Integrand size = 143, antiderivative size = 25 \[ \int \frac {50 x^2+20 x^3-10 x^4+e^4 \left (-100 x-30 x^2\right )+e^2 \left (-100 x^2-40 x^3\right )+\left (-10 x^2-4 x^3+2 x^4+e^4 \left (20 x+6 x^2\right )+e^2 \left (20 x^2+8 x^3\right )\right ) \log (9)}{e^8+4 e^6 x+x^2-2 x^3+x^4+e^4 \left (-2 x+6 x^2\right )+e^2 \left (-4 x^2+4 x^3\right )} \, dx=\frac {2 x^2 (5+x) (-5+\log (9))}{-x+\left (e^2+x\right )^2} \]
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {50 x^2+20 x^3-10 x^4+e^4 \left (-100 x-30 x^2\right )+e^2 \left (-100 x^2-40 x^3\right )+\left (-10 x^2-4 x^3+2 x^4+e^4 \left (20 x+6 x^2\right )+e^2 \left (20 x^2+8 x^3\right )\right ) \log (9)}{e^8+4 e^6 x+x^2-2 x^3+x^4+e^4 \left (-2 x+6 x^2\right )+e^2 \left (-4 x^2+4 x^3\right )} \, dx=2 \left (x+\frac {2 e^6+3 e^4 (-2+x)+6 x-14 e^2 x}{e^4+2 e^2 x+(-1+x) x}\right ) (-5+\log (9)) \]
Integrate[(50*x^2 + 20*x^3 - 10*x^4 + E^4*(-100*x - 30*x^2) + E^2*(-100*x^ 2 - 40*x^3) + (-10*x^2 - 4*x^3 + 2*x^4 + E^4*(20*x + 6*x^2) + E^2*(20*x^2 + 8*x^3))*Log[9])/(E^8 + 4*E^6*x + x^2 - 2*x^3 + x^4 + E^4*(-2*x + 6*x^2) + E^2*(-4*x^2 + 4*x^3)),x]
2*(x + (2*E^6 + 3*E^4*(-2 + x) + 6*x - 14*E^2*x)/(E^4 + 2*E^2*x + (-1 + x) *x))*(-5 + Log[9])
Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(25)=50\).
Time = 0.57 (sec) , antiderivative size = 125, normalized size of antiderivative = 5.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2459, 1380, 27, 2345, 27, 24}
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.
| \(\displaystyle \int \frac {-10 x^4+20 x^3+50 x^2+e^4 \left (-30 x^2-100 x\right )+e^2 \left (-40 x^3-100 x^2\right )+\left (2 x^4-4 x^3-10 x^2+e^4 \left (6 x^2+20 x\right )+e^2 \left (8 x^3+20 x^2\right )\right ) \log (9)}{x^4-2 x^3+x^2+e^4 \left (6 x^2-2 x\right )+e^2 \left (4 x^3-4 x^2\right )+4 e^6 x+e^8} \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {-2 \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^4 (5-\log (9))+\left (13-32 e^2+6 e^4\right ) \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^2 (5-\log (9))+2 \left (6-26 e^2+19 e^4-2 e^6\right ) \left (x+\frac {1}{4} \left (4 e^2-2\right )\right ) (5-\log (9))+\frac {1}{8} \left (23-144 e^2+220 e^4-48 e^6\right ) (5-\log (9))}{\left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^4-\frac {1}{2} \left (1-4 e^2\right ) \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^2+\frac {1}{16} \left (1-4 e^2\right )^2}d\left (x+\frac {1}{4} \left (4 e^2-2\right )\right )\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle \int \frac {2 \left (-16 \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^4 (5-\log (9))+8 \left (13-32 e^2+6 e^4\right ) \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^2 (5-\log (9))+16 \left (6-26 e^2+19 e^4-2 e^6\right ) \left (x+\frac {1}{4} \left (4 e^2-2\right )\right ) (5-\log (9))+\left (23-144 e^2+220 e^4-48 e^6\right ) (5-\log (9))\right )}{\left (-4 \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^2-4 e^2+1\right )^2}d\left (x+\frac {1}{4} \left (4 e^2-2\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {-16 (5-\log (9)) \left (x+\frac {1}{4} \left (-2+4 e^2\right )\right )^4+8 \left (13-32 e^2+6 e^4\right ) (5-\log (9)) \left (x+\frac {1}{4} \left (-2+4 e^2\right )\right )^2+16 \left (6-26 e^2+19 e^4-2 e^6\right ) (5-\log (9)) \left (x+\frac {1}{4} \left (-2+4 e^2\right )\right )+\left (23-144 e^2+220 e^4-48 e^6\right ) (5-\log (9))}{\left (-4 \left (x+\frac {1}{4} \left (-2+4 e^2\right )\right )^2-4 e^2+1\right )^2}d\left (x+\frac {1}{4} \left (-2+4 e^2\right )\right )\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle 2 \left (\frac {2 \left (2 \left (6-38 e^2+59 e^4-12 e^6\right ) \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )+\left (1-4 e^2\right ) \left (6-26 e^2+19 e^4-2 e^6\right )\right ) (5-\log (9))}{\left (1-4 e^2\right ) \left (-4 \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^2-4 e^2+1\right )}-\frac {\int \frac {2 \left (\left (1-4 e^2\right )^2-4 \left (1-4 e^2\right ) \left (x+\frac {1}{4} \left (-2+4 e^2\right )\right )^2\right ) (5-\log (9))}{-4 \left (x+\frac {1}{4} \left (-2+4 e^2\right )\right )^2-4 e^2+1}d\left (x+\frac {1}{4} \left (-2+4 e^2\right )\right )}{2 \left (1-4 e^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {2 \left (2 \left (6-38 e^2+59 e^4-12 e^6\right ) \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )+\left (1-4 e^2\right ) \left (6-26 e^2+19 e^4-2 e^6\right )\right ) (5-\log (9))}{\left (1-4 e^2\right ) \left (-4 \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^2-4 e^2+1\right )}-\frac {(5-\log (9)) \int \left (1-4 e^2\right )d\left (x+\frac {1}{4} \left (-2+4 e^2\right )\right )}{1-4 e^2}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle 2 \left (\frac {2 \left (2 \left (6-38 e^2+59 e^4-12 e^6\right ) \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )+\left (1-4 e^2\right ) \left (6-26 e^2+19 e^4-2 e^6\right )\right ) (5-\log (9))}{\left (1-4 e^2\right ) \left (-4 \left (x+\frac {1}{4} \left (4 e^2-2\right )\right )^2-4 e^2+1\right )}-\left (x+\frac {1}{4} \left (4 e^2-2\right )\right ) (5-\log (9))\right )\) |
Int[(50*x^2 + 20*x^3 - 10*x^4 + E^4*(-100*x - 30*x^2) + E^2*(-100*x^2 - 40 *x^3) + (-10*x^2 - 4*x^3 + 2*x^4 + E^4*(20*x + 6*x^2) + E^2*(20*x^2 + 8*x^ 3))*Log[9])/(E^8 + 4*E^6*x + x^2 - 2*x^3 + x^4 + E^4*(-2*x + 6*x^2) + E^2* (-4*x^2 + 4*x^3)),x]
