Integrand size = 11, antiderivative size = 24 \[ \int \frac {1}{b+a e^{n x}} \, dx=\frac {x}{b}-\frac {\log \left (b+a e^{n x}\right )}{b n} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2320, 36, 29, 31} \[ \int \frac {1}{b+a e^{n x}} \, dx=\frac {x}{b}-\frac {\log \left (a e^{n x}+b\right )}{b n} \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x (b+a x)} \, dx,x,e^{n x}\right )}{n} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{n x}\right )}{b n}-\frac {a \text {Subst}\left (\int \frac {1}{b+a x} \, dx,x,e^{n x}\right )}{b n} \\ & = \frac {x}{b}-\frac {\log \left (b+a e^{n x}\right )}{b n} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {1}{b+a e^{n x}} \, dx=\frac {\log \left (e^{n x}\right )}{b n}-\frac {\log \left (b^2 n+a b e^{n x} n\right )}{b n} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(-\frac {-n x +\ln \left (b +a \,{\mathrm e}^{n x}\right )}{b n}\) | \(23\) |
norman | \(\frac {x}{b}-\frac {\ln \left (b +a \,{\mathrm e}^{n x}\right )}{b n}\) | \(24\) |
risch | \(\frac {x}{b}-\frac {\ln \left ({\mathrm e}^{n x}+\frac {b}{a}\right )}{b n}\) | \(26\) |
derivativedivides | \(\frac {-\frac {\ln \left (b +a \,{\mathrm e}^{n x}\right )}{b}+\frac {\ln \left ({\mathrm e}^{n x}\right )}{b}}{n}\) | \(29\) |
default | \(\frac {-\frac {\ln \left (b +a \,{\mathrm e}^{n x}\right )}{b}+\frac {\ln \left ({\mathrm e}^{n x}\right )}{b}}{n}\) | \(29\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{b+a e^{n x}} \, dx=\frac {n x - \log \left (a e^{\left (n x\right )} + b\right )}{b n} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {1}{b+a e^{n x}} \, dx=\frac {x}{b} - \frac {\log {\left (e^{n x} + \frac {b}{a} \right )}}{b n} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {1}{b+a e^{n x}} \, dx=\frac {x}{b} - \frac {\log \left (a e^{\left (n x\right )} + b\right )}{b n} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{b+a e^{n x}} \, dx=\frac {\frac {n x}{b} - \frac {\log \left ({\left | a e^{\left (n x\right )} + b \right |}\right )}{b}}{n} \]
[In]
[Out]
Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{b+a e^{n x}} \, dx=-\frac {\ln \left (b+a\,{\mathrm {e}}^{n\,x}\right )-n\,x}{b\,n} \]
[In]
[Out]