Integrand size = 35, antiderivative size = 19 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+b c e x \log (F) \log (d x))}{x} \, dx=e F^{c (a+b x)} \log ^{1+n}(d x) \]
[Out]
Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2233} \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+b c e x \log (F) \log (d x))}{x} \, dx=e \log ^{n+1}(d x) F^{c (a+b x)} \]
[In]
[Out]
Rule 2233
Rubi steps \begin{align*} \text {integral}& = e F^{c (a+b x)} \log ^{1+n}(d x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+b c e x \log (F) \log (d x))}{x} \, dx=e F^{c (a+b x)} \log ^{1+n}(d x) \]
[In]
[Out]
Time = 15.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
method | result | size |
parallelrisch | \(\ln \left (d x \right ) \ln \left (d x \right )^{n} F^{c \left (b x +a \right )} e\) | \(22\) |
risch | \(\left (-\frac {i \pi e \,\operatorname {csgn}\left (i d \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i d x \right ) F^{c \left (b x +a \right )}}{2}+\frac {i \pi e \,\operatorname {csgn}\left (i d \right ) \operatorname {csgn}\left (i d x \right )^{2} F^{c \left (b x +a \right )}}{2}+\frac {i \pi e \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i d x \right )^{2} F^{c \left (b x +a \right )}}{2}-\frac {i \pi e \operatorname {csgn}\left (i d x \right )^{3} F^{c \left (b x +a \right )}}{2}+\ln \left (d \right ) e \,F^{c \left (b x +a \right )}+e \,F^{c \left (b x +a \right )} \ln \left (x \right )\right ) \left (\ln \left (d \right )+\ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i d x \right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i d \right )\right ) \left (-\operatorname {csgn}\left (i d x \right )+\operatorname {csgn}\left (i x \right )\right )}{2}\right )^{n}\) | \(180\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+b c e x \log (F) \log (d x))}{x} \, dx=F^{b c x + a c} e \log \left (d x\right )^{n} \log \left (d x\right ) \]
[In]
[Out]
\[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+b c e x \log (F) \log (d x))}{x} \, dx=e \left (\int \frac {F^{a c + b c x} \log {\left (d x \right )}^{n}}{x}\, dx + \int \frac {F^{a c + b c x} n \log {\left (d x \right )}^{n}}{x}\, dx + \int F^{a c + b c x} b c \log {\left (F \right )} \log {\left (d x \right )} \log {\left (d x \right )}^{n}\, dx\right ) \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+b c e x \log (F) \log (d x))}{x} \, dx={\left (F^{a c} e \log \left (d\right ) + F^{a c} e \log \left (x\right )\right )} e^{\left (b c x \log \left (F\right ) + n \log \left (\log \left (d\right ) + \log \left (x\right )\right )\right )} \]
[In]
[Out]
Exception generated. \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+b c e x \log (F) \log (d x))}{x} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {F^{c (a+b x)} \log ^n(d x) (e+e n+b c e x \log (F) \log (d x))}{x} \, dx=F^{a\,c+b\,c\,x}\,e\,{\ln \left (d\,x\right )}^{n+1} \]
[In]
[Out]