Integrand size = 122, antiderivative size = 22 \[ \int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 \left (49-14 x+x^2\right )+e^4 \left (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6\right )} \, dx=2-\frac {4}{(-7+x) \left (e^4+\left (20+x^2\right )^2\right )} \] Output:
2-4/(-7+x)/(exp(4)+(x^2+20)^2)
Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 \left (49-14 x+x^2\right )+e^4 \left (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6\right )} \, dx=-\frac {4}{(-7+x) \left (e^4+\left (20+x^2\right )^2\right )} \] Input:
Integrate[(1600 + 4*E^4 - 2240*x + 480*x^2 - 112*x^3 + 20*x^4)/(7840000 - 2240000*x + 1728000*x^2 - 448000*x^3 + 149600*x^4 - 33600*x^5 + 6320*x^6 - 1120*x^7 + 129*x^8 - 14*x^9 + x^10 + E^8*(49 - 14*x + x^2) + E^4*(39200 - 11200*x + 4720*x^2 - 1120*x^3 + 178*x^4 - 28*x^5 + 2*x^6)),x]
Output:
-4/((-7 + x)*(E^4 + (20 + x^2)^2))
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 1.40 (sec) , antiderivative size = 259, normalized size of antiderivative = 11.77, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2462, 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.
\(\displaystyle \int \frac {20 x^4-112 x^3+480 x^2-2240 x+4 e^4+1600}{x^{10}-14 x^9+129 x^8-1120 x^7+6320 x^6-33600 x^5+149600 x^4-448000 x^3+1728000 x^2+e^8 \left (x^2-14 x+49\right )+e^4 \left (2 x^6-28 x^5+178 x^4-1120 x^3+4720 x^2-11200 x+39200\right )-2240000 x+7840000} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (-\frac {4 \left (x^2+14 x+187\right )}{\left (4761+e^4\right ) \left (x^4+40 x^2+e^4+400\right )}+\frac {16 \left (-483 x^3+\left (1380+e^4\right ) x^2-7 \left (1380-e^4\right ) x+69 \left (400+e^4\right )\right )}{\left (4761+e^4\right ) \left (x^4+40 x^2+e^4+400\right )^2}+\frac {4}{\left (4761+e^4\right ) (x-7)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {2} \left (\sqrt {\sqrt {400+e^4}-20}+187 \sqrt {\frac {\sqrt {400+e^4}-20}{400+e^4}}\right ) \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {400+e^4}-20\right )}}{\sqrt {2 \left (20+\sqrt {400+e^4}\right )}}\right )}{e^2 \left (4761+e^4\right )}+\frac {\sqrt {\frac {2}{\left (400+e^4\right ) \left (20+\sqrt {400+e^4}\right )}} \left (187+\sqrt {400+e^4}\right ) \arctan \left (\frac {2 x+\sqrt {2 \left (\sqrt {400+e^4}-20\right )}}{\sqrt {2 \left (20+\sqrt {400+e^4}\right )}}\right )}{4761+e^4}+\frac {4 x \left (x^2+89\right )}{\left (4761+e^4\right ) \left (x^4+40 x^2+e^4+400\right )}+\frac {28 \left (x^2+89\right )}{\left (4761+e^4\right ) \left (x^4+40 x^2+e^4+400\right )}+\frac {4}{\left (4761+e^4\right ) (7-x)}\) |
