Integrand size = 9, antiderivative size = 19 \[ \int \frac {e^t}{(1+t)^2} \, dt=-\frac {e^t}{1+t}+\frac {\operatorname {ExpIntegralEi}(1+t)}{e} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2208, 2209} \[ \int \frac {e^t}{(1+t)^2} \, dt=\frac {\operatorname {ExpIntegralEi}(t+1)}{e}-\frac {e^t}{t+1} \]
[In]
[Out]
Rule 2208
Rule 2209
Rubi steps \begin{align*} \text {integral}& = -\frac {e^t}{1+t}+\int \frac {e^t}{1+t} \, dt \\ & = -\frac {e^t}{1+t}+\frac {\operatorname {ExpIntegralEi}(1+t)}{e} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^t}{(1+t)^2} \, dt=-\frac {e^t}{1+t}+\frac {\operatorname {ExpIntegralEi}(1+t)}{e} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
method | result | size |
default | \(-\frac {{\mathrm e}^{t}}{1+t}-{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (-1-t \right )\) | \(22\) |
risch | \(-\frac {{\mathrm e}^{t}}{1+t}-{\mathrm e}^{-1} \operatorname {Ei}_{1}\left (-1-t \right )\) | \(22\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {e^t}{(1+t)^2} \, dt=\frac {{\left ({\left (t + 1\right )} {\rm Ei}\left (t + 1\right ) - e^{\left (t + 1\right )}\right )} e^{\left (-1\right )}}{t + 1} \]
[In]
[Out]
\[ \int \frac {e^t}{(1+t)^2} \, dt=\int \frac {e^{t}}{\left (t + 1\right )^{2}}\, dt \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {e^t}{(1+t)^2} \, dt=-\frac {e^{\left (-1\right )} E_{2}\left (-t - 1\right )}{t + 1} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (17) = 34\).
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 4.21 \[ \int \frac {e^t}{(1+t)^2} \, dt=\frac {{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} {\rm Ei}\left (-{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} + 1\right ) - {\rm Ei}\left (-{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} + 1\right ) + e^{\left (-{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} + 1\right )}}{{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} e - e} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {e^t}{(1+t)^2} \, dt=\mathrm {ei}\left (t+1\right )\,{\mathrm {e}}^{-1}-\frac {{\mathrm {e}}^t}{t+1} \]
[In]
[Out]