Integrand size = 237, antiderivative size = 32 \[ \int \frac {-60000 x^3-50300 x^4+6904 x^5-292 x^6+4 x^7+e \left (-30000 x+32450 x^2-3748 x^3+150 x^4-2 x^5\right )+e^2 \left (-28800 x^3+3600 x^4-148 x^5+2 x^6\right )+\left (-62500 x^2-52500 x^3+7100 x^4-296 x^5+4 x^6+e \left (-31250+33750 x-3850 x^2+152 x^3-2 x^4\right )+e^2 \left (-30000 x^2+3700 x^3-150 x^4+2 x^5\right )\right ) \log \left (\frac {e-e^2 x-2 x^2}{x}\right )}{-31250 x^3+3750 x^4-150 x^5+2 x^6+e \left (15625 x-1875 x^2+75 x^3-x^4\right )+e^2 \left (-15625 x^2+1875 x^3-75 x^4+x^5\right )} \, dx=\left (-x+\frac {x}{25-x}-\log \left (-e^2+\frac {e}{x}-2 x\right )\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(32)=64\).
Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.00 \[ \int \frac {-60000 x^3-50300 x^4+6904 x^5-292 x^6+4 x^7+e \left (-30000 x+32450 x^2-3748 x^3+150 x^4-2 x^5\right )+e^2 \left (-28800 x^3+3600 x^4-148 x^5+2 x^6\right )+\left (-62500 x^2-52500 x^3+7100 x^4-296 x^5+4 x^6+e \left (-31250+33750 x-3850 x^2+152 x^3-2 x^4\right )+e^2 \left (-30000 x^2+3700 x^3-150 x^4+2 x^5\right )\right ) \log \left (\frac {e-e^2 x-2 x^2}{x}\right )}{-31250 x^3+3750 x^4-150 x^5+2 x^6+e \left (15625 x-1875 x^2+75 x^3-x^4\right )+e^2 \left (-15625 x^2+1875 x^3-75 x^4+x^5\right )} \, dx=2 \left (\frac {625}{2 (-25+x)^2}+\frac {650}{-25+x}+x+\frac {x^2}{2}+\frac {\left (25-25 x+x^2\right ) \log \left (-e^2+\frac {e}{x}-2 x\right )}{-25+x}+\frac {1}{2} \log ^2\left (-e^2+\frac {e}{x}-2 x\right )-\log (x)+\log \left (-e+e^2 x+2 x^2\right )\right ) \]
Integrate[(-60000*x^3 - 50300*x^4 + 6904*x^5 - 292*x^6 + 4*x^7 + E*(-30000 *x + 32450*x^2 - 3748*x^3 + 150*x^4 - 2*x^5) + E^2*(-28800*x^3 + 3600*x^4 - 148*x^5 + 2*x^6) + (-62500*x^2 - 52500*x^3 + 7100*x^4 - 296*x^5 + 4*x^6 + E*(-31250 + 33750*x - 3850*x^2 + 152*x^3 - 2*x^4) + E^2*(-30000*x^2 + 37 00*x^3 - 150*x^4 + 2*x^5))*Log[(E - E^2*x - 2*x^2)/x])/(-31250*x^3 + 3750* x^4 - 150*x^5 + 2*x^6 + E*(15625*x - 1875*x^2 + 75*x^3 - x^4) + E^2*(-1562 5*x^2 + 1875*x^3 - 75*x^4 + x^5)),x]
2*(625/(2*(-25 + x)^2) + 650/(-25 + x) + x + x^2/2 + ((25 - 25*x + x^2)*Lo g[-E^2 + E/x - 2*x])/(-25 + x) + Log[-E^2 + E/x - 2*x]^2/2 - Log[x] + Log[ -E + E^2*x + 2*x^2])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 15.15 (sec) , antiderivative size = 803, normalized size of antiderivative = 25.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2026, 2463, 7239, 27, 25, 7279, 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.
