3.69.47 $\int \frac{-244140000-e^{12}-585938000x-644531700x^2-429687600x^3-193359382x^4-61875000x^5-14437500x^6-2475000x^7-309375x^8-27500x^9-1650x^{10}-60x^{11}-x^{12}+e^8(-1875-1500x-450x^2-60x^3-3x^4)+e^4(-1171874-1875000x-1312500x^2-525000x^3-131250x^4-21000x^5-2100x^6-120x^7-3x^8)}{488280000x+2e^{12}x+1171874000x^2+1289062200x^3+859374960x^4+386718748x^5+123750000x^6+28875000x^7+4950000x^8+618750x^9+55000x^{10}+3300x^{11}+120x^{12}+2x^{13}+e^8(3750x+3000x^2+900x^3+120x^4+6x^5)+e^4(2343748x+3750000x^2+2625000x^3+1050000x^4+262500x^5+42000x^6+4200x^7+240x^8+6x^9)}dx$

Optimal. Leaf size=28 $$-3+\frac{1}{2}\log \left(\frac{4}{-x+\frac{x}{{(e^4+(5+x{)}^4)}^2}}\right)$$

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Rubi [B]  time = 1.55, antiderivative size = 81, normalized size of antiderivative = 2.89, number of steps used = 6, number of rules used = 3, integrand size = 282, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.011, Rules used = {6, 2074, 1587} $$\begin{array}{c}-\frac{1}{2}\log (x^4+20x^3+150x^2+500x+e^4+624)+\log (x^4+20x^3+150x^2+500x+e^4+625)-\frac{1}{2}\log (x^4+20x^3+150x^2+500x+e^4+626)-\frac{\log (x)}{2}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-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 + 1237
50000*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]

[Out]

-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

Rule 6

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

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-244140000-e^{12}-585938000x-644531700x^2-429687600x^3-193359382x^4-61875000x^5-14437500x^6-2475000x^7-309375x^8-27500x^9-1650x^{10}-60x^{11}-x^{12}+e^8(-1875-1500x-450x^2-60x^3-3x^4)+e^4(-1171874-1875000x-1312500x^2-525000x^3-131250x^4-21000x^5-2100x^6-120x^7-3x^8)}{(488280000+2e^{12})x+1171874000x^2+1289062200x^3+859374960x^4+386718748x^5+123750000x^6+28875000x^7+4950000x^8+618750x^9+55000x^{10}+3300x^{11}+120x^{12}+2x^{13}+e^8(3750x+3000x^2+900x^3+120x^4+6x^5)+e^4(2343748x+3750000x^2+2625000x^3+1050000x^4+262500x^5+42000x^6+4200x^7+240x^8+6x^9)}dx\\ & =\int (-\frac{1}{2x}-\frac{2(5+x{)}^3}{624+e^4+500x+150x^2+20x^3+x^4}+\frac{4(5+x{)}^3}{625+e^4+500x+150x^2+20x^3+x^4}-\frac{2(5+x{)}^3}{626+e^4+500x+150x^2+20x^3+x^4})dx\\ & =-\frac{\log (x)}{2}-2\int \frac{(5+x{)}^3}{624+e^4+500x+150x^2+20x^3+x^4}dx-2\int \frac{(5+x{)}^3}{626+e^4+500x+150x^2+20x^3+x^4}dx+4\int \frac{(5+x{)}^3}{625+e^4+500x+150x^2+20x^3+x^4}dx\\ & =-\frac{\log (x)}{2}-\frac{1}{2}\log (624+e^4+500x+150x^2+20x^3+x^4)+\log (625+e^4+500x+150x^2+20x^3+x^4)-\frac{1}{2}\log (626+e^4+500x+150x^2+20x^3+x^4)\end{array}\end{array}$$

