3.3.35 \(\int \frac {1600+4 e^4-2240 x+480 x^2-112 x^3+20 x^4}{7840000-2240000 x+1728000 x^2-448000 x^3+149600 x^4-33600 x^5+6320 x^6-1120 x^7+129 x^8-14 x^9+x^{10}+e^8 (49-14 x+x^2)+e^4 (39200-11200 x+4720 x^2-1120 x^3+178 x^4-28 x^5+2 x^6)} \, dx\) [235]

3.3.35.1 Optimal result

Integrand size = 122, antiderivative size = 22 \begin {dmath*} \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 )} \end {dmath*}

output

2-4/(-7+x)/(exp(4)+(x^2+20)^2)
 

3.3.35.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \begin {dmath*} \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 )} \end {dmath*}

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))
 

3.3.35.3 Rubi [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 1.14 (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))
 

3.3.35.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2462

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]
 

3.3.35.4 Maple [A] (verified)

Time = 0.40 (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)
 

3.3.35.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \begin {dmath*} \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} \end {dmath*}

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=\
 

output

-4/(x^5 - 7*x^4 + 40*x^3 - 280*x^2 + (x - 7)*e^4 + 400*x - 2800)
 

3.3.35.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (15) = 30\).

Time = 1.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \begin {dmath*} \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}} \end {dmath*}

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 
))
 

3.3.35.7 Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \begin {dmath*} \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} \end {dmath*}

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=\
 

output

-4/(x^5 - 7*x^4 + 40*x^3 - 280*x^2 + x*(e^4 + 400) - 7*e^4 - 2800)
 

3.3.35.8 Giac [F(-1)]

Timed out. \begin {dmath*} \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} \end {dmath*}

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=\
 

output

Timed out
 

3.3.35.9 Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \begin {dmath*} \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 )} \end {dmath*}

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))