| \(\displaystyle \int \frac {4 x^7-292 x^6+6904 x^5-50300 x^4-60000 x^3+e^2 \left (2 x^6-148 x^5+3600 x^4-28800 x^3\right )+e \left (-2 x^5+150 x^4-3748 x^3+32450 x^2-30000 x\right )+\left (4 x^6-296 x^5+7100 x^4-52500 x^3-62500 x^2+e \left (-2 x^4+152 x^3-3850 x^2+33750 x-31250\right )+e^2 \left (2 x^5-150 x^4+3700 x^3-30000 x^2\right )\right ) \log \left (\frac {-2 x^2-e^2 x+e}{x}\right )}{2 x^6-150 x^5+3750 x^4-31250 x^3+e \left (-x^4+75 x^3-1875 x^2+15625 x\right )+e^2 \left (x^5-75 x^4+1875 x^3-15625 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {4 x^7-292 x^6+6904 x^5-50300 x^4-60000 x^3+e^2 \left (2 x^6-148 x^5+3600 x^4-28800 x^3\right )+e \left (-2 x^5+150 x^4-3748 x^3+32450 x^2-30000 x\right )+\left (4 x^6-296 x^5+7100 x^4-52500 x^3-62500 x^2+e \left (-2 x^4+152 x^3-3850 x^2+33750 x-31250\right )+e^2 \left (2 x^5-150 x^4+3700 x^3-30000 x^2\right )\right ) \log \left (\frac {-2 x^2-e^2 x+e}{x}\right )}{x \left (2 x^5-\left (150-e^2\right ) x^4+\left (3750-e-75 e^2\right ) x^3-25 \left (1250-3 e-75 e^2\right ) x^2-625 e (3+25 e) x+15625 e\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (7500+2 e+150 e^2+e^4\right ) \left (4 x^7-292 x^6+6904 x^5-50300 x^4-60000 x^3+e \left (-2 x^5+150 x^4-3748 x^3+32450 x^2-30000 x\right )+e^2 \left (2 x^6-148 x^5+3600 x^4-28800 x^3\right )+\left (4 x^6-296 x^5+7100 x^4-52500 x^3-62500 x^2+e \left (-2 x^4+152 x^3-3850 x^2+33750 x-31250\right )+e^2 \left (2 x^5-150 x^4+3700 x^3-30000 x^2\right )\right ) \log \left (\frac {-2 x^2-e^2 x+e}{x}\right )\right )}{\left (1250-e+25 e^2\right )^3 (x-25) x}+\frac {\left (-100-e^2\right ) \left (4 x^7-292 x^6+6904 x^5-50300 x^4-60000 x^3+e \left (-2 x^5+150 x^4-3748 x^3+32450 x^2-30000 x\right )+e^2 \left (2 x^6-148 x^5+3600 x^4-28800 x^3\right )+\left (4 x^6-296 x^5+7100 x^4-52500 x^3-62500 x^2+e \left (-2 x^4+152 x^3-3850 x^2+33750 x-31250\right )+e^2 \left (2 x^5-150 x^4+3700 x^3-30000 x^2\right )\right ) \log \left (\frac {-2 x^2-e^2 x+e}{x}\right )\right )}{\left (1250-e+25 e^2\right )^2 (x-25)^2 x}+\frac {4 x^7-292 x^6+6904 x^5-50300 x^4-60000 x^3+e \left (-2 x^5+150 x^4-3748 x^3+32450 x^2-30000 x\right )+e^2 \left (2 x^6-148 x^5+3600 x^4-28800 x^3\right )+\left (4 x^6-296 x^5+7100 x^4-52500 x^3-62500 x^2+e \left (-2 x^4+152 x^3-3850 x^2+33750 x-31250\right )+e^2 \left (2 x^5-150 x^4+3700 x^3-30000 x^2\right )\right ) \log \left (\frac {-2 x^2-e^2 x+e}{x}\right )}{\left (1250-e+25 e^2\right ) (x-25)^3 x}+\frac {\left (2 \left (7500+2 e+150 e^2+e^4\right ) x+e^6+150 e^4+4 e^3+7500 e^2+300 e+125000\right ) \left (4 x^7-292 x^6+6904 x^5-50300 x^4-60000 x^3+e \left (-2 x^5+150 x^4-3748 x^3+32450 x^2-30000 x\right )+e^2 \left (2 x^6-148 x^5+3600 x^4-28800 x^3\right )+\left (4 x^6-296 x^5+7100 x^4-52500 x^3-62500 x^2+e \left (-2 x^4+152 x^3-3850 x^2+33750 x-31250\right )+e^2 \left (2 x^5-150 x^4+3700 x^3-30000 x^2\right )\right ) \log \left (\frac {-2 x^2-e^2 x+e}{x}\right )\right )}{\left (1250-e+25 e^2\right )^3 x \left (-2 x^2-e^2 x+e\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (e^2 \left (x^2-50 x+600\right ) x^2+2 \left (x^3-49 x^2+550 x+625\right ) x^2-e \left (x^3-51 x^2+650 x-625\right )\right ) \left ((x-24) x+(x-25) \log \left (-2 x+\frac {e}{x}-e^2\right )\right )}{(25-x)^3 x \left (-2 x^2-e^2 x+e\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\left (e^2 \left (x^2-50 x+600\right ) x^2+2 \left (x^3-49 x^2+550 x+625\right ) x^2+e \left (-x^3+51 x^2-650 x+625\right )\right ) \left ((24-x) x+(25-x) \log \left (-2 x-e^2+\frac {e}{x}\right )\right )}{(25-x)^3 x \left (-2 x^2-e^2 x+e\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\left (e^2 \left (x^2-50 x+600\right ) x^2+2 \left (x^3-49 x^2+550 x+625\right ) x^2+e \left (-x^3+51 x^2-650 x+625\right )\right ) \left ((24-x) x+(25-x) \log \left (-2 x-e^2+\frac {e}{x}\right )\right )}{(25-x)^3 x \left (-2 x^2-e^2 x+e\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -2 \int \left (\frac {\log \left (-2 x-e^2+\frac {e}{x}\right ) \left (2 x^5-98 \left (1-\frac {e^2}{98}\right ) x^4+1100 \left (1-\frac {e (1+50 e)}{1100}\right ) x^3+1250 \left (1+\frac {3 e (17+200 e)}{1250}\right ) x^2-650 e x+625 e\right )}{(25-x)^2 x \left (-2 x^2-e^2 x+e\right )}+\frac {(24-x) \left (2 x^5-98 \left (1-\frac {e^2}{98}\right ) x^4+1100 \left (1-\frac {e (1+50 e)}{1100}\right ) x^3+1250 \left (1+\frac {3 e (17+200 e)}{1250}\right ) x^2-650 e x+625 e\right )}{(25-x)^3 \left (-2 x^2-e^2 x+e\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (-\frac {x^2}{2}-\log \left (-2 x-e^2+\frac {e}{x}\right ) x-x+\frac {\log ^2(x)}{2}+\frac {1}{2} \log ^2\left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right )+\frac {1}{2} \log ^2\left (4 x+\sqrt {e \left (8+e^3\right )}+e^2\right )+\frac {25 \sqrt {e \left (8+e^3\right )} \text {arctanh}\left (\frac {4 x+e^2}{\sqrt {e \left (8+e^3\right )}}\right )}{1250-e+25 e^2}-\frac {1}{2} \sqrt {e \left (8+e^3\right )} \text {arctanh}\left (\frac {4 x+e^2}{\sqrt {e \left (8+e^3\right )}}\right )+\frac {(2+e) \left (4-2 e+e^2\right ) \left (1200-e+25 e^2\right ) \sqrt {\frac {e}{8+e^3}} \text {arctanh}\left (\frac {4 x+e^2}{\sqrt {e \left (8+e^3\right )}}\right )}{2 \left (1250-e+25 e^2\right )}+\frac {25 \log \left (-2 x-e^2+\frac {e}{x}\right )}{25-x}+\log \left (-2 x-e^2+\frac {e}{x}\right ) \log (x)-\log \left (e^2+\sqrt {e \left (8+e^3\right )}\right ) \log (x)+\log (x)-\log \left (-2 x-e^2+\frac {e}{x}\right ) \log \left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right )-\log \left (-\frac {4 x}{e^2-\sqrt {e \left (8+e^3\right )}}\right ) \log \left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right )-\log \left (\frac {1}{4} \left (-e^2+\sqrt {e \left (8+e^3\right )}\right )\right ) \log \left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right )-\log \left (-2 x-e^2+\frac {e}{x}\right ) \log \left (4 x+\sqrt {e \left (8+e^3\right )}+e^2\right )+\log \left (-\frac {4 x-\sqrt {e \left (8+e^3\right )}+e^2}{2 \sqrt {e \left (8+e^3\right )}}\right ) \log \left (4 x+\sqrt {e \left (8+e^3\right )}+e^2\right )+\log \left (4 x-\sqrt {e \left (8+e^3\right )}+e^2\right ) \log \left (\frac {4 x+\sqrt {e \left (8+e^3\right )}+e^2}{2 \sqrt {e \left (8+e^3\right )}}\right )-\log (x) \log \left (\frac {4 x}{e^2+\sqrt {e \left (8+e^3\right )}}+1\right )+\frac {e \left (4+1200 e-e^2+25 e^3\right ) \log \left (-2 x^2-e^2 x+e\right )}{4 \left (1250-e+25 e^2\right )}-\frac {25 \left (100+e^2\right ) \log \left (-2 x^2-e^2 x+e\right )}{2 \left (1250-e+25 e^2\right )}-\frac {1}{4} e^2 \log \left (-2 x^2-e^2 x+e\right )+\operatorname {PolyLog}\left (2,-\frac {4 x-\sqrt {e \left (8+e^3\right )}+e^2}{2 \sqrt {e \left (8+e^3\right )}}\right )+\operatorname {PolyLog}\left (2,\frac {4 x+\sqrt {e \left (8+e^3\right )}+e^2}{2 \sqrt {e \left (8+e^3\right )}}\right )+\frac {650}{25-x}-\frac {625}{2 (25-x)^2}\right )\) |
Int[(-60000*x^3 - 50300*x^4 + 6904*x^5 - 292*x^6 + 4*x^7 + E*(-30000*x + 3 2450*x^2 - 3748*x^3 + 150*x^4 - 2*x^5) + E^2*(-28800*x^3 + 3600*x^4 - 148* x^5 + 2*x^6) + (-62500*x^2 - 52500*x^3 + 7100*x^4 - 296*x^5 + 4*x^6 + E*(- 31250 + 33750*x - 3850*x^2 + 152*x^3 - 2*x^4) + E^2*(-30000*x^2 + 3700*x^3 - 150*x^4 + 2*x^5))*Log[(E - E^2*x - 2*x^2)/x])/(-31250*x^3 + 3750*x^4 - 150*x^5 + 2*x^6 + E*(15625*x - 1875*x^2 + 75*x^3 - x^4) + E^2*(-15625*x^2 + 1875*x^3 - 75*x^4 + x^5)),x]
-2*(-625/(2*(25 - x)^2) + 650/(25 - x) - x - x^2/2 + ((2 + E)*(4 - 2*E + E ^2)*(1200 - E + 25*E^2)*Sqrt[E/(8 + E^3)]*ArcTanh[(E^2 + 4*x)/Sqrt[E*(8 + E^3)]])/(2*(1250 - E + 25*E^2)) - (Sqrt[E*(8 + E^3)]*ArcTanh[(E^2 + 4*x)/S qrt[E*(8 + E^3)]])/2 + (25*Sqrt[E*(8 + E^3)]*ArcTanh[(E^2 + 4*x)/Sqrt[E*(8 + E^3)]])/(1250 - E + 25*E^2) + (25*Log[-E^2 + E/x - 2*x])/(25 - x) - x*L og[-E^2 + E/x - 2*x] + Log[x] - Log[E^2 + Sqrt[E*(8 + E^3)]]*Log[x] + Log[ -E^2 + E/x - 2*x]*Log[x] + Log[x]^2/2 - Log[(-E^2 + Sqrt[E*(8 + E^3)])/4]* Log[E^2 - Sqrt[E*(8 + E^3)] + 4*x] - Log[-E^2 + E/x - 2*x]*Log[E^2 - Sqrt[ E*(8 + E^3)] + 4*x] - Log[(-4*x)/(E^2 - Sqrt[E*(8 + E^3)])]*Log[E^2 - Sqrt [E*(8 + E^3)] + 4*x] + Log[E^2 - Sqrt[E*(8 + E^3)] + 4*x]^2/2 - Log[-E^2 + E/x - 2*x]*Log[E^2 + Sqrt[E*(8 + E^3)] + 4*x] + Log[-1/2*(E^2 - Sqrt[E*(8 + E^3)] + 4*x)/Sqrt[E*(8 + E^3)]]*Log[E^2 + Sqrt[E*(8 + E^3)] + 4*x] + Lo g[E^2 + Sqrt[E*(8 + E^3)] + 4*x]^2/2 + Log[E^2 - Sqrt[E*(8 + E^3)] + 4*x]* Log[(E^2 + Sqrt[E*(8 + E^3)] + 4*x)/(2*Sqrt[E*(8 + E^3)])] - Log[x]*Log[1 + (4*x)/(E^2 + Sqrt[E*(8 + E^3)])] - (E^2*Log[E - E^2*x - 2*x^2])/4 - (25* (100 + E^2)*Log[E - E^2*x - 2*x^2])/(2*(1250 - E + 25*E^2)) + (E*(4 + 1200 *E - E^2 + 25*E^3)*Log[E - E^2*x - 2*x^2])/(4*(1250 - E + 25*E^2)) + PolyL og[2, -1/2*(E^2 - Sqrt[E*(8 + E^3)] + 4*x)/Sqrt[E*(8 + E^3)]] + PolyLog[2, (E^2 + Sqrt[E*(8 + E^3)] + 4*x)/(2*Sqrt[E*(8 + E^3)])])
3.14.19.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(155\) vs. \(2(32)=64\).
