\(\int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 (-1875-1500 x-450 x^2-60 x^3-3 x^4)+e^4 (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8)}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5)+e^4 (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9)} \, dx\) [847]

Optimal result

Integrand size = 282, antiderivative size = 28 \[ \int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx=-3+\frac {1}{2} \log \left (\frac {4}{-x+\frac {x}{\left (e^4+(5+x)^4\right )^2}}\right ) \] Output:

1/2*ln(4/(x/((5+x)^4+exp(4))^2-x))-3
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(112\) vs. \(2(28)=56\).

Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.00 \[ \int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx=\frac {1}{2} \left (-\log (x)+2 \log \left (625+e^4+500 x+150 x^2+20 x^3+x^4\right )-\log \left (390624+1250 e^4+e^8+625000 x+1000 e^4 x+437500 x^2+300 e^4 x^2+175000 x^3+40 e^4 x^3+43750 x^4+2 e^4 x^4+7000 x^5+700 x^6+40 x^7+x^8\right )\right ) \] Input:

Integrate[(-244140000 - E^12 - 585938000*x - 644531700*x^2 - 429687600*x^3 
 - 193359382*x^4 - 61875000*x^5 - 14437500*x^6 - 2475000*x^7 - 309375*x^8 
- 27500*x^9 - 1650*x^10 - 60*x^11 - x^12 + E^8*(-1875 - 1500*x - 450*x^2 - 
 60*x^3 - 3*x^4) + E^4*(-1171874 - 1875000*x - 1312500*x^2 - 525000*x^3 - 
131250*x^4 - 21000*x^5 - 2100*x^6 - 120*x^7 - 3*x^8))/(488280000*x + 2*E^1 
2*x + 1171874000*x^2 + 1289062200*x^3 + 859374960*x^4 + 386718748*x^5 + 12 
3750000*x^6 + 28875000*x^7 + 4950000*x^8 + 618750*x^9 + 55000*x^10 + 3300* 
x^11 + 120*x^12 + 2*x^13 + E^8*(3750*x + 3000*x^2 + 900*x^3 + 120*x^4 + 6* 
x^5) + E^4*(2343748*x + 3750000*x^2 + 2625000*x^3 + 1050000*x^4 + 262500*x 
^5 + 42000*x^6 + 4200*x^7 + 240*x^8 + 6*x^9)),x]
 

Output:

(-Log[x] + 2*Log[625 + E^4 + 500*x + 150*x^2 + 20*x^3 + x^4] - Log[390624 
+ 1250*E^4 + E^8 + 625000*x + 1000*E^4*x + 437500*x^2 + 300*E^4*x^2 + 1750 
00*x^3 + 40*E^4*x^3 + 43750*x^4 + 2*E^4*x^4 + 7000*x^5 + 700*x^6 + 40*x^7 
+ x^8])/2
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(28)=56\).

Time = 1.65 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.89, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6, 2026, 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.

