Integrand size = 17, antiderivative size = 40 \[ \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {x}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2320, 272, 36, 29, 31} \[ \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {x}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 272
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^n\right )} \, dx,x,F^{g (e+f x)}\right )}{f g \log (F)} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{f g n \log (F)} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,\left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)} \\ & = \frac {x}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a f g n \log (F)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.38 \[ \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )-\log \left (a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)\right )}{a f g n \log (F)} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.10
method | result | size |
norman | \(\frac {x}{a}-\frac {\ln \left (a +b \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}\right )}{\ln \left (F \right ) a f g n}\) | \(44\) |
parallelrisch | \(\frac {n g f \ln \left (F \right ) x -\ln \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{\ln \left (F \right ) f g a n}\) | \(44\) |
derivativedivides | \(\frac {-\frac {\ln \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a}+\frac {\ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a}}{g f \ln \left (F \right ) n}\) | \(53\) |
default | \(\frac {-\frac {\ln \left (a +b \left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a}+\frac {\ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}\right )}{a}}{g f \ln \left (F \right ) n}\) | \(53\) |
risch | \(\frac {\ln \left (F^{g \left (f x +e \right )}\right )}{\ln \left (F \right ) a f g}-\frac {\ln \left (\left (F^{g \left (f x +e \right )}\right )^{n}+\frac {a}{b}\right )}{\ln \left (F \right ) a f g n}\) | \(62\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.10 \[ \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {f g n x \log \left (F\right ) - \log \left (F^{f g n x + e g n} b + a\right )}{a f g n \log \left (F\right )} \]
[In]
[Out]
Time = 1.89 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.28 \[ \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {2 \operatorname {atan}{\left (\frac {2 \left (\frac {a}{2 b} + \left (F^{g \left (e + f x\right )}\right )^{n}\right )}{\sqrt {- \frac {a^{2}}{b^{2}}}} \right )}}{b f g n \sqrt {- \frac {a^{2}}{b^{2}}} \log {\left (F \right )}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.48 \[ \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {f g n x + e g n}{a f g n} - \frac {\log \left (F^{f g n x + e g n} b + a\right )}{a f g n \log \left (F\right )} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.80 \[ \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=\frac {\log \left ({\left | F \right |}^{f g n x} {\left | F \right |}^{e g n}\right )}{a f g n \log \left (F\right )} - \frac {\log \left ({\left | F^{f g n x} F^{e g n} b + a \right |}\right )}{a f g n \log \left (F\right )} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.10 \[ \int \frac {1}{a+b \left (F^{g (e+f x)}\right )^n} \, dx=-\frac {\ln \left (a+b\,{\left (F^{e\,g+f\,g\,x}\right )}^n\right )-f\,g\,n\,x\,\ln \left (F\right )}{a\,f\,g\,n\,\ln \left (F\right )} \]
[In]
[Out]