2*(-(((-2 + 4*E^2)/4 + x)*(5 - Log[9])) + (2*((1 - 4*E^2)*(6 - 26*E^2 + 19 *E^4 - 2*E^6) + 2*(6 - 38*E^2 + 59*E^4 - 12*E^6)*((-2 + 4*E^2)/4 + x))*(5 - Log[9]))/((1 - 4*E^2)*(1 - 4*E^2 - 4*((-2 + 4*E^2)/4 + x)^2)))
3.10.11.3.1 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[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Time = 0.41 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88
| method | result | size |
| gosper | \(-\frac {2 \left (-x^{3}+5 \,{\mathrm e}^{4}+10 \,{\mathrm e}^{2} x -5 x \right ) \left (2 \ln \left (3\right )-5\right )}{{\mathrm e}^{4}+2 \,{\mathrm e}^{2} x +x^{2}-x}\) | \(47\) |
| norman | \(\frac {-10 \left (2 \,{\mathrm e}^{2}-1\right ) \left (2 \ln \left (3\right )-5\right ) x +\left (4 \ln \left (3\right )-10\right ) x^{3}-10 \,{\mathrm e}^{4} \left (2 \ln \left (3\right )-5\right )}{{\mathrm e}^{4}+2 \,{\mathrm e}^{2} x +x^{2}-x}\) | \(58\) |
| parallelrisch | \(-\frac {-4 x^{3} \ln \left (3\right )+20 \,{\mathrm e}^{4} \ln \left (3\right )+40 x \,{\mathrm e}^{2} \ln \left (3\right )+10 x^{3}-50 \,{\mathrm e}^{4}-100 \,{\mathrm e}^{2} x -20 x \ln \left (3\right )+50 x}{{\mathrm e}^{4}+2 \,{\mathrm e}^{2} x +x^{2}-x}\) | \(68\) |
| risch | \(4 x \ln \left (3\right )-10 x +\frac {\left (12 \,{\mathrm e}^{4} \ln \left (3\right )-30 \,{\mathrm e}^{4}-56 \,{\mathrm e}^{2} \ln \left (3\right )+140 \,{\mathrm e}^{2}+24 \ln \left (3\right )-60\right ) x +4 \left (2 \,{\mathrm e}^{2} \ln \left (3\right )-5 \,{\mathrm e}^{2}-6 \ln \left (3\right )+15\right ) {\mathrm e}^{4}}{{\mathrm e}^{4}+2 \,{\mathrm e}^{2} x +x^{2}-x}\) | \(76\) |
| default | \(\left (4 \ln \left (3\right )-10\right ) \left (x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\left (4 \,{\mathrm e}^{2}-2\right ) \textit {\_Z}^{3}+\left (-4 \,{\mathrm e}^{2}+6 \,{\mathrm e}^{4}+1\right ) \textit {\_Z}^{2}+\left (-2 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{6}\right ) \textit {\_Z} +{\mathrm e}^{8}\right )}{\sum }\frac {\left (\left (14 \,{\mathrm e}^{2}-3 \,{\mathrm e}^{4}-6\right ) \textit {\_R}^{2}+4 \left (3 \,{\mathrm e}^{4}-{\mathrm e}^{6}\right ) \textit {\_R} -{\mathrm e}^{8}\right ) \ln \left (x -\textit {\_R} \right )}{2 \,{\mathrm e}^{6}+6 \textit {\_R} \,{\mathrm e}^{4}+6 \textit {\_R}^{2} {\mathrm e}^{2}+2 \textit {\_R}^{3}-{\mathrm e}^{4}-4 \textit {\_R} \,{\mathrm e}^{2}-3 \textit {\_R}^{2}+\textit {\_R}}\right )}{2}\right )\) | \(134\) |
int((2*((6*x^2+20*x)*exp(2)^2+(8*x^3+20*x^2)*exp(2)+2*x^4-4*x^3-10*x^2)*ln (3)+(-30*x^2-100*x)*exp(2)^2+(-40*x^3-100*x^2)*exp(2)-10*x^4+20*x^3+50*x^2 )/(exp(2)^4+4*x*exp(2)^3+(6*x^2-2*x)*exp(2)^2+(4*x^3-4*x^2)*exp(2)+x^4-2*x ^3+x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (26) = 52\).
Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 3.84 \[ \int \frac {50 x^2+20 x^3-10 x^4+e^4 \left (-100 x-30 x^2\right )+e^2 \left (-100 x^2-40 x^3\right )+\left (-10 x^2-4 x^3+2 x^4+e^4 \left (20 x+6 x^2\right )+e^2 \left (20 x^2+8 x^3\right )\right ) \log (9)}{e^8+4 e^6 x+x^2-2 x^3+x^4+e^4 \left (-2 x+6 x^2\right )+e^2 \left (-4 x^2+4 x^3\right )} \, dx=-\frac {2 \, {\left (5 \, x^{3} - 5 \, x^{2} + 10 \, {\left (2 \, x - 3\right )} e^{4} + 10 \, {\left (x^{2} - 7 \, x\right )} e^{2} - 2 \, {\left (x^{3} - x^{2} + 2 \, {\left (2 \, x - 3\right )} e^{4} + 2 \, {\left (x^{2} - 7 \, x\right )} e^{2} + 6 \, x + 2 \, e^{6}\right )} \log \left (3\right ) + 30 \, x + 10 \, e^{6}\right )}}{x^{2} + 2 \, x e^{2} - x + e^{4}} \]
integrate((2*((6*x^2+20*x)*exp(2)^2+(8*x^3+20*x^2)*exp(2)+2*x^4-4*x^3-10*x ^2)*log(3)+(-30*x^2-100*x)*exp(2)^2+(-40*x^3-100*x^2)*exp(2)-10*x^4+20*x^3 +50*x^2)/(exp(2)^4+4*x*exp(2)^3+(6*x^2-2*x)*exp(2)^2+(4*x^3-4*x^2)*exp(2)+ x^4-2*x^3+x^2),x, algorithm=\
-2*(5*x^3 - 5*x^2 + 10*(2*x - 3)*e^4 + 10*(x^2 - 7*x)*e^2 - 2*(x^3 - x^2 + 2*(2*x - 3)*e^4 + 2*(x^2 - 7*x)*e^2 + 6*x + 2*e^6)*log(3) + 30*x + 10*e^6 )/(x^2 + 2*x*e^2 - x + e^4)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (20) = 40\).
Time = 2.55 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.48 \[ \int \frac {50 x^2+20 x^3-10 x^4+e^4 \left (-100 x-30 x^2\right )+e^2 \left (-100 x^2-40 x^3\right )+\left (-10 x^2-4 x^3+2 x^4+e^4 \left (20 x+6 x^2\right )+e^2 \left (20 x^2+8 x^3\right )\right ) \log (9)}{e^8+4 e^6 x+x^2-2 x^3+x^4+e^4 \left (-2 x+6 x^2\right )+e^2 \left (-4 x^2+4 x^3\right )} \, dx=- x \left (10 - 4 \log {\left (3 \right )}\right ) - \frac {x \left (- 140 e^{2} - 12 e^{4} \log {\left (3 \right )} - 24 \log {\left (3 \right )} + 60 + 56 e^{2} \log {\left (3 \right )} + 30 e^{4}\right ) - 8 e^{6} \log {\left (3 \right )} - 60 e^{4} + 24 e^{4} \log {\left (3 \right )} + 20 e^{6}}{x^{2} + x \left (-1 + 2 e^{2}\right ) + e^{4}} \]
integrate((2*((6*x**2+20*x)*exp(2)**2+(8*x**3+20*x**2)*exp(2)+2*x**4-4*x** 3-10*x**2)*ln(3)+(-30*x**2-100*x)*exp(2)**2+(-40*x**3-100*x**2)*exp(2)-10* x**4+20*x**3+50*x**2)/(exp(2)**4+4*x*exp(2)**3+(6*x**2-2*x)*exp(2)**2+(4*x **3-4*x**2)*exp(2)+x**4-2*x**3+x**2),x)
-x*(10 - 4*log(3)) - (x*(-140*exp(2) - 12*exp(4)*log(3) - 24*log(3) + 60 + 56*exp(2)*log(3) + 30*exp(4)) - 8*exp(6)*log(3) - 60*exp(4) + 24*exp(4)*l og(3) + 20*exp(6))/(x**2 + x*(-1 + 2*exp(2)) + exp(4))
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (26) = 52\).
