Integrand size = 31, antiderivative size = 79 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b}+\frac {B n \operatorname {PolyLog}\left (2,1+\frac {b c-a d}{d (a+b x)}\right )}{b} \]
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Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2542, 2458, 2378, 2370, 2352} \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=\frac {B n \operatorname {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{b}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{b} \]
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Rule 2352
Rule 2370
Rule 2378
Rule 2458
Rule 2542
Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b}+\frac {(B (b c-a d) n) \int \frac {\log \left (\frac {-b c+a d}{d (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{b} \\ & = -\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b}+\frac {(B (b c-a d) n) \text {Subst}\left (\int \frac {\log \left (\frac {-b c+a d}{d x}\right )}{x \left (\frac {b c-a d}{b}+\frac {d x}{b}\right )} \, dx,x,a+b x\right )}{b^2} \\ & = -\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b}-\frac {(B (b c-a d) n) \text {Subst}\left (\int \frac {\log \left (\frac {(-b c+a d) x}{d}\right )}{\left (\frac {b c-a d}{b}+\frac {d}{b x}\right ) x} \, dx,x,\frac {1}{a+b x}\right )}{b^2} \\ & = -\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b}-\frac {(B (b c-a d) n) \text {Subst}\left (\int \frac {\log \left (\frac {(-b c+a d) x}{d}\right )}{\frac {d}{b}+\frac {(b c-a d) x}{b}} \, dx,x,\frac {1}{a+b x}\right )}{b^2} \\ & = -\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b}+\frac {B n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.63 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=\frac {-B n \log ^2\left (\frac {-b c+a d}{d (a+b x)}\right )+2 A \log (a+b x)-2 B \log \left (\frac {-b c+a d}{d (a+b x)}\right ) \left (n \log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+2 B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{2 b} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.59 (sec) , antiderivative size = 523, normalized size of antiderivative = 6.62
method | result | size |
risch | \(-\frac {B \ln \left (b x +a \right ) \ln \left (\left (d x +c \right )^{n}\right )}{b}+\frac {B n \operatorname {dilog}\left (\frac {-a d +c b +d \left (b x +a \right )}{-a d +c b}\right )}{b}+\frac {B n \ln \left (b x +a \right ) \ln \left (\frac {-a d +c b +d \left (b x +a \right )}{-a d +c b}\right )}{b}+\frac {i B \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2} \ln \left (b x +a \right )}{2 b}+\frac {i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} \ln \left (b x +a \right )}{2 b}+\frac {i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} \ln \left (b x +a \right )}{2 b}+\frac {i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2} \ln \left (b x +a \right )}{2 b}+\frac {A \ln \left (b x +a \right )}{b}+\frac {B \ln \left (e \right ) \ln \left (b x +a \right )}{b}+\frac {B \ln \left (\left (b x +a \right )^{n}\right )^{2}}{2 b n}-\frac {i B \pi \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3} \ln \left (b x +a \right )}{2 b}-\frac {i B \pi \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{3} \ln \left (b x +a \right )}{2 b}-\frac {i B \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right ) \ln \left (b x +a \right )}{2 b}-\frac {i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \ln \left (b x +a \right )}{2 b}\) | \(523\) |
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\[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{b x + a} \,d x } \]
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\[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=\int \frac {A + B \log {\left (e \left (a + b x\right )^{n} \left (c + d x\right )^{- n} \right )}}{a + b x}\, dx \]
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\[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{b x + a} \,d x } \]
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\[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A}{b x + a} \,d x } \]
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Timed out. \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{a+b x} \, dx=\int \frac {A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{a+b\,x} \,d x \]
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