\(\displaystyle \int \frac {-x^{12}-60 x^{11}-1650 x^{10}-27500 x^9-309375 x^8-2475000 x^7-14437500 x^6-61875000 x^5-193359382 x^4-429687600 x^3-644531700 x^2+e^8 \left (-3 x^4-60 x^3-450 x^2-1500 x-1875\right )+e^4 \left (-3 x^8-120 x^7-2100 x^6-21000 x^5-131250 x^4-525000 x^3-1312500 x^2-1875000 x-1171874\right )-585938000 x-e^{12}-244140000}{2 x^{13}+120 x^{12}+3300 x^{11}+55000 x^{10}+618750 x^9+4950000 x^8+28875000 x^7+123750000 x^6+386718748 x^5+859374960 x^4+1289062200 x^3+1171874000 x^2+e^8 \left (6 x^5+120 x^4+900 x^3+3000 x^2+3750 x\right )+e^4 \left (6 x^9+240 x^8+4200 x^7+42000 x^6+262500 x^5+1050000 x^4+2625000 x^3+3750000 x^2+2343748 x\right )+2 e^{12} x+488280000 x} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-x^{12}-60 x^{11}-1650 x^{10}-27500 x^9-309375 x^8-2475000 x^7-14437500 x^6-61875000 x^5-193359382 x^4-429687600 x^3-644531700 x^2+e^8 \left (-3 x^4-60 x^3-450 x^2-1500 x-1875\right )+e^4 \left (-3 x^8-120 x^7-2100 x^6-21000 x^5-131250 x^4-525000 x^3-1312500 x^2-1875000 x-1171874\right )-585938000 x-e^{12}-244140000}{2 x^{13}+120 x^{12}+3300 x^{11}+55000 x^{10}+618750 x^9+4950000 x^8+28875000 x^7+123750000 x^6+386718748 x^5+859374960 x^4+1289062200 x^3+1171874000 x^2+e^8 \left (6 x^5+120 x^4+900 x^3+3000 x^2+3750 x\right )+e^4 \left (6 x^9+240 x^8+4200 x^7+42000 x^6+262500 x^5+1050000 x^4+2625000 x^3+3750000 x^2+2343748 x\right )+\left (488280000+2 e^{12}\right ) x}dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {-x^{12}-60 x^{11}-1650 x^{10}-27500 x^9-309375 x^8-2475000 x^7-14437500 x^6-61875000 x^5-193359382 x^4-429687600 x^3-644531700 x^2+e^8 \left (-3 x^4-60 x^3-450 x^2-1500 x-1875\right )+e^4 \left (-3 x^8-120 x^7-2100 x^6-21000 x^5-131250 x^4-525000 x^3-1312500 x^2-1875000 x-1171874\right )-585938000 x-e^{12}-244140000}{x \left (2 x^{12}+120 x^{11}+3300 x^{10}+55000 x^9+6 \left (103125+e^4\right ) x^8+240 \left (20625+e^4\right ) x^7+4200 \left (6875+e^4\right ) x^6+6000 \left (20625+7 e^4\right ) x^5+2 \left (193359374+131250 e^4+3 e^8\right ) x^4+120 \left (7161458+8750 e^4+e^8\right ) x^3+300 \left (4296874+8750 e^4+3 e^8\right ) x^2+1000 \left (1171874+3750 e^4+3 e^8\right ) x+2 \left (244140000+1171874 e^4+1875 e^8+e^{12}\right )\right )}dx\)

\(\Big \downarrow \) 2462

\(\displaystyle \int \left (-\frac {2 (x+5)^3}{x^4+20 x^3+150 x^2+500 x+e^4+624}+\frac {4 (x+5)^3}{x^4+20 x^3+150 x^2+500 x+e^4+625}-\frac {2 (x+5)^3}{x^4+20 x^3+150 x^2+500 x+e^4+626}-\frac {1}{2 x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{2} \log \left (x^4+20 x^3+150 x^2+500 x+e^4+624\right )+\log \left (x^4+20 x^3+150 x^2+500 x+e^4+625\right )-\frac {1}{2} \log \left (x^4+20 x^3+150 x^2+500 x+e^4+626\right )-\frac {\log (x)}{2}\)

Input:

Int[(-244140000 - E^12 - 585938000*x - 644531700*x^2 - 429687600*x^3 - 193 
359382*x^4 - 61875000*x^5 - 14437500*x^6 - 2475000*x^7 - 309375*x^8 - 2750 
0*x^9 - 1650*x^10 - 60*x^11 - x^12 + E^8*(-1875 - 1500*x - 450*x^2 - 60*x^ 
3 - 3*x^4) + E^4*(-1171874 - 1875000*x - 1312500*x^2 - 525000*x^3 - 131250 
*x^4 - 21000*x^5 - 2100*x^6 - 120*x^7 - 3*x^8))/(488280000*x + 2*E^12*x + 
1171874000*x^2 + 1289062200*x^3 + 859374960*x^4 + 386718748*x^5 + 12375000 
0*x^6 + 28875000*x^7 + 4950000*x^8 + 618750*x^9 + 55000*x^10 + 3300*x^11 + 
 120*x^12 + 2*x^13 + E^8*(3750*x + 3000*x^2 + 900*x^3 + 120*x^4 + 6*x^5) + 
 E^4*(2343748*x + 3750000*x^2 + 2625000*x^3 + 1050000*x^4 + 262500*x^5 + 4 
2000*x^6 + 4200*x^7 + 240*x^8 + 6*x^9)),x]
 