Time = 0.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.96 \[ \int \frac {50 x^2+20 x^3-10 x^4+e^4 \left (-100 x-30 x^2\right )+e^2 \left (-100 x^2-40 x^3\right )+\left (-10 x^2-4 x^3+2 x^4+e^4 \left (20 x+6 x^2\right )+e^2 \left (20 x^2+8 x^3\right )\right ) \log (9)}{e^8+4 e^6 x+x^2-2 x^3+x^4+e^4 \left (-2 x+6 x^2\right )+e^2 \left (-4 x^2+4 x^3\right )} \, dx=2 \, x {\left (2 \, \log \left (3\right ) - 5\right )} + \frac {2 \, {\left ({\left (2 \, {\left (3 \, e^{4} - 14 \, e^{2} + 6\right )} \log \left (3\right ) - 15 \, e^{4} + 70 \, e^{2} - 30\right )} x + 4 \, {\left (e^{6} - 3 \, e^{4}\right )} \log \left (3\right ) - 10 \, e^{6} + 30 \, e^{4}\right )}}{x^{2} + x {\left (2 \, e^{2} - 1\right )} + e^{4}} \]
integrate((2*((6*x^2+20*x)*exp(2)^2+(8*x^3+20*x^2)*exp(2)+2*x^4-4*x^3-10*x ^2)*log(3)+(-30*x^2-100*x)*exp(2)^2+(-40*x^3-100*x^2)*exp(2)-10*x^4+20*x^3 +50*x^2)/(exp(2)^4+4*x*exp(2)^3+(6*x^2-2*x)*exp(2)^2+(4*x^3-4*x^2)*exp(2)+ x^4-2*x^3+x^2),x, algorithm=\
2*x*(2*log(3) - 5) + 2*((2*(3*e^4 - 14*e^2 + 6)*log(3) - 15*e^4 + 70*e^2 - 30)*x + 4*(e^6 - 3*e^4)*log(3) - 10*e^6 + 30*e^4)/(x^2 + x*(2*e^2 - 1) + e^4)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.20 \[ \int \frac {50 x^2+20 x^3-10 x^4+e^4 \left (-100 x-30 x^2\right )+e^2 \left (-100 x^2-40 x^3\right )+\left (-10 x^2-4 x^3+2 x^4+e^4 \left (20 x+6 x^2\right )+e^2 \left (20 x^2+8 x^3\right )\right ) \log (9)}{e^8+4 e^6 x+x^2-2 x^3+x^4+e^4 \left (-2 x+6 x^2\right )+e^2 \left (-4 x^2+4 x^3\right )} \, dx=4 \, x \log \left (3\right ) - 10 \, x + \frac {2 \, {\left (6 \, x e^{4} \log \left (3\right ) - 28 \, x e^{2} \log \left (3\right ) - 15 \, x e^{4} + 70 \, x e^{2} + 12 \, x \log \left (3\right ) + 4 \, e^{6} \log \left (3\right ) - 12 \, e^{4} \log \left (3\right ) - 30 \, x - 10 \, e^{6} + 30 \, e^{4}\right )}}{x^{2} + 2 \, x e^{2} - x + e^{4}} \]
integrate((2*((6*x^2+20*x)*exp(2)^2+(8*x^3+20*x^2)*exp(2)+2*x^4-4*x^3-10*x ^2)*log(3)+(-30*x^2-100*x)*exp(2)^2+(-40*x^3-100*x^2)*exp(2)-10*x^4+20*x^3 +50*x^2)/(exp(2)^4+4*x*exp(2)^3+(6*x^2-2*x)*exp(2)^2+(4*x^3-4*x^2)*exp(2)+ x^4-2*x^3+x^2),x, algorithm=\