Input:
Int[(1600 + 4*E^4 - 2240*x + 480*x^2 - 112*x^3 + 20*x^4)/(7840000 - 224000 0*x + 1728000*x^2 - 448000*x^3 + 149600*x^4 - 33600*x^5 + 6320*x^6 - 1120* x^7 + 129*x^8 - 14*x^9 + x^10 + E^8*(49 - 14*x + x^2) + E^4*(39200 - 11200 *x + 4720*x^2 - 1120*x^3 + 178*x^4 - 28*x^5 + 2*x^6)),x]
Output:
4/((4761 + E^4)*(7 - x)) + (28*(89 + x^2))/((4761 + E^4)*(400 + E^4 + 40*x ^2 + x^4)) + (4*x*(89 + x^2))/((4761 + E^4)*(400 + E^4 + 40*x^2 + x^4)) + (Sqrt[2/((400 + E^4)*(20 + Sqrt[400 + E^4]))]*(187 + Sqrt[400 + E^4])*ArcT an[(Sqrt[2*(-20 + Sqrt[400 + E^4])] + 2*x)/Sqrt[2*(20 + Sqrt[400 + E^4])]] )/(4761 + E^4) - (Sqrt[2]*(Sqrt[-20 + Sqrt[400 + E^4]] + 187*Sqrt[(-20 + S qrt[400 + E^4])/(400 + E^4)])*ArcTan[(Sqrt[2*(-20 + Sqrt[400 + E^4])] + 2* x)/Sqrt[2*(20 + Sqrt[400 + E^4])]])/(E^2*(4761 + E^4))
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
method | result | size |
norman | \(-\frac {4}{\left (-7+x \right ) \left (x^{4}+40 x^{2}+{\mathrm e}^{4}+400\right )}\) | \(22\) |
gosper | \(-\frac {4}{x^{5}-7 x^{4}+40 x^{3}+x \,{\mathrm e}^{4}-280 x^{2}-7 \,{\mathrm e}^{4}+400 x -2800}\) | \(36\) |
risch | \(-\frac {4}{x^{5}-7 x^{4}+40 x^{3}+x \,{\mathrm e}^{4}-280 x^{2}-7 \,{\mathrm e}^{4}+400 x -2800}\) | \(36\) |
parallelrisch | \(-\frac {4}{x^{5}-7 x^{4}+40 x^{3}+x \,{\mathrm e}^{4}-280 x^{2}-7 \,{\mathrm e}^{4}+400 x -2800}\) | \(36\) |
Input:
int((4*exp(4)+20*x^4-112*x^3+480*x^2-2240*x+1600)/((x^2-14*x+49)*exp(4)^2+ (2*x^6-28*x^5+178*x^4-1120*x^3+4720*x^2-11200*x+39200)*exp(4)+x^10-14*x^9+ 129*x^8-1120*x^7+6320*x^6-33600*x^5+149600*x^4-448000*x^3+1728000*x^2-2240 000*x+7840000),x,method=_RETURNVERBOSE)
Output:
-4/(-7+x)/(x^4+40*x^2+exp(4)+400)
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 \left (49-14 x+x^2\right )+e^4 \left (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6\right )} \, dx=-\frac {4}{x^{5} - 7 \, x^{4} + 40 \, x^{3} - 280 \, x^{2} + {\left (x - 7\right )} e^{4} + 400 \, x - 2800} \] Input:
integrate((4*exp(4)+20*x^4-112*x^3+480*x^2-2240*x+1600)/((x^2-14*x+49)*exp (4)^2+(2*x^6-28*x^5+178*x^4-1120*x^3+4720*x^2-11200*x+39200)*exp(4)+x^10-1 4*x^9+129*x^8-1120*x^7+6320*x^6-33600*x^5+149600*x^4-448000*x^3+1728000*x^ 2-2240000*x+7840000),x, algorithm="fricas")
Output:
-4/(x^5 - 7*x^4 + 40*x^3 - 280*x^2 + (x - 7)*e^4 + 400*x - 2800)
Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (15) = 30\).