Output:

-1/2*Log[x] - Log[624 + E^4 + 500*x + 150*x^2 + 20*x^3 + x^4]/2 + Log[625 
+ E^4 + 500*x + 150*x^2 + 20*x^3 + x^4] - Log[626 + E^4 + 500*x + 150*x^2 
+ 20*x^3 + x^4]/2
 

Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

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

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(25)=50\).

Time = 3.68 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61

method	result	size
norman	\(-\frac {\ln \left (x \right )}{2}-\frac {\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +624\right )}{2}-\frac {\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +626\right )}{2}+\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +625\right )\)	\(73\)
parallelrisch	\(-\frac {\ln \left (x \right )}{2}-\frac {\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +624\right )}{2}-\frac {\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +626\right )}{2}+\ln \left (x^{4}+20 x^{3}+150 x^{2}+{\mathrm e}^{4}+500 x +625\right )\)	\(73\)
risch	\(\ln \left (-x^{4}-20 x^{3}-150 x^{2}-{\mathrm e}^{4}-500 x -625\right )-\frac {\ln \left (x^{9}+40 x^{8}+700 x^{7}+7000 x^{6}+\left (2 \,{\mathrm e}^{4}+43750\right ) x^{5}+\left (40 \,{\mathrm e}^{4}+175000\right ) x^{4}+\left (437500+300 \,{\mathrm e}^{4}\right ) x^{3}+\left (1000 \,{\mathrm e}^{4}+625000\right ) x^{2}+\left ({\mathrm e}^{8}+1250 \,{\mathrm e}^{4}+390624\right ) x \right )}{2}\)	\(99\)
default	\(-\frac {\ln \left (x \right )}{2}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{12}+60 \textit {\_Z}^{11}+1650 \textit {\_Z}^{10}+27500 \textit {\_Z}^{9}+\left (3 \,{\mathrm e}^{4}+309375\right ) \textit {\_Z}^{8}+\left (120 \,{\mathrm e}^{4}+2475000\right ) \textit {\_Z}^{7}+\left (2100 \,{\mathrm e}^{4}[phone]\right ) \textit {\_Z}^{6}+\left (21000 \,{\mathrm e}^{4}[phone]\right ) \textit {\_Z}^{5}+\left (131250 \,{\mathrm e}^{4}+3 \,{\mathrm e}^{8}[phone]\right ) \textit {\_Z}^{4}+\left (525000 \,{\mathrm e}^{4}+60 \,{\mathrm e}^{8}[phone]\right ) \textit {\_Z}^{3}+\left (1312500 \,{\mathrm e}^{4}+450 \,{\mathrm e}^{8}[phone]\right ) \textit {\_Z}^{2}+\left (1875000 \,{\mathrm e}^{4}+1500 \,{\mathrm e}^{8}[phone]\right ) \textit {\_Z} +{\mathrm e}^{12}+1875 \,{\mathrm e}^{8}+1171874 \,{\mathrm e}^{4}[phone]\right )}{\sum }\frac {\left (-\textit {\_R}^{3}-15 \textit {\_R}^{2}-75 \textit {\_R} -125\right ) \ln \left (x -\textit {\_R} \right )}{146484250+26250 \textit {\_R}^{4} {\mathrm e}^{4}+225 \textit {\_R} \,{\mathrm e}^{8}+393750 \textit {\_R}^{2} {\mathrm e}^{4}+656250 \textit {\_R} \,{\mathrm e}^{4}+468750 \,{\mathrm e}^{4}+131250 \textit {\_R}^{3} {\mathrm e}^{4}+165 \textit {\_R}^{10}+618750 \textit {\_R}^{7}[phone] \textit {\_R}^{4}+4331250 \textit {\_R}^{6}[phone] \textit {\_R}^{5}+4125 \textit {\_R}^{9}+61875 \textit {\_R}^{8}+3 \textit {\_R}^{11}[phone] \textit {\_R} [phone] \textit {\_R}^{2}[phone] \textit {\_R}^{3}+3 \textit {\_R}^{3} {\mathrm e}^{8}+45 \textit {\_R}^{2} {\mathrm e}^{8}+210 \,{\mathrm e}^{4} \textit {\_R}^{6}+375 \,{\mathrm e}^{8}+6 \,{\mathrm e}^{4} \textit {\_R}^{7}+3150 \textit {\_R}^{5} {\mathrm e}^{4}}\right )\)	\(287\)

