Integrand size = 95, antiderivative size = 20 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=-e^{4-x}+e^{2 x}+\frac {1}{(-3+x)^4} \]
Time = 1.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=-e^{4-x}+e^{2 x}+\frac {1}{(-3+x)^4} \]
Integrate[(-4*E^x + E^4*(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5) + E^(3*x)*(-486 + 810*x - 540*x^2 + 180*x^3 - 30*x^4 + 2*x^5))/(E^x*(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5)),x]
Time = 0.67 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2007, 7239, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{-x} \left (e^4 \left (x^5-15 x^4+90 x^3-270 x^2+405 x-243\right )+e^{3 x} \left (2 x^5-30 x^4+180 x^3-540 x^2+810 x-486\right )-4 e^x\right )}{x^5-15 x^4+90 x^3-270 x^2+405 x-243} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{-x} \left (e^4 \left (x^5-15 x^4+90 x^3-270 x^2+405 x-243\right )+e^{3 x} \left (2 x^5-30 x^4+180 x^3-540 x^2+810 x-486\right )-4 e^x\right )}{(x-3)^5}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{-x} \left (-2 e^{3 x} (x-3)^5-e^4 (x-3)^5+4 e^x\right )}{(3-x)^5}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (e^{4-x}+2 e^{2 x}-\frac {4}{(x-3)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -e^{4-x}+e^{2 x}+\frac {1}{(x-3)^4}\) |
Int[(-4*E^x + E^4*(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5) + E^(3* x)*(-486 + 810*x - 540*x^2 + 180*x^3 - 30*x^4 + 2*x^5))/(E^x*(-243 + 405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5)),x]
3.24.98.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
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 | \({\mathrm e}^{4} \left (-{\mathrm e}^{-x}-\frac {9 \,{\mathrm e}^{-x} \left (11 x^{3}-30 x^{2}-15 x +72\right )}{8 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}-\frac {21 \,{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{8}\right )+\frac {1}{\left (-3+x \right )^{4}}-243 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-x} \left (x^{3}-10 x^{2}+35 x -48\right )}{24 x^{4}-288 x^{3}+1296 x^{2}-2592 x +1944}-\frac {{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{24}\right )+{\mathrm e}^{2 x}+405 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-x} \left (x^{3}-10 x^{2}+35 x -24\right )}{24 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}+\frac {{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{24}\right )-270 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-x} \left (x^{3}-10 x^{2}+43 x -48\right )}{8 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}+\frac {{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{8}\right )+90 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-x} x \left (x^{2}-18 x +27\right )}{8 x^{4}-96 x^{3}+432 x^{2}-864 x +648}-\frac {{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{8}\right )-15 \,{\mathrm e}^{4} \left (\frac {3 \,{\mathrm e}^{-x} \left (x^{3}-42 x^{2}+147 x -144\right )}{8 \left (x^{4}-12 x^{3}+54 x^{2}-108 x +81\right )}-\frac {11 \,{\mathrm e}^{-3} \operatorname {Ei}_{1}\left (-3+x \right )}{8}\right )\) | \(332\) |
int(((2*x^5-30*x^4+180*x^3-540*x^2+810*x-486)*exp(x)^3-4*exp(x)+(x^5-15*x^ 4+90*x^3-270*x^2+405*x-243)*exp(4))/(x^5-15*x^4+90*x^3-270*x^2+405*x-243)/ exp(x),x,method=_RETURNVERBOSE)
