Integrand size = 38, antiderivative size = 22 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-2+b c x \log (F)) \log (d x))}{x^3} \, dx=\frac {e F^{c (a+b x)} \log ^{1+n}(d x)}{x^2} \]
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Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2233} \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-2+b c x \log (F)) \log (d x))}{x^3} \, dx=\frac {e \log ^{n+1}(d x) F^{c (a+b x)}}{x^2} \]
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Rule 2233
Rubi steps \begin{align*} \text {integral}& = \frac {e F^{c (a+b x)} \log ^{1+n}(d x)}{x^2} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-2+b c x \log (F)) \log (d x))}{x^3} \, dx=\frac {e F^{a c+b c x} \log ^{1+n}(d x)}{x^2} \]
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Time = 15.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(\frac {\ln \left (d x \right ) \ln \left (d x \right )^{n} F^{c \left (b x +a \right )} e}{x^{2}}\) | \(25\) |
risch | \(\frac {F^{c \left (b x +a \right )} e \left (-i \pi \,\operatorname {csgn}\left (i d \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i d x \right )+i \pi \,\operatorname {csgn}\left (i d \right ) \operatorname {csgn}\left (i d x \right )^{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i d x \right )^{2}-i \pi \operatorname {csgn}\left (i d x \right )^{3}+2 \ln \left (d \right )+2 \ln \left (x \right )\right ) \left (\ln \left (d \right )+\ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i d x \right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i d \right )\right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i x \right )\right )}{2}\right )^{n}}{2 x^{2}}\) | \(136\) |
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-2+b c x \log (F)) \log (d x))}{x^3} \, dx=\frac {F^{b c x + a c} e \log \left (d x\right )^{n} \log \left (d x\right )}{x^{2}} \]
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\[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-2+b c x \log (F)) \log (d x))}{x^3} \, dx=e \left (\int \frac {F^{a c + b c x} \log {\left (d x \right )}^{n}}{x^{3}}\, dx + \int \frac {F^{a c + b c x} n \log {\left (d x \right )}^{n}}{x^{3}}\, dx + \int \left (- \frac {2 F^{a c + b c x} \log {\left (d x \right )} \log {\left (d x \right )}^{n}}{x^{3}}\right )\, dx + \int \frac {F^{a c + b c x} b c \log {\left (F \right )} \log {\left (d x \right )} \log {\left (d x \right )}^{n}}{x^{2}}\, dx\right ) \]
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-2+b c x \log (F)) \log (d x))}{x^3} \, dx=\frac {{\left (F^{a c} e \log \left (d\right ) + F^{a c} e \log \left (x\right )\right )} e^{\left (b c x \log \left (F\right ) + n \log \left (\log \left (d\right ) + \log \left (x\right )\right )\right )}}{x^{2}} \]
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\[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-2+b c x \log (F)) \log (d x))}{x^3} \, dx=\int { \frac {{\left ({\left (b c x \log \left (F\right ) - 2\right )} e \log \left (d x\right ) + e n + e\right )} F^{{\left (b x + a\right )} c} \log \left (d x\right )^{n}}{x^{3}} \,d x } \]
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Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+e (-2+b c x \log (F)) \log (d x))}{x^3} \, dx=\frac {F^{a\,c+b\,c\,x}\,e\,{\ln \left (d\,x\right )}^{n+1}}{x^2} \]
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