Integrand size = 45, antiderivative size = 496 \[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}+\frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}+\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {2 m n^2 \operatorname {PolyLog}\left (4,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {2 m n^2 \operatorname {PolyLog}\left (4,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d} \]
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Time = 0.40 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2589, 2553, 2404, 2354, 2421, 2430, 6724} \[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\frac {\log \left (h (f+g x)^m\right ) \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 n (b c-a d)}-\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{3 n (b c-a d)}+\frac {m \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c-a d}-\frac {2 m n \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b c-a d}+\frac {m \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 n (b c-a d)}-\frac {2 m n^2 \operatorname {PolyLog}\left (4,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {2 m n^2 \operatorname {PolyLog}\left (4,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d} \]
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Rule 2354
Rule 2404
Rule 2421
Rule 2430
Rule 2553
Rule 2589
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {(g m) \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx}{3 (b c-a d) n} \\ & = \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {(g m) \text {Subst}\left (\int \frac {\log ^3\left (e x^n\right )}{(b-d x) (b f-a g-(d f-c g) x)} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 n} \\ & = \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {(g m) \text {Subst}\left (\int \left (\frac {d \log ^3\left (e x^n\right )}{(b c-a d) g (b-d x)}+\frac {(-d f+c g) \log ^3\left (e x^n\right )}{(b c-a d) g (b f-a g-(d f-c g) x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{3 n} \\ & = \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {(d m) \text {Subst}\left (\int \frac {\log ^3\left (e x^n\right )}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 (b c-a d) n}+\frac {((d f-c g) m) \text {Subst}\left (\int \frac {\log ^3\left (e x^n\right )}{b f-a g+(-d f+c g) x} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 (b c-a d) n} \\ & = \frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}+\frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}-\frac {m \text {Subst}\left (\int \frac {\log ^2\left (e x^n\right ) \log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b c-a d}+\frac {m \text {Subst}\left (\int \frac {\log ^2\left (e x^n\right ) \log \left (1+\frac {(-d f+c g) x}{b f-a g}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b c-a d} \\ & = \frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}+\frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}+\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {(2 m n) \text {Subst}\left (\int \frac {\log \left (e x^n\right ) \text {Li}_2\left (\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b c-a d}+\frac {(2 m n) \text {Subst}\left (\int \frac {\log \left (e x^n\right ) \text {Li}_2\left (-\frac {(-d f+c g) x}{b f-a g}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b c-a d} \\ & = \frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}+\frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}+\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {\left (2 m n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b c-a d}-\frac {\left (2 m n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {(-d f+c g) x}{b f-a g}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{b c-a d} \\ & = \frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 (b c-a d) n}+\frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{3 (b c-a d) n}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b c-a d) n}+\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac {2 m n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_3\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {2 m n^2 \text {Li}_4\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {2 m n^2 \text {Li}_4\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(25557\) vs. \(2(496)=992\).
Time = 7.15 (sec) , antiderivative size = 25557, normalized size of antiderivative = 51.53 \[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\text {Result too large to show} \]
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\[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} \ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right )}d x\]
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\[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {\log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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