4*x*log(3) - 10*x + 2*(6*x*e^4*log(3) - 28*x*e^2*log(3) - 15*x*e^4 + 70*x* e^2 + 12*x*log(3) + 4*e^6*log(3) - 12*e^4*log(3) - 30*x - 10*e^6 + 30*e^4) /(x^2 + 2*x*e^2 - x + e^4)
Time = 12.49 (sec) , antiderivative size = 187, normalized size of antiderivative = 7.48 \[ \int \frac {50 x^2+20 x^3-10 x^4+e^4 \left (-100 x-30 x^2\right )+e^2 \left (-100 x^2-40 x^3\right )+\left (-10 x^2-4 x^3+2 x^4+e^4 \left (20 x+6 x^2\right )+e^2 \left (20 x^2+8 x^3\right )\right ) \log (9)}{e^8+4 e^6 x+x^2-2 x^3+x^4+e^4 \left (-2 x+6 x^2\right )+e^2 \left (-4 x^2+4 x^3\right )} \, dx=x\,\left (\ln \left (81\right )-10\right )-\frac {\frac {60\,{\mathrm {e}}^4-260\,{\mathrm {e}}^6+80\,{\mathrm {e}}^8+\ln \left (\frac {9^{14\,{\mathrm {e}}^8}\,9^{40\,{\mathrm {e}}^6}\,{81}^{6\,{\mathrm {e}}^6}\,{81}^{14\,{\mathrm {e}}^{10}}}{9^{10\,{\mathrm {e}}^4}\,9^{28\,{\mathrm {e}}^{10}}\,{81}^{{\mathrm {e}}^4}\,{81}^{15\,{\mathrm {e}}^8}}\right )}{4\,{\mathrm {e}}^2-1}+\frac {x\,\left (380\,{\mathrm {e}}^2-590\,{\mathrm {e}}^4+120\,{\mathrm {e}}^6+\ln \left (\frac {282429536481\,9^{40\,{\mathrm {e}}^6}\,9^{70\,{\mathrm {e}}^4}\,{81}^{14\,{\mathrm {e}}^8}\,{81}^{24\,{\mathrm {e}}^4}}{9^{28\,{\mathrm {e}}^8}\,9^{60\,{\mathrm {e}}^2}\,{81}^{8\,{\mathrm {e}}^2}\,{81}^{32\,{\mathrm {e}}^6}}\right )-60\right )}{4\,{\mathrm {e}}^2-1}}{x^2+\left (2\,{\mathrm {e}}^2-1\right )\,x+{\mathrm {e}}^4} \]
int(-(exp(4)*(100*x + 30*x^2) + exp(2)*(100*x^2 + 40*x^3) - 2*log(3)*(exp( 4)*(20*x + 6*x^2) + exp(2)*(20*x^2 + 8*x^3) - 10*x^2 - 4*x^3 + 2*x^4) - 50 *x^2 - 20*x^3 + 10*x^4)/(exp(8) - exp(4)*(2*x - 6*x^2) + 4*x*exp(6) - exp( 2)*(4*x^2 - 4*x^3) + x^2 - 2*x^3 + x^4),x)
x*(log(81) - 10) - ((60*exp(4) - 260*exp(6) + 80*exp(8) + log((9^(14*exp(8 ))*9^(40*exp(6))*81^(6*exp(6))*81^(14*exp(10)))/(9^(10*exp(4))*9^(28*exp(1 0))*81^exp(4)*81^(15*exp(8)))))/(4*exp(2) - 1) + (x*(380*exp(2) - 590*exp( 4) + 120*exp(6) + log((282429536481*9^(40*exp(6))*9^(70*exp(4))*81^(14*exp (8))*81^(24*exp(4)))/(9^(28*exp(8))*9^(60*exp(2))*81^(8*exp(2))*81^(32*exp (6)))) - 60))/(4*exp(2) - 1))/(exp(4) + x^2 + x*(2*exp(2) - 1))