Integrand size = 95, antiderivative size = 20 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=-e^{4-x}+e^{2 x}+\frac {1}{(-3+x)^4} \]
[Out]
Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6820, 6874, 2225} \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=-e^{4-x}+e^{2 x}+\frac {1}{(x-3)^4} \]
[In]
[Out]
Rule 2225
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (4 e^x-e^4 (-3+x)^5-2 e^{3 x} (-3+x)^5\right )}{(3-x)^5} \, dx \\ & = \int \left (e^{4-x}+2 e^{2 x}-\frac {4}{(-3+x)^5}\right ) \, dx \\ & = \frac {1}{(-3+x)^4}+2 \int e^{2 x} \, dx+\int e^{4-x} \, dx \\ & = -e^{4-x}+e^{2 x}+\frac {1}{(-3+x)^4} \\ \end{align*}
Time = 1.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=-e^{4-x}+e^{2 x}+\frac {1}{(-3+x)^4} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
parts | \(\frac {1}{\left (-3+x \right )^{4}}-{\mathrm e}^{4} {\mathrm e}^{-x}+{\mathrm e}^{2 x}\) | \(19\) |
risch | \(\frac {1}{x^{4}-12 x^{3}+54 x^{2}-108 x +81}+{\mathrm e}^{2 x}-{\mathrm e}^{-x +4}\) | \(34\) |
norman | \(\frac {\left (x^{4} {\mathrm e}^{3 x}+{\mathrm e}^{x}+81 \,{\mathrm e}^{3 x}+108 x \,{\mathrm e}^{4}-108 x \,{\mathrm e}^{3 x}-54 x^{2} {\mathrm e}^{4}+54 x^{2} {\mathrm e}^{3 x}+12 x^{3} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3 x}-x^{4} {\mathrm e}^{4}-81 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-x}}{\left (-3+x \right )^{4}}\) | \(83\) |
parallelrisch | \(-\frac {\left (-x^{4} {\mathrm e}^{3 x}-54 x^{2} {\mathrm e}^{3 x}-12 x^{3} {\mathrm e}^{4}+54 x^{2} {\mathrm e}^{4}+108 x \,{\mathrm e}^{3 x}+12 x^{3} {\mathrm e}^{3 x}+x^{4} {\mathrm e}^{4}-108 x \,{\mathrm e}^{4}+81 \,{\mathrm e}^{4}-81 \,{\mathrm e}^{3 x}-{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x^{4}-12 x^{3}+54 x^{2}-108 x +81}\) | \(101\) |
default | \({\mathrm e}^{4} \left (-{\mathrm e}^{-x}-\frac {9 \,{\mathrm e}^{-x} \left (11 x^{3}-30 x^{2}-15 x +72\right )}{8 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}-\frac {21 \,{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{8}\right )+\frac {1}{\left (-3+x \right )^{4}}-243 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-x} \left (x^{3}-10 x^{2}+35 x -48\right )}{24 x^{4}-288 x^{3}+1296 x^{2}-2592 x +1944}-\frac {{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{24}\right )+{\mathrm e}^{2 x}+405 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-x} \left (x^{3}-10 x^{2}+35 x -24\right )}{24 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}+\frac {{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{24}\right )-270 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-x} \left (x^{3}-10 x^{2}+43 x -48\right )}{8 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}+\frac {{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{8}\right )+90 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-x} x \left (x^{2}-18 x +27\right )}{8 x^{4}-96 x^{3}+432 x^{2}-864 x +648}-\frac {{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{8}\right )-15 \,{\mathrm e}^{4} \left (\frac {3 \,{\mathrm e}^{-x} \left (x^{3}-42 x^{2}+147 x -144\right )}{8 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}-\frac {11 \,{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{8}\right )\) | \(332\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (18) = 36\).
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.80 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=-\frac {{\left ({\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )} e^{4} - {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )} e^{\left (3 \, x\right )} - e^{x}\right )} e^{\left (-x\right )}}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=e^{2 x} - e^{4} e^{- x} + \frac {4}{4 x^{4} - 48 x^{3} + 216 x^{2} - 432 x + 324} \]
[In]
[Out]
\[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=\int { \frac {{\left ({\left (x^{5} - 15 \, x^{4} + 90 \, x^{3} - 270 \, x^{2} + 405 \, x - 243\right )} e^{4} + 2 \, {\left (x^{5} - 15 \, x^{4} + 90 \, x^{3} - 270 \, x^{2} + 405 \, x - 243\right )} e^{\left (3 \, x\right )} - 4 \, e^{x}\right )} e^{\left (-x\right )}}{x^{5} - 15 \, x^{4} + 90 \, x^{3} - 270 \, x^{2} + 405 \, x - 243} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.60 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=\frac {x^{4} e^{\left (2 \, x\right )} - x^{4} e^{\left (-x + 4\right )} - 12 \, x^{3} e^{\left (2 \, x\right )} + 12 \, x^{3} e^{\left (-x + 4\right )} + 54 \, x^{2} e^{\left (2 \, x\right )} - 54 \, x^{2} e^{\left (-x + 4\right )} - 108 \, x e^{\left (2 \, x\right )} + 108 \, x e^{\left (-x + 4\right )} + 81 \, e^{\left (2 \, x\right )} - 81 \, e^{\left (-x + 4\right )} + 1}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81} \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.75 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=\frac {{\mathrm {e}}^x}{81\,{\mathrm {e}}^x+54\,x^2\,{\mathrm {e}}^x-12\,x^3\,{\mathrm {e}}^x+x^4\,{\mathrm {e}}^x-108\,x\,{\mathrm {e}}^x}-{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^4-{\mathrm {e}}^{3\,x}+\frac {{\mathrm {e}}^x}{81}\right ) \]
[In]
[Out]