Time = 3.45 (sec) , antiderivative size = 156, normalized size of antiderivative = 4.88
| method | result | size |
| norman | \(\frac {x^{4}+x^{2} \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right )^{2}+28800 x -98 \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right ) x^{2}+1200 x \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right )-48 x^{3}+625 \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right )^{2}-50 x \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right )^{2}-360000+2 \ln \left (\frac {-{\mathrm e}^{2} x +{\mathrm e}-2 x^{2}}{x}\right ) x^{3}}{\left (x -25\right )^{2}}\) | \(156\) |
| parallelrisch | \(-\frac {\left (-625 x^{4} {\mathrm e}^{2}-1250 \,{\mathrm e}^{2} \ln \left (-\frac {{\mathrm e}^{2} x +2 x^{2}-{\mathrm e}}{x}\right ) x^{3}-625 \,{\mathrm e}^{2} x^{2} \ln \left (-\frac {{\mathrm e}^{2} x +2 x^{2}-{\mathrm e}}{x}\right )^{2}+30000 x^{3} {\mathrm e}^{2}+61250 \,{\mathrm e}^{2} \ln \left (-\frac {{\mathrm e}^{2} x +2 x^{2}-{\mathrm e}}{x}\right ) x^{2}+31250 \,{\mathrm e}^{2} x \ln \left (-\frac {{\mathrm e}^{2} x +2 x^{2}-{\mathrm e}}{x}\right )^{2}-360000 x^{2} {\mathrm e}^{2}-750000 \,{\mathrm e}^{2} \ln \left (-\frac {{\mathrm e}^{2} x +2 x^{2}-{\mathrm e}}{x}\right ) x -390625 \,{\mathrm e}^{2} \ln \left (-\frac {{\mathrm e}^{2} x +2 x^{2}-{\mathrm e}}{x}\right )^{2}\right ) {\mathrm e}^{-2}}{625 \left (x^{2}-50 x +625\right )}\) | \(218\) |
int((((2*x^5-150*x^4+3700*x^3-30000*x^2)*exp(2)+(-2*x^4+152*x^3-3850*x^2+3 3750*x-31250)*exp(1)+4*x^6-296*x^5+7100*x^4-52500*x^3-62500*x^2)*ln((-exp( 2)*x+exp(1)-2*x^2)/x)+(2*x^6-148*x^5+3600*x^4-28800*x^3)*exp(2)+(-2*x^5+15 0*x^4-3748*x^3+32450*x^2-30000*x)*exp(1)+4*x^7-292*x^6+6904*x^5-50300*x^4- 60000*x^3)/((x^5-75*x^4+1875*x^3-15625*x^2)*exp(2)+(-x^4+75*x^3-1875*x^2+1 5625*x)*exp(1)+2*x^6-150*x^5+3750*x^4-31250*x^3),x,method=_RETURNVERBOSE)
(x^4+x^2*ln((-exp(2)*x+exp(1)-2*x^2)/x)^2+28800*x-98*ln((-exp(2)*x+exp(1)- 2*x^2)/x)*x^2+1200*x*ln((-exp(2)*x+exp(1)-2*x^2)/x)-48*x^3+625*ln((-exp(2) *x+exp(1)-2*x^2)/x)^2-50*x*ln((-exp(2)*x+exp(1)-2*x^2)/x)^2-360000+2*ln((- exp(2)*x+exp(1)-2*x^2)/x)*x^3)/(x-25)^2
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.94 \[ \int \frac {-60000 x^3-50300 x^4+6904 x^5-292 x^6+4 x^7+e \left (-30000 x+32450 x^2-3748 x^3+150 x^4-2 x^5\right )+e^2 \left (-28800 x^3+3600 x^4-148 x^5+2 x^6\right )+\left (-62500 x^2-52500 x^3+7100 x^4-296 x^5+4 x^6+e \left (-31250+33750 x-3850 x^2+152 x^3-2 x^4\right )+e^2 \left (-30000 x^2+3700 x^3-150 x^4+2 x^5\right )\right ) \log \left (\frac {e-e^2 x-2 x^2}{x}\right )}{-31250 x^3+3750 x^4-150 x^5+2 x^6+e \left (15625 x-1875 x^2+75 x^3-x^4\right )+e^2 \left (-15625 x^2+1875 x^3-75 x^4+x^5\right )} \, dx=\frac {x^{4} - 48 \, x^{3} + {\left (x^{2} - 50 \, x + 625\right )} \log \left (-\frac {2 \, x^{2} + x e^{2} - e}{x}\right )^{2} + 525 \, x^{2} + 2 \, {\left (x^{3} - 49 \, x^{2} + 600 \, x\right )} \log \left (-\frac {2 \, x^{2} + x e^{2} - e}{x}\right ) + 2550 \, x - 31875}{x^{2} - 50 \, x + 625} \]
integrate((((2*x^5-150*x^4+3700*x^3-30000*x^2)*exp(2)+(-2*x^4+152*x^3-3850 *x^2+33750*x-31250)*exp(1)+4*x^6-296*x^5+7100*x^4-52500*x^3-62500*x^2)*log ((-exp(2)*x+exp(1)-2*x^2)/x)+(2*x^6-148*x^5+3600*x^4-28800*x^3)*exp(2)+(-2 *x^5+150*x^4-3748*x^3+32450*x^2-30000*x)*exp(1)+4*x^7-292*x^6+6904*x^5-503 00*x^4-60000*x^3)/((x^5-75*x^4+1875*x^3-15625*x^2)*exp(2)+(-x^4+75*x^3-187 5*x^2+15625*x)*exp(1)+2*x^6-150*x^5+3750*x^4-31250*x^3),x, algorithm=\
(x^4 - 48*x^3 + (x^2 - 50*x + 625)*log(-(2*x^2 + x*e^2 - e)/x)^2 + 525*x^2 + 2*(x^3 - 49*x^2 + 600*x)*log(-(2*x^2 + x*e^2 - e)/x) + 2550*x - 31875)/ (x^2 - 50*x + 625)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (20) = 40\).
