Integrand size = 23, antiderivative size = 75 \[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}+\frac {2 p \log \left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{e m^2}-\frac {2 p^2 \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{e m^3} \]
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Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2375, 2421, 6724} \[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\frac {2 p \log \left (f x^p\right ) \operatorname {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{e m^2}+\frac {\log ^2\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{e m}-\frac {2 p^2 \operatorname {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{e m^3} \]
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Rule 2375
Rule 2421
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}-\frac {(2 p) \int \frac {\log \left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{x} \, dx}{e m} \\ & = \frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}+\frac {2 p \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{e m^2}-\frac {\left (2 p^2\right ) \int \frac {\text {Li}_2\left (-\frac {e x^m}{d}\right )}{x} \, dx}{e m^2} \\ & = \frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{e m}+\frac {2 p \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{e m^2}-\frac {2 p^2 \text {Li}_3\left (-\frac {e x^m}{d}\right )}{e m^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(220\) vs. \(2(75)=150\).
Time = 0.18 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.93 \[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\frac {p^2 \log ^3(x)+3 p \log ^2(x) \left (-p \log (x)+\log \left (f x^p\right )\right )+3 \log (x) \left (-p \log (x)+\log \left (f x^p\right )\right )^2-\frac {3 \left (-p \log (x)+\log \left (f x^p\right )\right )^2 \left (\log \left (x^m\right )-\log \left (d m \left (d+e x^m\right )\right )\right )}{m}-\frac {6 p \left (-p \log (x)+\log \left (f x^p\right )\right ) \left (\frac {1}{2} m^2 \log ^2(x)+\left (-m \log (x)+\log \left (-\frac {e x^m}{d}\right )\right ) \log \left (d+e x^m\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^m}{d}\right )\right )}{m^2}+\frac {3 p^2 \left (m^2 \log ^2(x) \log \left (1+\frac {d x^{-m}}{e}\right )-2 m \log (x) \operatorname {PolyLog}\left (2,-\frac {d x^{-m}}{e}\right )-2 \operatorname {PolyLog}\left (3,-\frac {d x^{-m}}{e}\right )\right )}{m^3}}{3 e} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.42 (sec) , antiderivative size = 496, normalized size of antiderivative = 6.61
method | result | size |
risch | \(\frac {\ln \left (d +e \,x^{m}\right ) \ln \left (x \right )^{2} p^{2}}{m e}-\frac {2 \ln \left (d +e \,x^{m}\right ) \ln \left (x \right ) \ln \left (x^{p}\right ) p}{m e}+\frac {\ln \left (d +e \,x^{m}\right ) \ln \left (x^{p}\right )^{2}}{m e}+\frac {p^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e \,x^{m}}{d}\right )}{m e}+\frac {2 p^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e \,x^{m}}{d}\right )}{m^{2} e}-\frac {2 p^{2} \operatorname {Li}_{3}\left (-\frac {e \,x^{m}}{d}\right )}{e \,m^{3}}-\frac {2 p^{2} \operatorname {dilog}\left (\frac {d +e \,x^{m}}{d}\right ) \ln \left (x \right )}{m^{2} e}+\frac {2 p \operatorname {dilog}\left (\frac {d +e \,x^{m}}{d}\right ) \ln \left (x^{p}\right )}{m^{2} e}-\frac {2 p^{2} \ln \left (x \right )^{2} \ln \left (\frac {d +e \,x^{m}}{d}\right )}{m e}+\frac {2 p \ln \left (x \right ) \ln \left (\frac {d +e \,x^{m}}{d}\right ) \ln \left (x^{p}\right )}{m e}+\left (-i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right )+i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{p}\right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right )^{2}-i \pi \operatorname {csgn}\left (i f \,x^{p}\right )^{3}+2 \ln \left (f \right )\right ) \left (\frac {\left (\ln \left (x^{p}\right )-p \ln \left (x \right )\right ) \ln \left (d +e \,x^{m}\right )}{m e}+\frac {p \operatorname {dilog}\left (\frac {d +e \,x^{m}}{d}\right )}{m^{2} e}+\frac {p \ln \left (x \right ) \ln \left (\frac {d +e \,x^{m}}{d}\right )}{m e}\right )+\frac {{\left (-i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right )+i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{p}\right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{p}\right ) \operatorname {csgn}\left (i f \,x^{p}\right )^{2}-i \pi \operatorname {csgn}\left (i f \,x^{p}\right )^{3}+2 \ln \left (f \right )\right )}^{2} \ln \left (d +e \,x^{m}\right )}{4 m e}\) | \(496\) |
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Time = 0.34 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.40 \[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\frac {m^{2} \log \left (e x^{m} + d\right ) \log \left (f\right )^{2} - 2 \, p^{2} {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right ) + 2 \, {\left (m p^{2} \log \left (x\right ) + m p \log \left (f\right )\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (m^{2} p^{2} \log \left (x\right )^{2} + 2 \, m^{2} p \log \left (f\right ) \log \left (x\right )\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{e m^{3}} \]
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\[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\int \frac {x^{m - 1} \log {\left (f x^{p} \right )}^{2}}{d + e x^{m}}\, dx \]
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\[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\int { \frac {x^{m - 1} \log \left (f x^{p}\right )^{2}}{e x^{m} + d} \,d x } \]
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\[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\int { \frac {x^{m - 1} \log \left (f x^{p}\right )^{2}}{e x^{m} + d} \,d x } \]
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Timed out. \[ \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx=\int \frac {x^{m-1}\,{\ln \left (f\,x^p\right )}^2}{d+e\,x^m} \,d x \]
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