Time = 1.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 \left (49-14 x+x^2\right )+e^4 \left (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6\right )} \, dx=- \frac {4}{x^{5} - 7 x^{4} + 40 x^{3} - 280 x^{2} + x \left (e^{4} + 400\right ) - 2800 - 7 e^{4}} \] Input:
integrate((4*exp(4)+20*x**4-112*x**3+480*x**2-2240*x+1600)/((x**2-14*x+49) *exp(4)**2+(2*x**6-28*x**5+178*x**4-1120*x**3+4720*x**2-11200*x+39200)*exp (4)+x**10-14*x**9+129*x**8-1120*x**7+6320*x**6-33600*x**5+149600*x**4-4480 00*x**3+1728000*x**2-2240000*x+7840000),x)
Output:
-4/(x**5 - 7*x**4 + 40*x**3 - 280*x**2 + x*(exp(4) + 400) - 2800 - 7*exp(4 ))
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 \left (49-14 x+x^2\right )+e^4 \left (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6\right )} \, dx=-\frac {4}{x^{5} - 7 \, x^{4} + 40 \, x^{3} - 280 \, x^{2} + x {\left (e^{4} + 400\right )} - 7 \, e^{4} - 2800} \] Input:
integrate((4*exp(4)+20*x^4-112*x^3+480*x^2-2240*x+1600)/((x^2-14*x+49)*exp (4)^2+(2*x^6-28*x^5+178*x^4-1120*x^3+4720*x^2-11200*x+39200)*exp(4)+x^10-1 4*x^9+129*x^8-1120*x^7+6320*x^6-33600*x^5+149600*x^4-448000*x^3+1728000*x^ 2-2240000*x+7840000),x, algorithm="maxima")
Output:
-4/(x^5 - 7*x^4 + 40*x^3 - 280*x^2 + x*(e^4 + 400) - 7*e^4 - 2800)
Timed out. \[ \int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 \left (49-14 x+x^2\right )+e^4 \left (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6\right )} \, dx=\text {Timed out} \] Input:
integrate((4*exp(4)+20*x^4-112*x^3+480*x^2-2240*x+1600)/((x^2-14*x+49)*exp (4)^2+(2*x^6-28*x^5+178*x^4-1120*x^3+4720*x^2-11200*x+39200)*exp(4)+x^10-1 4*x^9+129*x^8-1120*x^7+6320*x^6-33600*x^5+149600*x^4-448000*x^3+1728000*x^ 2-2240000*x+7840000),x, algorithm="giac")
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 \left (49-14 x+x^2\right )+e^4 \left (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6\right )} \, dx=-\frac {4}{\left (x-7\right )\,\left (x^4+40\,x^2+{\mathrm {e}}^4+400\right )} \] Input:
int((4*exp(4) - 2240*x + 480*x^2 - 112*x^3 + 20*x^4 + 1600)/(exp(4)*(4720* x^2 - 11200*x - 1120*x^3 + 178*x^4 - 28*x^5 + 2*x^6 + 39200) - 2240000*x + exp(8)*(x^2 - 14*x + 49) + 1728000*x^2 - 448000*x^3 + 149600*x^4 - 33600* x^5 + 6320*x^6 - 1120*x^7 + 129*x^8 - 14*x^9 + x^10 + 7840000),x)
Output:
-4/((x - 7)*(exp(4) + 40*x^2 + x^4 + 400))
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 \left (49-14 x+x^2\right )+e^4 \left (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6\right )} \, dx=-\frac {4}{e^{4} x +x^{5}-7 e^{4}-7 x^{4}+40 x^{3}-280 x^{2}+400 x -2800} \] Input:
int((4*exp(4)+20*x^4-112*x^3+480*x^2-2240*x+1600)/((x^2-14*x+49)*exp(4)^2+ (2*x^6-28*x^5+178*x^4-1120*x^3+4720*x^2-11200*x+39200)*exp(4)+x^10-14*x^9+ 129*x^8-1120*x^7+6320*x^6-33600*x^5+149600*x^4-448000*x^3+1728000*x^2-2240 000*x+7840000),x)
Output:
( - 4)/(e**4*x - 7*e**4 + x**5 - 7*x**4 + 40*x**3 - 280*x**2 + 400*x - 280 0)