Time = 2.97 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.88 \[ \int \frac {-60000 x^3-50300 x^4+6904 x^5-292 x^6+4 x^7+e \left (-30000 x+32450 x^2-3748 x^3+150 x^4-2 x^5\right )+e^2 \left (-28800 x^3+3600 x^4-148 x^5+2 x^6\right )+\left (-62500 x^2-52500 x^3+7100 x^4-296 x^5+4 x^6+e \left (-31250+33750 x-3850 x^2+152 x^3-2 x^4\right )+e^2 \left (-30000 x^2+3700 x^3-150 x^4+2 x^5\right )\right ) \log \left (\frac {e-e^2 x-2 x^2}{x}\right )}{-31250 x^3+3750 x^4-150 x^5+2 x^6+e \left (15625 x-1875 x^2+75 x^3-x^4\right )+e^2 \left (-15625 x^2+1875 x^3-75 x^4+x^5\right )} \, dx=x^{2} + 2 x + \frac {1300 x - 31875}{x^{2} - 50 x + 625} - 2 \log {\left (x \right )} + \log {\left (\frac {- 2 x^{2} - x e^{2} + e}{x} \right )}^{2} + 2 \log {\left (x^{2} + \frac {x e^{2}}{2} - \frac {e}{2} \right )} + \frac {\left (2 x^{2} - 50 x + 50\right ) \log {\left (\frac {- 2 x^{2} - x e^{2} + e}{x} \right )}}{x - 25} \]
integrate((((2*x**5-150*x**4+3700*x**3-30000*x**2)*exp(2)+(-2*x**4+152*x** 3-3850*x**2+33750*x-31250)*exp(1)+4*x**6-296*x**5+7100*x**4-52500*x**3-625 00*x**2)*ln((-exp(2)*x+exp(1)-2*x**2)/x)+(2*x**6-148*x**5+3600*x**4-28800* x**3)*exp(2)+(-2*x**5+150*x**4-3748*x**3+32450*x**2-30000*x)*exp(1)+4*x**7 -292*x**6+6904*x**5-50300*x**4-60000*x**3)/((x**5-75*x**4+1875*x**3-15625* x**2)*exp(2)+(-x**4+75*x**3-1875*x**2+15625*x)*exp(1)+2*x**6-150*x**5+3750 *x**4-31250*x**3),x)
x**2 + 2*x + (1300*x - 31875)/(x**2 - 50*x + 625) - 2*log(x) + log((-2*x** 2 - x*exp(2) + E)/x)**2 + 2*log(x**2 + x*exp(2)/2 - E/2) + (2*x**2 - 50*x + 50)*log((-2*x**2 - x*exp(2) + E)/x)/(x - 25)
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (26) = 52\).