Input:

int((-exp(4)^3+(-3*x^4-60*x^3-450*x^2-1500*x-1875)*exp(4)^2+(-3*x^8-120*x^ 
7-2100*x^6-21000*x^5-131250*x^4-525000*x^3-1312500*x^2-1875000*x-1171874)* 
exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-14437500*x^ 
6-61875000*x^5-193359382*x^4-429687600*x^3-644531700*x^2-585938000*x-24414 
0000)/(2*x*exp(4)^3+(6*x^5+120*x^4+900*x^3+3000*x^2+3750*x)*exp(4)^2+(6*x^ 
9+240*x^8+4200*x^7+42000*x^6+262500*x^5+1050000*x^4+2625000*x^3+3750000*x^ 
2+2343748*x)*exp(4)+2*x^13+120*x^12+3300*x^11+55000*x^10+618750*x^9+495000 
0*x^8+28875000*x^7+123750000*x^6+386718748*x^5+859374960*x^4+1289062200*x^ 
3+1171874000*x^2+488280000*x),x,method=_RETURNVERBOSE)
 

Output:

-1/2*ln(x)-1/2*ln(x^4+20*x^3+150*x^2+exp(4)+500*x+624)-1/2*ln(x^4+20*x^3+1 
50*x^2+exp(4)+500*x+626)+ln(x^4+20*x^3+150*x^2+exp(4)+500*x+625)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (24) = 48\).

Time = 0.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.46 \[ \int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx=-\frac {1}{2} \, \log \left (x^{9} + 40 \, x^{8} + 700 \, x^{7} + 7000 \, x^{6} + 43750 \, x^{5} + 175000 \, x^{4} + 437500 \, x^{3} + 625000 \, x^{2} + x e^{8} + 2 \, {\left (x^{5} + 20 \, x^{4} + 150 \, x^{3} + 500 \, x^{2} + 625 \, x\right )} e^{4} + 390624 \, x\right ) + \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 625\right ) \] Input:

integrate((-exp(4)^3+(-3*x^4-60*x^3-450*x^2-1500*x-1875)*exp(4)^2+(-3*x^8- 
120*x^7-2100*x^6-21000*x^5-131250*x^4-525000*x^3-1312500*x^2-1875000*x-117 
1874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-14437 
500*x^6-61875000*x^5-193359382*x^4-429687600*x^3-644531700*x^2-585938000*x 
-244140000)/(2*x*exp(4)^3+(6*x^5+120*x^4+900*x^3+3000*x^2+3750*x)*exp(4)^2 
+(6*x^9+240*x^8+4200*x^7+42000*x^6+262500*x^5+1050000*x^4+2625000*x^3+3750 
000*x^2+2343748*x)*exp(4)+2*x^13+120*x^12+3300*x^11+55000*x^10+618750*x^9+ 
4950000*x^8+28875000*x^7+123750000*x^6+386718748*x^5+859374960*x^4+1289062 
200*x^3+1171874000*x^2+488280000*x),x, algorithm="fricas")
 

Output:

-1/2*log(x^9 + 40*x^8 + 700*x^7 + 7000*x^6 + 43750*x^5 + 175000*x^4 + 4375 
00*x^3 + 625000*x^2 + x*e^8 + 2*(x^5 + 20*x^4 + 150*x^3 + 500*x^2 + 625*x) 
*e^4 + 390624*x) + log(x^4 + 20*x^3 + 150*x^2 + 500*x + e^4 + 625)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (19) = 38\).

