Integrand size = 42, antiderivative size = 20 \[ \int \frac {-12+2 x^3+e^x \left (-8 x+8 x^2+2 x^3+e^4 \left (-2 x+2 x^2\right )\right )}{x^3} \, dx=\frac {2 \left (3+x \left (4+e^4+x\right ) \left (e^x+x\right )\right )}{x^2} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {14, 2230, 2225, 2208, 2209} \[ \int \frac {-12+2 x^3+e^x \left (-8 x+8 x^2+2 x^3+e^4 \left (-2 x+2 x^2\right )\right )}{x^3} \, dx=\frac {6}{x^2}+2 x+2 e^x+\frac {2 \left (4+e^4\right ) e^x}{x} \]
[In]
[Out]
Rule 14
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 e^x \left (-4-e^4+\left (4+e^4\right ) x+x^2\right )}{x^2}+\frac {2 \left (-6+x^3\right )}{x^3}\right ) \, dx \\ & = 2 \int \frac {e^x \left (-4-e^4+\left (4+e^4\right ) x+x^2\right )}{x^2} \, dx+2 \int \frac {-6+x^3}{x^3} \, dx \\ & = 2 \int \left (1-\frac {6}{x^3}\right ) \, dx+2 \int \left (e^x+\frac {e^x \left (-4-e^4\right )}{x^2}+\frac {e^x \left (4+e^4\right )}{x}\right ) \, dx \\ & = \frac {6}{x^2}+2 x+2 \int e^x \, dx-\left (2 \left (4+e^4\right )\right ) \int \frac {e^x}{x^2} \, dx+\left (2 \left (4+e^4\right )\right ) \int \frac {e^x}{x} \, dx \\ & = 2 e^x+\frac {6}{x^2}+\frac {2 e^x \left (4+e^4\right )}{x}+2 x+2 \left (4+e^4\right ) \operatorname {ExpIntegralEi}(x)-\left (2 \left (4+e^4\right )\right ) \int \frac {e^x}{x} \, dx \\ & = 2 e^x+\frac {6}{x^2}+\frac {2 e^x \left (4+e^4\right )}{x}+2 x \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {-12+2 x^3+e^x \left (-8 x+8 x^2+2 x^3+e^4 \left (-2 x+2 x^2\right )\right )}{x^3} \, dx=2 \left (\frac {3}{x^2}+x+\frac {e^x \left (\left (4+e^4\right ) x+x^2\right )}{x^2}\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10
method | result | size |
risch | \(2 x +\frac {6}{x^{2}}+\frac {2 \left (x +4+{\mathrm e}^{4}\right ) {\mathrm e}^{x}}{x}\) | \(22\) |
norman | \(\frac {6+\left (2 \,{\mathrm e}^{4}+8\right ) x \,{\mathrm e}^{x}+2 x^{3}+2 \,{\mathrm e}^{x} x^{2}}{x^{2}}\) | \(29\) |
parallelrisch | \(\frac {2 x \,{\mathrm e}^{4} {\mathrm e}^{x}+2 x^{3}+2 \,{\mathrm e}^{x} x^{2}+6+8 \,{\mathrm e}^{x} x}{x^{2}}\) | \(31\) |
default | \(2 x +\frac {6}{x^{2}}+\frac {8 \,{\mathrm e}^{x}}{x}-2 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{x}}{x}-\operatorname {Ei}_{1}\left (-x \right )\right )-2 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-x \right )+2 \,{\mathrm e}^{x}\) | \(49\) |
parts | \(2 x +\frac {6}{x^{2}}+\frac {8 \,{\mathrm e}^{x}}{x}-2 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{x}}{x}-\operatorname {Ei}_{1}\left (-x \right )\right )-2 \,{\mathrm e}^{4} \operatorname {Ei}_{1}\left (-x \right )+2 \,{\mathrm e}^{x}\) | \(49\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {-12+2 x^3+e^x \left (-8 x+8 x^2+2 x^3+e^4 \left (-2 x+2 x^2\right )\right )}{x^3} \, dx=\frac {2 \, {\left (x^{3} + {\left (x^{2} + x e^{4} + 4 \, x\right )} e^{x} + 3\right )}}{x^{2}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {-12+2 x^3+e^x \left (-8 x+8 x^2+2 x^3+e^4 \left (-2 x+2 x^2\right )\right )}{x^3} \, dx=2 x + \frac {\left (2 x + 8 + 2 e^{4}\right ) e^{x}}{x} + \frac {6}{x^{2}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \frac {-12+2 x^3+e^x \left (-8 x+8 x^2+2 x^3+e^4 \left (-2 x+2 x^2\right )\right )}{x^3} \, dx=2 \, {\rm Ei}\left (x\right ) e^{4} - 2 \, e^{4} \Gamma \left (-1, -x\right ) + 2 \, x + \frac {6}{x^{2}} + 8 \, {\rm Ei}\left (x\right ) + 2 \, e^{x} - 8 \, \Gamma \left (-1, -x\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {-12+2 x^3+e^x \left (-8 x+8 x^2+2 x^3+e^4 \left (-2 x+2 x^2\right )\right )}{x^3} \, dx=\frac {2 \, {\left (x^{3} + x^{2} e^{x} + x e^{\left (x + 4\right )} + 4 \, x e^{x} + 3\right )}}{x^{2}} \]
[In]
[Out]
Time = 8.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {-12+2 x^3+e^x \left (-8 x+8 x^2+2 x^3+e^4 \left (-2 x+2 x^2\right )\right )}{x^3} \, dx=2\,x+2\,{\mathrm {e}}^x+\frac {x\,{\mathrm {e}}^x\,\left (2\,{\mathrm {e}}^4+8\right )+6}{x^2} \]
[In]
[Out]