Integrand size = 28, antiderivative size = 21 \[ \int \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}+x^2} \left (x+4 e^2 x\right ) \, dx=\left (2+\frac {1}{2 e^2}\right ) e^{\frac {e^{x^2}}{2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6, 12, 6847, 2320, 2225} \[ \int \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}+x^2} \left (x+4 e^2 x\right ) \, dx=\frac {1}{2} \left (1+4 e^2\right ) e^{\frac {e^{x^2}}{2}-2} \]
[In]
[Out]
Rule 6
Rule 12
Rule 2225
Rule 2320
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}+x^2} \left (1+4 e^2\right ) x \, dx \\ & = \frac {1}{2} \left (1+4 e^2\right ) \int e^{-2+\frac {e^{x^2}}{2}+x^2} x \, dx \\ & = \frac {1}{4} \left (1+4 e^2\right ) \text {Subst}\left (\int e^{-2+\frac {e^x}{2}+x} \, dx,x,x^2\right ) \\ & = \frac {1}{4} \left (1+4 e^2\right ) \text {Subst}\left (\int e^{-2+\frac {x}{2}} \, dx,x,e^{x^2}\right ) \\ & = \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}} \left (1+4 e^2\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}+x^2} \left (x+4 e^2 x\right ) \, dx=\frac {1}{2} e^{-2+\frac {e^{x^2}}{2}} \left (1+4 e^2\right ) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05
method | result | size |
norman | \(\frac {\left (4 \,{\mathrm e}^{2}+1\right ) {\mathrm e}^{-2} {\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{2}}}{2}\) | \(22\) |
risch | \(2 \,{\mathrm e}^{-2+\frac {{\mathrm e}^{x^{2}}}{2}} {\mathrm e}^{2}+\frac {{\mathrm e}^{-2+\frac {{\mathrm e}^{x^{2}}}{2}}}{2}\) | \(26\) |
default | \(\frac {{\mathrm e}^{-2} \left ({\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{2}}+4 \,{\mathrm e}^{2} {\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{2}}\right )}{2}\) | \(28\) |
parallelrisch | \(\frac {{\mathrm e}^{-2} \left ({\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{2}}+4 \,{\mathrm e}^{2} {\mathrm e}^{\frac {{\mathrm e}^{x^{2}}}{2}}\right )}{2}\) | \(28\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}+x^2} \left (x+4 e^2 x\right ) \, dx=\frac {1}{2} \, {\left (4 \, e^{2} + 1\right )} e^{\left (\frac {1}{2} \, e^{\left (x^{2}\right )} - 2\right )} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}+x^2} \left (x+4 e^2 x\right ) \, dx=\frac {\left (1 + 4 e^{2}\right ) e^{\frac {e^{x^{2}}}{2}}}{2 e^{2}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}+x^2} \left (x+4 e^2 x\right ) \, dx=2 \, e^{\left (\frac {1}{2} \, e^{\left (x^{2}\right )}\right )} + \frac {1}{2} \, e^{\left (\frac {1}{2} \, e^{\left (x^{2}\right )} - 2\right )} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}+x^2} \left (x+4 e^2 x\right ) \, dx=\frac {1}{2} \, {\left (e^{\left (\frac {1}{2} \, e^{\left (x^{2}\right )}\right )} + 4 \, e^{\left (\frac {1}{2} \, e^{\left (x^{2}\right )} + 2\right )}\right )} e^{\left (-2\right )} \]
[In]
[Out]
Time = 12.99 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {1}{2} e^{-2+\frac {e^{x^2}}{2}+x^2} \left (x+4 e^2 x\right ) \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{x^2}}{2}-2}\,\left (2\,{\mathrm {e}}^2+\frac {1}{2}\right ) \]
[In]
[Out]