Time = 18.55 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.46 \[ \int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx=\log {\left (x^{4} + 20 x^{3} + 150 x^{2} + 500 x + e^{4} + 625 \right )} - \frac {\log {\left (x^{9} + 40 x^{8} + 700 x^{7} + 7000 x^{6} + x^{5} \cdot \left (2 e^{4} + 43750\right ) + x^{4} \cdot \left (40 e^{4} + 175000\right ) + x^{3} \cdot \left (300 e^{4} + 437500\right ) + x^{2} \cdot \left (1000 e^{4} + 625000\right ) + x \left (e^{8} + 1250 e^{4} + 390624\right ) \right )}}{2} \] Input:

integrate((-exp(4)**3+(-3*x**4-60*x**3-450*x**2-1500*x-1875)*exp(4)**2+(-3 
*x**8-120*x**7-2100*x**6-21000*x**5-131250*x**4-525000*x**3-1312500*x**2-1 
875000*x-1171874)*exp(4)-x**12-60*x**11-1650*x**10-27500*x**9-309375*x**8- 
2475000*x**7-14437500*x**6-61875000*x**5-193359382*x**4-429687600*x**3-644 
531700*x**2-585938000*x-244140000)/(2*x*exp(4)**3+(6*x**5+120*x**4+900*x** 
3+3000*x**2+3750*x)*exp(4)**2+(6*x**9+240*x**8+4200*x**7+42000*x**6+262500 
*x**5+1050000*x**4+2625000*x**3+3750000*x**2+2343748*x)*exp(4)+2*x**13+120 
*x**12+3300*x**11+55000*x**10+618750*x**9+4950000*x**8+28875000*x**7+12375 
0000*x**6+386718748*x**5+859374960*x**4+1289062200*x**3+1171874000*x**2+48 
8280000*x),x)
 

Output:

log(x**4 + 20*x**3 + 150*x**2 + 500*x + exp(4) + 625) - log(x**9 + 40*x**8 
 + 700*x**7 + 7000*x**6 + x**5*(2*exp(4) + 43750) + x**4*(40*exp(4) + 1750 
00) + x**3*(300*exp(4) + 437500) + x**2*(1000*exp(4) + 625000) + x*(exp(8) 
 + 1250*exp(4) + 390624))/2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (24) = 48\).

Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.57 \[ \int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx=-\frac {1}{2} \, \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 626\right ) + \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 625\right ) - \frac {1}{2} \, \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 624\right ) - \frac {1}{2} \, \log \left (x\right ) \] Input:

integrate((-exp(4)^3+(-3*x^4-60*x^3-450*x^2-1500*x-1875)*exp(4)^2+(-3*x^8- 
120*x^7-2100*x^6-21000*x^5-131250*x^4-525000*x^3-1312500*x^2-1875000*x-117 
1874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-14437 
500*x^6-61875000*x^5-193359382*x^4-429687600*x^3-644531700*x^2-585938000*x 
-244140000)/(2*x*exp(4)^3+(6*x^5+120*x^4+900*x^3+3000*x^2+3750*x)*exp(4)^2 
+(6*x^9+240*x^8+4200*x^7+42000*x^6+262500*x^5+1050000*x^4+2625000*x^3+3750 
000*x^2+2343748*x)*exp(4)+2*x^13+120*x^12+3300*x^11+55000*x^10+618750*x^9+ 
4950000*x^8+28875000*x^7+123750000*x^6+386718748*x^5+859374960*x^4+1289062 
200*x^3+1171874000*x^2+488280000*x),x, algorithm="maxima")
 

Output:

-1/2*log(x^4 + 20*x^3 + 150*x^2 + 500*x + e^4 + 626) + log(x^4 + 20*x^3 + 
150*x^2 + 500*x + e^4 + 625) - 1/2*log(x^4 + 20*x^3 + 150*x^2 + 500*x + e^ 
4 + 624) - 1/2*log(x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).

Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx=-\frac {1}{2} \, \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 626\right ) + \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 625\right ) - \frac {1}{2} \, \log \left (x^{4} + 20 \, x^{3} + 150 \, x^{2} + 500 \, x + e^{4} + 624\right ) - \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \] Input:

integrate((-exp(4)^3+(-3*x^4-60*x^3-450*x^2-1500*x-1875)*exp(4)^2+(-3*x^8- 
120*x^7-2100*x^6-21000*x^5-131250*x^4-525000*x^3-1312500*x^2-1875000*x-117 
1874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-14437 
500*x^6-61875000*x^5-193359382*x^4-429687600*x^3-644531700*x^2-585938000*x 
-244140000)/(2*x*exp(4)^3+(6*x^5+120*x^4+900*x^3+3000*x^2+3750*x)*exp(4)^2 
+(6*x^9+240*x^8+4200*x^7+42000*x^6+262500*x^5+1050000*x^4+2625000*x^3+3750 
000*x^2+2343748*x)*exp(4)+2*x^13+120*x^12+3300*x^11+55000*x^10+618750*x^9+ 
4950000*x^8+28875000*x^7+123750000*x^6+386718748*x^5+859374960*x^4+1289062 
200*x^3+1171874000*x^2+488280000*x),x, algorithm="giac")
 

Output:

-1/2*log(x^4 + 20*x^3 + 150*x^2 + 500*x + e^4 + 626) + log(x^4 + 20*x^3 + 
150*x^2 + 500*x + e^4 + 625) - 1/2*log(x^4 + 20*x^3 + 150*x^2 + 500*x + e^ 
4 + 624) - 1/2*log(abs(x))
 

Mupad [B] (verification not implemented)

Time = 3.94 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.46 \[ \int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx=\ln \left (x^4+20\,x^3+150\,x^2+500\,x+{\mathrm {e}}^4+625\right )-\frac {\ln \left (x\,\left (625000\,x+1250\,{\mathrm {e}}^4+{\mathrm {e}}^8+1000\,x\,{\mathrm {e}}^4+300\,x^2\,{\mathrm {e}}^4+40\,x^3\,{\mathrm {e}}^4+2\,x^4\,{\mathrm {e}}^4+437500\,x^2+175000\,x^3+43750\,x^4+7000\,x^5+700\,x^6+40\,x^7+x^8+390624\right )\right )}{2} \] Input:

int(-(585938000*x + exp(12) + exp(4)*(1875000*x + 1312500*x^2 + 525000*x^3 
 + 131250*x^4 + 21000*x^5 + 2100*x^6 + 120*x^7 + 3*x^8 + 1171874) + exp(8) 
*(1500*x + 450*x^2 + 60*x^3 + 3*x^4 + 1875) + 644531700*x^2 + 429687600*x^ 
3 + 193359382*x^4 + 61875000*x^5 + 14437500*x^6 + 2475000*x^7 + 309375*x^8 
 + 27500*x^9 + 1650*x^10 + 60*x^11 + x^12 + 244140000)/(488280000*x + 2*x* 
exp(12) + exp(4)*(2343748*x + 3750000*x^2 + 2625000*x^3 + 1050000*x^4 + 26 
2500*x^5 + 42000*x^6 + 4200*x^7 + 240*x^8 + 6*x^9) + exp(8)*(3750*x + 3000 
*x^2 + 900*x^3 + 120*x^4 + 6*x^5) + 1171874000*x^2 + 1289062200*x^3 + 8593 
74960*x^4 + 386718748*x^5 + 123750000*x^6 + 28875000*x^7 + 4950000*x^8 + 6 
18750*x^9 + 55000*x^10 + 3300*x^11 + 120*x^12 + 2*x^13),x)
 

