Optimal. Leaf size=823 \[ \frac {1}{2} m \log ^2(x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+\log (x) \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+2 b n \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \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 )+2 b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {1}{2} \log ^2(x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\log (x) \text {Li}_2\left (-\frac {e x}{d}\right )+\text {Li}_3\left (-\frac {e x}{d}\right )\right )-b^2 n^2 \left (m \log (x)-\log \left (f x^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)+2 \log (d+e x) \text {Li}_2\left (1+\frac {e x}{d}\right )-2 \text {Li}_3\left (1+\frac {e x}{d}\right )\right )+\frac {1}{12} b^2 m n^2 \left (\log ^4\left (-\frac {e x}{d}\right )+6 \log ^2\left (-\frac {e x}{d}\right ) \log ^2\left (-\frac {e x}{d+e x}\right )-4 \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {d}{d+e x}\right )\right ) \log ^3\left (-\frac {e x}{d+e x}\right )+\log ^4\left (-\frac {e x}{d+e x}\right )+6 \log ^2(x) \log ^2(d+e x)+4 \left (2 \log ^3\left (-\frac {e x}{d}\right )-3 \log ^2(x) \log (d+e x)\right ) \log \left (1+\frac {e x}{d}\right )+6 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (x)+3 \log \left (-\frac {e x}{d}\right )\right ) \log ^2\left (1+\frac {e x}{d}\right )-4 \log ^2\left (-\frac {e x}{d}\right ) \log \left (-\frac {e x}{d+e x}\right ) \left (\log \left (-\frac {e x}{d}\right )+3 \log \left (1+\frac {e x}{d}\right )\right )+12 \left (\log ^2\left (-\frac {e x}{d}\right )-2 \log \left (-\frac {e x}{d}\right ) \left (\log \left (-\frac {e x}{d+e x}\right )+\log \left (1+\frac {e x}{d}\right )\right )+2 \log (x) \left (-\log (d+e x)+\log \left (1+\frac {e x}{d}\right )\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )-12 \log ^2\left (-\frac {e x}{d+e x}\right ) \text {Li}_2\left (\frac {e x}{d+e x}\right )+12 \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {e x}{d+e x}\right )\right )^2 \text {Li}_2\left (1+\frac {e x}{d}\right )+24 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log \left (1+\frac {e x}{d}\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )+24 \left (\log \left (-\frac {e x}{d+e x}\right )+\log (d+e x)\right ) \text {Li}_3\left (-\frac {e x}{d}\right )+24 \log \left (-\frac {e x}{d+e x}\right ) \text {Li}_3\left (\frac {e x}{d+e x}\right )+24 \left (-\log (x)+\log \left (-\frac {e x}{d+e x}\right )\right ) \text {Li}_3\left (1+\frac {e x}{d}\right )-24 \left (\text {Li}_4\left (-\frac {e x}{d}\right )+\text {Li}_4\left (\frac {e x}{d+e x}\right )-\text {Li}_4\left (1+\frac {e x}{d}\right )\right )\right ) \]
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Rubi [F]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx &=\frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 m}-\frac {(b e n) \int \frac {\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{m}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 823, normalized size = 1.00 \begin {gather*} \frac {1}{2} m \log ^2(x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+\log (x) \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+2 b n \left (-m \log (x)+\log \left (f x^m\right )\right ) \left (a-b n \log (d+e x)+b \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 )+2 b m n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (\frac {1}{2} \log ^2(x) \left (\log (d+e x)-\log \left (1+\frac {e x}{d}\right )\right )-\log (x) \text {Li}_2\left (-\frac {e x}{d}\right )+\text {Li}_3\left (-\frac {e x}{d}\right )\right )-b^2 n^2 \left (m \log (x)-\log \left (f x^m\right )\right ) \left (\log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)+2 \log (d+e x) \text {Li}_2\left (1+\frac {e x}{d}\right )-2 \text {Li}_3\left (1+\frac {e x}{d}\right )\right )+\frac {1}{12} b^2 m n^2 \left (\log ^4\left (-\frac {e x}{d}\right )+6 \log ^2\left (-\frac {e x}{d}\right ) \log ^2\left (-\frac {e x}{d+e x}\right )-4 \left (\log \left (-\frac {e x}{d}\right )+\log \left (\frac {d}{d+e x}\right )\right ) \log ^3\left (-\frac {e x}{d+e x}\right )+\log ^4\left (-\frac {e x}{d+e x}\right )+6 \log ^2(x) \log ^2(d+e x)+4 \left (2 \log ^3\left (-\frac {e x}{d}\right )-3 \log ^2(x) \log (d+e x)\right ) \log \left (1+\frac {e x}{d}\right )+6 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (x)+3 \log \left (-\frac {e x}{d}\right )\right ) \log ^2\left (1+\frac {e x}{d}\right )-4 \log ^2\left (-\frac {e x}{d}\right ) \log \left (-\frac {e x}{d+e x}\right ) \left (\log \left (-\frac {e x}{d}\right )+3 \log \left (1+\frac {e x}{d}\right )\right )+12 \left (\log ^2\left (-\frac {e x}{d}\right )-2 \log \left (-\frac {e x}{d}\right ) \left (\log \left (-\frac {e x}{d+e x}\right )+\log \left (1+\frac {e x}{d}\right )\right )+2 \log (x) \left (-\log (d+e x)+\log \left (1+\frac {e x}{d}\right )\right )\right ) \text {Li}_2\left (-\frac {e x}{d}\right )-12 \log ^2\left (-\frac {e x}{d+e x}\right ) \text {Li}_2\left (\frac {e x}{d+e x}\right )+12 \left (\log \left (-\frac {e x}{d}\right )-\log \left (-\frac {e x}{d+e x}\right )\right )^2 \text {Li}_2\left (1+\frac {e x}{d}\right )+24 \left (\log (x)-\log \left (-\frac {e x}{d}\right )\right ) \log \left (1+\frac {e x}{d}\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )+24 \left (\log \left (-\frac {e x}{d+e x}\right )+\log (d+e x)\right ) \text {Li}_3\left (-\frac {e x}{d}\right )+24 \log \left (-\frac {e x}{d+e x}\right ) \text {Li}_3\left (\frac {e x}{d+e x}\right )+24 \left (-\log (x)+\log \left (-\frac {e x}{d+e x}\right )\right ) \text {Li}_3\left (1+\frac {e x}{d}\right )-24 \left (\text {Li}_4\left (-\frac {e x}{d}\right )+\text {Li}_4\left (\frac {e x}{d+e x}\right )-\text {Li}_4\left (1+\frac {e x}{d}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (f \,x^{m}\right ) \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}{x}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \frac {\ln \left (f\,x^m\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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