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 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=-\frac {{\left ({\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )} e^{4} - {\left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )} e^{\left (3 \, x\right )} - e^{x}\right )} e^{\left (-x\right )}}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81} \]
integrate(((2*x^5-30*x^4+180*x^3-540*x^2+810*x-486)*exp(x)^3-4*exp(x)+(x^5 -15*x^4+90*x^3-270*x^2+405*x-243)*exp(4))/(x^5-15*x^4+90*x^3-270*x^2+405*x -243)/exp(x),x, algorithm=\
-((x^4 - 12*x^3 + 54*x^2 - 108*x + 81)*e^4 - (x^4 - 12*x^3 + 54*x^2 - 108* x + 81)*e^(3*x) - e^x)*e^(-x)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81)
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=e^{2 x} - e^{4} e^{- x} + \frac {4}{4 x^{4} - 48 x^{3} + 216 x^{2} - 432 x + 324} \]
integrate(((2*x**5-30*x**4+180*x**3-540*x**2+810*x-486)*exp(x)**3-4*exp(x) +(x**5-15*x**4+90*x**3-270*x**2+405*x-243)*exp(4))/(x**5-15*x**4+90*x**3-2 70*x**2+405*x-243)/exp(x),x)
\[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=\int { \frac {{\left ({\left (x^{5} - 15 \, x^{4} + 90 \, x^{3} - 270 \, x^{2} + 405 \, x - 243\right )} e^{4} + 2 \, {\left (x^{5} - 15 \, x^{4} + 90 \, x^{3} - 270 \, x^{2} + 405 \, x - 243\right )} e^{\left (3 \, x\right )} - 4 \, e^{x}\right )} e^{\left (-x\right )}}{x^{5} - 15 \, x^{4} + 90 \, x^{3} - 270 \, x^{2} + 405 \, x - 243} \,d x } \]
integrate(((2*x^5-30*x^4+180*x^3-540*x^2+810*x-486)*exp(x)^3-4*exp(x)+(x^5 -15*x^4+90*x^3-270*x^2+405*x-243)*exp(4))/(x^5-15*x^4+90*x^3-270*x^2+405*x -243)/exp(x),x, algorithm=\
((x^5 - 15*x^4 + 90*x^3 - 270*x^2 + 405*x - 243)*e^(2*x) - (x^5*e^4 - 15*x ^4*e^4 + 90*x^3*e^4 - 270*x^2*e^4 + 405*x*e^4)*e^(-x) + x - 3)/(x^5 - 15*x ^4 + 90*x^3 - 270*x^2 + 405*x - 243) + 243*e*exp_integral_e(5, x - 3)/(x - 3)^4 - 1215*integrate(e^(-x + 4)/(x^6 - 18*x^5 + 135*x^4 - 540*x^3 + 1215 *x^2 - 1458*x + 729), x)
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 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=\frac {x^{4} e^{\left (2 \, x\right )} - x^{4} e^{\left (-x + 4\right )} - 12 \, x^{3} e^{\left (2 \, x\right )} + 12 \, x^{3} e^{\left (-x + 4\right )} + 54 \, x^{2} e^{\left (2 \, x\right )} - 54 \, x^{2} e^{\left (-x + 4\right )} - 108 \, x e^{\left (2 \, x\right )} + 108 \, x e^{\left (-x + 4\right )} + 81 \, e^{\left (2 \, x\right )} - 81 \, e^{\left (-x + 4\right )} + 1}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81} \]
integrate(((2*x^5-30*x^4+180*x^3-540*x^2+810*x-486)*exp(x)^3-4*exp(x)+(x^5 -15*x^4+90*x^3-270*x^2+405*x-243)*exp(4))/(x^5-15*x^4+90*x^3-270*x^2+405*x -243)/exp(x),x, algorithm=\
(x^4*e^(2*x) - x^4*e^(-x + 4) - 12*x^3*e^(2*x) + 12*x^3*e^(-x + 4) + 54*x^ 2*e^(2*x) - 54*x^2*e^(-x + 4) - 108*x*e^(2*x) + 108*x*e^(-x + 4) + 81*e^(2 *x) - 81*e^(-x + 4) + 1)/(x^4 - 12*x^3 + 54*x^2 - 108*x + 81)
Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.75 \[ \int \frac {e^{-x} \left (-4 e^x+e^4 \left (-243+405 x-270 x^2+90 x^3-15 x^4+x^5\right )+e^{3 x} \left (-486+810 x-540 x^2+180 x^3-30 x^4+2 x^5\right )\right )}{-243+405 x-270 x^2+90 x^3-15 x^4+x^5} \, dx=\frac {{\mathrm {e}}^x}{81\,{\mathrm {e}}^x+54\,x^2\,{\mathrm {e}}^x-12\,x^3\,{\mathrm {e}}^x+x^4\,{\mathrm {e}}^x-108\,x\,{\mathrm {e}}^x}-{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^4-{\mathrm {e}}^{3\,x}+\frac {{\mathrm {e}}^x}{81}\right ) \]
int((exp(-x)*(exp(4)*(405*x - 270*x^2 + 90*x^3 - 15*x^4 + x^5 - 243) - 4*e xp(x) + exp(3*x)*(810*x - 540*x^2 + 180*x^3 - 30*x^4 + 2*x^5 - 486)))/(405 *x - 270*x^2 + 90*x^3 - 15*x^4 + x^5 - 243),x)