Time = 0.48 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.84 \[ \int \frac {-60000 x^3-50300 x^4+6904 x^5-292 x^6+4 x^7+e \left (-30000 x+32450 x^2-3748 x^3+150 x^4-2 x^5\right )+e^2 \left (-28800 x^3+3600 x^4-148 x^5+2 x^6\right )+\left (-62500 x^2-52500 x^3+7100 x^4-296 x^5+4 x^6+e \left (-31250+33750 x-3850 x^2+152 x^3-2 x^4\right )+e^2 \left (-30000 x^2+3700 x^3-150 x^4+2 x^5\right )\right ) \log \left (\frac {e-e^2 x-2 x^2}{x}\right )}{-31250 x^3+3750 x^4-150 x^5+2 x^6+e \left (15625 x-1875 x^2+75 x^3-x^4\right )+e^2 \left (-15625 x^2+1875 x^3-75 x^4+x^5\right )} \, dx=\frac {x^{4} - 48 \, x^{3} + {\left (x^{2} - 50 \, x + 625\right )} \log \left (-2 \, x^{2} - x e^{2} + e\right )^{2} + {\left (x^{2} - 50 \, x + 625\right )} \log \left (x\right )^{2} + 525 \, x^{2} + 2 \, {\left (x^{3} - 49 \, x^{2} - {\left (x^{2} - 50 \, x + 625\right )} \log \left (x\right ) + 600 \, x\right )} \log \left (-2 \, x^{2} - x e^{2} + e\right ) - 2 \, {\left (x^{3} - 49 \, x^{2} + 600 \, x\right )} \log \left (x\right ) + 2550 \, x - 31875}{x^{2} - 50 \, x + 625} \]
integrate((((2*x^5-150*x^4+3700*x^3-30000*x^2)*exp(2)+(-2*x^4+152*x^3-3850 *x^2+33750*x-31250)*exp(1)+4*x^6-296*x^5+7100*x^4-52500*x^3-62500*x^2)*log ((-exp(2)*x+exp(1)-2*x^2)/x)+(2*x^6-148*x^5+3600*x^4-28800*x^3)*exp(2)+(-2 *x^5+150*x^4-3748*x^3+32450*x^2-30000*x)*exp(1)+4*x^7-292*x^6+6904*x^5-503 00*x^4-60000*x^3)/((x^5-75*x^4+1875*x^3-15625*x^2)*exp(2)+(-x^4+75*x^3-187 5*x^2+15625*x)*exp(1)+2*x^6-150*x^5+3750*x^4-31250*x^3),x, algorithm=\
(x^4 - 48*x^3 + (x^2 - 50*x + 625)*log(-2*x^2 - x*e^2 + e)^2 + (x^2 - 50*x + 625)*log(x)^2 + 525*x^2 + 2*(x^3 - 49*x^2 - (x^2 - 50*x + 625)*log(x) + 600*x)*log(-2*x^2 - x*e^2 + e) - 2*(x^3 - 49*x^2 + 600*x)*log(x) + 2550*x - 31875)/(x^2 - 50*x + 625)
\[ \int \frac {-60000 x^3-50300 x^4+6904 x^5-292 x^6+4 x^7+e \left (-30000 x+32450 x^2-3748 x^3+150 x^4-2 x^5\right )+e^2 \left (-28800 x^3+3600 x^4-148 x^5+2 x^6\right )+\left (-62500 x^2-52500 x^3+7100 x^4-296 x^5+4 x^6+e \left (-31250+33750 x-3850 x^2+152 x^3-2 x^4\right )+e^2 \left (-30000 x^2+3700 x^3-150 x^4+2 x^5\right )\right ) \log \left (\frac {e-e^2 x-2 x^2}{x}\right )}{-31250 x^3+3750 x^4-150 x^5+2 x^6+e \left (15625 x-1875 x^2+75 x^3-x^4\right )+e^2 \left (-15625 x^2+1875 x^3-75 x^4+x^5\right )} \, dx=\int { \frac {2 \, {\left (2 \, x^{7} - 146 \, x^{6} + 3452 \, x^{5} - 25150 \, x^{4} - 30000 \, x^{3} + {\left (x^{6} - 74 \, x^{5} + 1800 \, x^{4} - 14400 \, x^{3}\right )} e^{2} - {\left (x^{5} - 75 \, x^{4} + 1874 \, x^{3} - 16225 \, x^{2} + 15000 \, x\right )} e + {\left (2 \, x^{6} - 148 \, x^{5} + 3550 \, x^{4} - 26250 \, x^{3} - 31250 \, x^{2} + {\left (x^{5} - 75 \, x^{4} + 1850 \, x^{3} - 15000 \, x^{2}\right )} e^{2} - {\left (x^{4} - 76 \, x^{3} + 1925 \, x^{2} - 16875 \, x + 15625\right )} e\right )} \log \left (-\frac {2 \, x^{2} + x e^{2} - e}{x}\right )\right )}}{2 \, x^{6} - 150 \, x^{5} + 3750 \, x^{4} - 31250 \, x^{3} + {\left (x^{5} - 75 \, x^{4} + 1875 \, x^{3} - 15625 \, x^{2}\right )} e^{2} - {\left (x^{4} - 75 \, x^{3} + 1875 \, x^{2} - 15625 \, x\right )} e} \,d x } \]
