 \text {Li}_3(-e x)}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (e x +1\right )\, dx\]
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 \left (a + b \log {\left (c x^{n} \right )}\right )^{2} \log {\left (e x + 1 \right )}\, 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 \ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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