Output:

log(500*x + exp(4) + 150*x^2 + 20*x^3 + x^4 + 625) - log(x*(625000*x + 125 
0*exp(4) + exp(8) + 1000*x*exp(4) + 300*x^2*exp(4) + 40*x^3*exp(4) + 2*x^4 
*exp(4) + 437500*x^2 + 175000*x^3 + 43750*x^4 + 7000*x^5 + 700*x^6 + 40*x^ 
7 + x^8 + 390624))/2
 

Reduce [F]

\[ \int \frac {-244140000-e^{12}-585938000 x-644531700 x^2-429687600 x^3-193359382 x^4-61875000 x^5-14437500 x^6-2475000 x^7-309375 x^8-27500 x^9-1650 x^{10}-60 x^{11}-x^{12}+e^8 \left (-1875-1500 x-450 x^2-60 x^3-3 x^4\right )+e^4 \left (-1171874-1875000 x-1312500 x^2-525000 x^3-131250 x^4-21000 x^5-2100 x^6-120 x^7-3 x^8\right )}{488280000 x+2 e^{12} x+1171874000 x^2+1289062200 x^3+859374960 x^4+386718748 x^5+123750000 x^6+28875000 x^7+4950000 x^8+618750 x^9+55000 x^{10}+3300 x^{11}+120 x^{12}+2 x^{13}+e^8 \left (3750 x+3000 x^2+900 x^3+120 x^4+6 x^5\right )+e^4 \left (2343748 x+3750000 x^2+2625000 x^3+1050000 x^4+262500 x^5+42000 x^6+4200 x^7+240 x^8+6 x^9\right )} \, dx=-4 \left (\int \frac {x^{3}}{e^{12}+3 e^{8} x^{4}+3 e^{4} x^{8}+x^{12}+60 e^{8} x^{3}+120 e^{4} x^{7}+60 x^{11}+450 e^{8} x^{2}+2100 e^{4} x^{6}+1650 x^{10}+1500 e^{8} x +21000 e^{4} x^{5}+27500 x^{9}+1875 e^{8}+131250 e^{4} x^{4}+309375 x^{8}+525000 e^{4} x^{3}+2475000 x^{7}+1312500 e^{4} x^{2}[phone] x^{6}+1875000 e^{4} x [phone] x^{5}+1171874 e^{4}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}d x \right )-60 \left (\int \frac {x^{2}}{e^{12}+3 e^{8} x^{4}+3 e^{4} x^{8}+x^{12}+60 e^{8} x^{3}+120 e^{4} x^{7}+60 x^{11}+450 e^{8} x^{2}+2100 e^{4} x^{6}+1650 x^{10}+1500 e^{8} x +21000 e^{4} x^{5}+27500 x^{9}+1875 e^{8}+131250 e^{4} x^{4}+309375 x^{8}+525000 e^{4} x^{3}+2475000 x^{7}+1312500 e^{4} x^{2}[phone] x^{6}+1875000 e^{4} x [phone] x^{5}+1171874 e^{4}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}d x \right )-300 \left (\int \frac {x}{e^{12}+3 e^{8} x^{4}+3 e^{4} x^{8}+x^{12}+60 e^{8} x^{3}+120 e^{4} x^{7}+60 x^{11}+450 e^{8} x^{2}+2100 e^{4} x^{6}+1650 x^{10}+1500 e^{8} x +21000 e^{4} x^{5}+27500 x^{9}+1875 e^{8}+131250 e^{4} x^{4}+309375 x^{8}+525000 e^{4} x^{3}+2475000 x^{7}+1312500 e^{4} x^{2}[phone] x^{6}+1875000 e^{4} x [phone] x^{5}+1171874 e^{4}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}d x \right )-500 \left (\int \frac {1}{e^{12}+3 e^{8} x^{4}+3 e^{4} x^{8}+x^{12}+60 e^{8} x^{3}+120 e^{4} x^{7}+60 x^{11}+450 e^{8} x^{2}+2100 e^{4} x^{6}+1650 x^{10}+1500 e^{8} x +21000 e^{4} x^{5}+27500 x^{9}+1875 e^{8}+131250 e^{4} x^{4}+309375 x^{8}+525000 e^{4} x^{3}+2475000 x^{7}+1312500 e^{4} x^{2}[phone] x^{6}+1875000 e^{4} x [phone] x^{5}+1171874 e^{4}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}d x \right )-\frac {\mathrm {log}\left (x \right )}{2} \] Input:

int((-exp(4)^3+(-3*x^4-60*x^3-450*x^2-1500*x-1875)*exp(4)^2+(-3*x^8-120*x^ 
7-2100*x^6-21000*x^5-131250*x^4-525000*x^3-1312500*x^2-1875000*x-1171874)* 
exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-14437500*x^ 
6-61875000*x^5-193359382*x^4-429687600*x^3-644531700*x^2-585938000*x-24414 
0000)/(2*x*exp(4)^3+(6*x^5+120*x^4+900*x^3+3000*x^2+3750*x)*exp(4)^2+(6*x^ 
9+240*x^8+4200*x^7+42000*x^6+262500*x^5+1050000*x^4+2625000*x^3+3750000*x^ 
2+2343748*x)*exp(4)+2*x^13+120*x^12+3300*x^11+55000*x^10+618750*x^9+495000 
0*x^8+28875000*x^7+123750000*x^6+386718748*x^5+859374960*x^4+1289062200*x^ 
3+1171874000*x^2+488280000*x),x)
 

Output:

( - 8*int(x**3/(e**12 + 3*e**8*x**4 + 60*e**8*x**3 + 450*e**8*x**2 + 1500* 
e**8*x + 1875*e**8 + 3*e**4*x**8 + 120*e**4*x**7 + 2100*e**4*x**6 + 21000* 
e**4*x**5 + 131250*e**4*x**4 + 525000*e**4*x**3 + 1312500*e**4*x**2 + 1875 
000*e**4*x + 1171874*e**4 + x**12 + 60*x**11 + 1650*x**10 + 27500*x**9 + 3 
09375*x**8 + 2475000*x**7 + 14437500*x**6 + 61875000*x**5 + 193359374*x**4 
 + 429687480*x**3 + 644531100*x**2 + 585937000*x + 244140000),x) - 120*int 
(x**2/(e**12 + 3*e**8*x**4 + 60*e**8*x**3 + 450*e**8*x**2 + 1500*e**8*x + 
1875*e**8 + 3*e**4*x**8 + 120*e**4*x**7 + 2100*e**4*x**6 + 21000*e**4*x**5 
 + 131250*e**4*x**4 + 525000*e**4*x**3 + 1312500*e**4*x**2 + 1875000*e**4* 
x + 1171874*e**4 + x**12 + 60*x**11 + 1650*x**10 + 27500*x**9 + 309375*x** 
8 + 2475000*x**7 + 14437500*x**6 + 61875000*x**5 + 193359374*x**4 + 429687 
480*x**3 + 644531100*x**2 + 585937000*x + 244140000),x) - 600*int(x/(e**12 
 + 3*e**8*x**4 + 60*e**8*x**3 + 450*e**8*x**2 + 1500*e**8*x + 1875*e**8 + 
3*e**4*x**8 + 120*e**4*x**7 + 2100*e**4*x**6 + 21000*e**4*x**5 + 131250*e* 
*4*x**4 + 525000*e**4*x**3 + 1312500*e**4*x**2 + 1875000*e**4*x + 1171874* 
e**4 + x**12 + 60*x**11 + 1650*x**10 + 27500*x**9 + 309375*x**8 + 2475000* 
x**7 + 14437500*x**6 + 61875000*x**5 + 193359374*x**4 + 429687480*x**3 + 6 
44531100*x**2 + 585937000*x + 244140000),x) - 1000*int(1/(e**12 + 3*e**8*x 
**4 + 60*e**8*x**3 + 450*e**8*x**2 + 1500*e**8*x + 1875*e**8 + 3*e**4*x**8 
 + 120*e**4*x**7 + 2100*e**4*x**6 + 21000*e**4*x**5 + 131250*e**4*x**4 ...