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Mathematica [B]  time = 0.27, size = 112, normalized size = 4.00 $$\begin{array}{c}\frac{1}{2}(-\log (x)+2\log (625+e^4+500x+150x^2+20x^3+x^4)-\log (390624+1250e^4+e^8+625000x+1000e^{4}x+437500x^2+300e^4{x}^2+175000x^3+40e^4{x}^3+43750x^4+2e^4{x}^4+7000x^5+700x^6+40x^7+x^8))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-244140000 - E^12 - 585938000*x - 644531700*x^2 - 429687600*x^3 - 193359382*x^4 - 61875000*x^5 - 14
437500*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 + 26
2500*x^5 + 42000*x^6 + 4200*x^7 + 240*x^8 + 6*x^9)),x]

[Out]

(-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 + 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])/2

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fricas [B]  time = 0.82, size = 97, normalized size = 3.46 $$\begin{array}{c}-\frac{1}{2}\log (x^9+40x^8+700x^7+7000x^6+43750x^5+175000x^4+437500x^3+625000x^2+x{e}^8+2(x^5+20x^4+150x^3+500x^2+625x)e^4+390624x)+\log (x^4+20x^3+150x^2+500x+e^4+625)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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-1171874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-144
37500*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+1
20*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+1237500
00*x^6+386718748*x^5+859374960*x^4+1289062200*x^3+1171874000*x^2+488280000*x),x, algorithm="fricas")

[Out]

-1/2*log(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*(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)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\text{Timed out}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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-1171874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-144
37500*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+1
20*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+1237500
00*x^6+386718748*x^5+859374960*x^4+1289062200*x^3+1171874000*x^2+488280000*x),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 1.34, size = 73, normalized size = 2.61