integrate((((2*x^5-150*x^4+3700*x^3-30000*x^2)*exp(2)+(-2*x^4+152*x^3-3850 *x^2+33750*x-31250)*exp(1)+4*x^6-296*x^5+7100*x^4-52500*x^3-62500*x^2)*log ((-exp(2)*x+exp(1)-2*x^2)/x)+(2*x^6-148*x^5+3600*x^4-28800*x^3)*exp(2)+(-2 *x^5+150*x^4-3748*x^3+32450*x^2-30000*x)*exp(1)+4*x^7-292*x^6+6904*x^5-503 00*x^4-60000*x^3)/((x^5-75*x^4+1875*x^3-15625*x^2)*exp(2)+(-x^4+75*x^3-187 5*x^2+15625*x)*exp(1)+2*x^6-150*x^5+3750*x^4-31250*x^3),x, algorithm=\
integrate(2*(2*x^7 - 146*x^6 + 3452*x^5 - 25150*x^4 - 30000*x^3 + (x^6 - 7 4*x^5 + 1800*x^4 - 14400*x^3)*e^2 - (x^5 - 75*x^4 + 1874*x^3 - 16225*x^2 + 15000*x)*e + (2*x^6 - 148*x^5 + 3550*x^4 - 26250*x^3 - 31250*x^2 + (x^5 - 75*x^4 + 1850*x^3 - 15000*x^2)*e^2 - (x^4 - 76*x^3 + 1925*x^2 - 16875*x + 15625)*e)*log(-(2*x^2 + x*e^2 - e)/x))/(2*x^6 - 150*x^5 + 3750*x^4 - 3125 0*x^3 + (x^5 - 75*x^4 + 1875*x^3 - 15625*x^2)*e^2 - (x^4 - 75*x^3 + 1875*x ^2 - 15625*x)*e), x)
Time = 14.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.06 \[ \int \frac {-60000 x^3-50300 x^4+6904 x^5-292 x^6+4 x^7+e \left (-30000 x+32450 x^2-3748 x^3+150 x^4-2 x^5\right )+e^2 \left (-28800 x^3+3600 x^4-148 x^5+2 x^6\right )+\left (-62500 x^2-52500 x^3+7100 x^4-296 x^5+4 x^6+e \left (-31250+33750 x-3850 x^2+152 x^3-2 x^4\right )+e^2 \left (-30000 x^2+3700 x^3-150 x^4+2 x^5\right )\right ) \log \left (\frac {e-e^2 x-2 x^2}{x}\right )}{-31250 x^3+3750 x^4-150 x^5+2 x^6+e \left (15625 x-1875 x^2+75 x^3-x^4\right )+e^2 \left (-15625 x^2+1875 x^3-75 x^4+x^5\right )} \, dx=2\,x-48\,\ln \left (x^2+\frac {{\mathrm {e}}^2\,x}{2}-\frac {\mathrm {e}}{2}\right )+48\,\ln \left (x\right )+\frac {1300\,x-31875}{x^2-50\,x+625}+{\ln \left (-\frac {2\,x^2+{\mathrm {e}}^2\,x-\mathrm {e}}{x}\right )}^2+x^2+\frac {\ln \left (-\frac {2\,x^2+{\mathrm {e}}^2\,x-\mathrm {e}}{x}\right )\,\left (x^2-600\right )}{\frac {x}{2}-\frac {25}{2}} \]
int(-(log(-(x*exp(2) - exp(1) + 2*x^2)/x)*(exp(1)*(3850*x^2 - 33750*x - 15 2*x^3 + 2*x^4 + 31250) + 62500*x^2 + 52500*x^3 - 7100*x^4 + 296*x^5 - 4*x^ 6 + exp(2)*(30000*x^2 - 3700*x^3 + 150*x^4 - 2*x^5)) + exp(1)*(30000*x - 3 2450*x^2 + 3748*x^3 - 150*x^4 + 2*x^5) + 60000*x^3 + 50300*x^4 - 6904*x^5 + 292*x^6 - 4*x^7 + exp(2)*(28800*x^3 - 3600*x^4 + 148*x^5 - 2*x^6))/(exp( 1)*(15625*x - 1875*x^2 + 75*x^3 - x^4) - 31250*x^3 + 3750*x^4 - 150*x^5 + 2*x^6 - exp(2)*(15625*x^2 - 1875*x^3 + 75*x^4 - x^5)),x)