method	result	size
norman	$-\frac{\ln (x)}{2}-\frac{\ln (x^4+20x^3+150x^2+{\mathrm{e}}^4+500x+624)}{2}-\frac{\ln (x^4+20x^3+150x^2+{\mathrm{e}}^4+500x+626)}{2}+\ln (x^4+20x^3+150x^2+{\mathrm{e}}^4+500x+625)$	$73$
risch	$\ln (-x^4-20x^3-150x^2-{\mathrm{e}}^4-500x-625)-\frac{\ln (x^9+40x^8+700x^7+7000x^6+(2{\mathrm{e}}^4+43750)x^5+(40{\mathrm{e}}^4+175000)x^4+(300{\mathrm{e}}^4+437500)x^3+(1000{\mathrm{e}}^4+625000)x^2+({\mathrm{e}}^8+1250{\mathrm{e}}^4+390624)x)}{2}$	$99$
default	$-\frac{\ln (x)}{2}+(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}({\mathit{\text{\_Z}}}^{12}+60{\mathit{\text{\_Z}}}^{11}+1650{\mathit{\text{\_Z}}}^{10}+27500{\mathit{\text{\_Z}}}^9+(3{\mathrm{e}}^4+309375){\mathit{\text{\_Z}}}^8+(120{\mathrm{e}}^4+2475000){\mathit{\text{\_Z}}}^7+(2100{\mathrm{e}}^4+14437500){\mathit{\text{\_Z}}}^6+(21000{\mathrm{e}}^4+61875000){\mathit{\text{\_Z}}}^5+(3{\mathrm{e}}^8+131250{\mathrm{e}}^4+193359374){\mathit{\text{\_Z}}}^4+(60{\mathrm{e}}^8+525000{\mathrm{e}}^4+429687480){\mathit{\text{\_Z}}}^3+(450{\mathrm{e}}^8+1312500{\mathrm{e}}^4+644531100){\mathit{\text{\_Z}}}^2+(1500{\mathrm{e}}^8+1875000{\mathrm{e}}^4+585937000)\mathit{\text{\_Z}}+{\mathrm{e}}^{12}+1875{\mathrm{e}}^8+1171874{\mathrm{e}}^4+244140000)}\frac{(-{\mathit{\text{\_R}}}^3-15{\mathit{\text{\_R}}}^2-75\mathit{\text{\_R}}-125)\ln (x-\mathit{\text{\_R}})}{146484250+375{\mathrm{e}}^8+225\mathit{\text{\_R}}{\mathrm{e}}^8+468750{\mathrm{e}}^4+165{\mathit{\text{\_R}}}^{10}+3{\mathit{\text{\_R}}}^{11}+4331250{\mathit{\text{\_R}}}^6+618750{\mathit{\text{\_R}}}^7+4125{\mathit{\text{\_R}}}^9+61875{\mathit{\text{\_R}}}^8+21656250{\mathit{\text{\_R}}}^5+77343750{\mathit{\text{\_R}}}^4+193359374{\mathit{\text{\_R}}}^3+322265610{\mathit{\text{\_R}}}^2+322265550\mathit{\text{\_R}}+210{\mathit{\text{\_R}}}^6{\mathrm{e}}^4+6{\mathit{\text{\_R}}}^7{\mathrm{e}}^4+393750{\mathit{\text{\_R}}}^2{\mathrm{e}}^4+656250\mathit{\text{\_R}}{\mathrm{e}}^4+3{\mathit{\text{\_R}}}^3{\mathrm{e}}^8+45{\mathit{\text{\_R}}}^2{\mathrm{e}}^8+131250{\mathit{\text{\_R}}}^3{\mathrm{e}}^4+26250{\mathit{\text{\_R}}}^4{\mathrm{e}}^4+3150{\mathit{\text{\_R}}}^5{\mathrm{e}}^4})$	$287$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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^4-5250
00*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-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+375000
0*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
[phone]*x^5+859374960*x^4+1289062200*x^3+1171874000*x^2+488280000*x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.37, size = 72, normalized size = 2.57 $$\begin{array}{c}-\frac{1}{2}\log (x^4+20x^3+150x^2+500x+e^4+626)+\log (x^4+20x^3+150x^2+500x+e^4+625)-\frac{1}{2}\log (x^4+20x^3+150x^2+500x+e^4+624)-\frac{1}{2}\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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-1171874)*exp(4)-x^12-60*x^11-1650*x^10-27500*x^9-309375*x^8-2475000*x^7-144
37500*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+1
20*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+1237500
00*x^6+386718748*x^5+859374960*x^4+1289062200*x^3+1171874000*x^2+488280000*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.49, size = 97, normalized size = 3.46 $$\begin{array}{c}\ln (x^4+20x^3+150x^2+500x+{\mathrm{e}}^4+625)-\frac{\ln \left(x(625000x+1250{\mathrm{e}}^4+{\mathrm{e}}^8+1000x{\mathrm{e}}^4+300x^2{\mathrm{e}}^4+40x^3{\mathrm{e}}^4+2x^4{\mathrm{e}}^4+437500x^2+175000x^3+43750x^4+7000x^5+700x^6+40x^7+x^8+390624)\right)}{2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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
+ 262500*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 + 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),x)

[Out]

log(500*x + exp(4) + 150*x^2 + 20*x^3 + x^4 + 625) - log(x*(625000*x + 1250*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

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sympy [B]  time = 41.52, size = 97, normalized size = 3.46 $$\begin{array}{c}\log (x^4+20x^3+150x^2+500x+e^4+625)-\frac{\log (x^9+40x^8+700x^7+7000x^6+x^5(2e^4+43750)+x^4(40e^4+175000)+x^3(300e^4+437500)+x^2(1000e^4+625000)+x(e^8+1250e^4+390624))}{2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

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-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-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+26
2500*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+61
8750*x**9+4950000*x**8+28875000*x**7+123750000*x**6+386718748*x**5+859374960*x**4+1289062200*x**3+1171874000*x
**2+488280000*x),x)

[Out]

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*ex
p(4) + 43750) + x**4*(40*exp(4) + 175000) + x**3*(300*exp(4) + 437500) + x**2*(1000*exp(4) + 625000) + x*(exp(
8) + 1250*exp(4) + 390624))/2

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