Integrand size = 124, antiderivative size = 29 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {1}{16} (-3+x)^2 x^2+\frac {1}{x^4 \left (5+2 e^{-e}+x\right )} \] Output:
1/x^4/(2/exp(exp(1))+5+x)+x^2*(1/4*x-3/4)^2
Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {2 (-3+x)^2 x^6+e^e \left (16+45 x^6-21 x^7-x^8+x^9\right )}{16 x^4 \left (2+e^e (5+x)\right )} \] Input:
Integrate[(36*x^6 - 36*x^7 + 8*x^8 + E^E*(-64 + 180*x^6 - 144*x^7 + 4*x^8 + 8*x^9) + E^(2*E)*(-160 - 40*x + 225*x^6 - 135*x^7 - 31*x^8 + 11*x^9 + 2* x^10))/(32*x^5 + E^E*(160*x^5 + 32*x^6) + E^(2*E)*(200*x^5 + 80*x^6 + 8*x^ 7)),x]
Output:
(2*(-3 + x)^2*x^6 + E^E*(16 + 45*x^6 - 21*x^7 - x^8 + x^9))/(16*x^4*(2 + E ^E*(5 + x)))
Leaf count is larger than twice the leaf count of optimal. \(123\) vs. \(2(29)=58\).
Time = 0.73 (sec) , antiderivative size = 123, normalized size of antiderivative = 4.24, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2026, 2007, 2123, 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.
| \(\displaystyle \int \frac {8 x^8-36 x^7+36 x^6+e^e \left (8 x^9+4 x^8-144 x^7+180 x^6-64\right )+e^{2 e} \left (2 x^{10}+11 x^9-31 x^8-135 x^7+225 x^6-40 x-160\right )}{32 x^5+e^e \left (32 x^6+160 x^5\right )+e^{2 e} \left (8 x^7+80 x^6+200 x^5\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {8 x^8-36 x^7+36 x^6+e^e \left (8 x^9+4 x^8-144 x^7+180 x^6-64\right )+e^{2 e} \left (2 x^{10}+11 x^9-31 x^8-135 x^7+225 x^6-40 x-160\right )}{x^5 \left (8 e^{2 e} x^2+16 e^e \left (2+5 e^e\right ) x+8 \left (2+5 e^e\right )^2\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {8 x^8-36 x^7+36 x^6+e^e \left (8 x^9+4 x^8-144 x^7+180 x^6-64\right )+e^{2 e} \left (2 x^{10}+11 x^9-31 x^8-135 x^7+225 x^6-40 x-160\right )}{x^5 \left (2 \sqrt {2} e^e x+2 \sqrt {2} \left (2+5 e^e\right )\right )^2}dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (-\frac {4 e^e}{\left (2+5 e^e\right ) x^5}+\frac {3 e^{2 e}}{\left (2+5 e^e\right )^2 x^4}+\frac {x^3}{4}-\frac {2 e^{3 e}}{\left (2+5 e^e\right )^3 x^3}-\frac {9 x^2}{8}+\frac {e^{4 e}}{\left (2+5 e^e\right )^4 x^2}+\frac {9 x}{8}-\frac {e^{6 e}}{\left (2+5 e^e\right )^4 \left (e^e x+5 e^e+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^4}{16}+\frac {e^e}{\left (2+5 e^e\right ) x^4}-\frac {3 x^3}{8}-\frac {e^{2 e}}{\left (2+5 e^e\right )^2 x^3}+\frac {9 x^2}{16}+\frac {e^{3 e}}{\left (2+5 e^e\right )^3 x^2}+\frac {e^{5 e}}{\left (2+5 e^e\right )^4 \left (e^e x+5 e^e+2\right )}-\frac {e^{4 e}}{\left (2+5 e^e\right )^4 x}\) |
Input:
Int[(36*x^6 - 36*x^7 + 8*x^8 + E^E*(-64 + 180*x^6 - 144*x^7 + 4*x^8 + 8*x^ 9) + E^(2*E)*(-160 - 40*x + 225*x^6 - 135*x^7 - 31*x^8 + 11*x^9 + 2*x^10)) /(32*x^5 + E^E*(160*x^5 + 32*x^6) + E^(2*E)*(200*x^5 + 80*x^6 + 8*x^7)),x]
Output:
E^E/((2 + 5*E^E)*x^4) - E^(2*E)/((2 + 5*E^E)^2*x^3) + E^(3*E)/((2 + 5*E^E) ^3*x^2) - E^(4*E)/((2 + 5*E^E)^4*x) + (9*x^2)/16 - (3*x^3)/8 + x^4/16 + E^ (5*E)/((2 + 5*E^E)^4*(2 + 5*E^E + E^E*x))
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 1.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(\frac {x^{4}}{16}-\frac {3 x^{3}}{8}+\frac {9 x^{2}}{16}+\frac {{\mathrm e}^{{\mathrm e}}}{x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\) | \(38\) |
| norman | \(\frac {\left (-\frac {21 \,{\mathrm e}^{{\mathrm e}}}{16}-\frac {3}{4}\right ) x^{7}+\left (-\frac {{\mathrm e}^{{\mathrm e}}}{16}+\frac {1}{8}\right ) x^{8}+\left (\frac {45 \,{\mathrm e}^{{\mathrm e}}}{16}+\frac {9}{8}\right ) x^{6}+\frac {{\mathrm e}^{{\mathrm e}} x^{9}}{16}+{\mathrm e}^{{\mathrm e}}}{x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\) | \(64\) |
| gosper | \(\frac {{\mathrm e}^{{\mathrm e}} x^{9}-{\mathrm e}^{{\mathrm e}} x^{8}-21 \,{\mathrm e}^{{\mathrm e}} x^{7}+2 x^{8}+45 \,{\mathrm e}^{{\mathrm e}} x^{6}-12 x^{7}+18 x^{6}+16 \,{\mathrm e}^{{\mathrm e}}}{16 x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\) | \(72\) |
| parallelrisch | \(\frac {\left ({\mathrm e}^{2 \,{\mathrm e}} x^{9}-{\mathrm e}^{2 \,{\mathrm e}} x^{8}-21 \,{\mathrm e}^{2 \,{\mathrm e}} x^{7}+2 \,{\mathrm e}^{{\mathrm e}} x^{8}+45 \,{\mathrm e}^{2 \,{\mathrm e}} x^{6}-12 \,{\mathrm e}^{{\mathrm e}} x^{7}+18 \,{\mathrm e}^{{\mathrm e}} x^{6}+16 \,{\mathrm e}^{2 \,{\mathrm e}}\right ) {\mathrm e}^{-{\mathrm e}}}{16 x^{4} \left (x \,{\mathrm e}^{{\mathrm e}}+5 \,{\mathrm e}^{{\mathrm e}}+2\right )}\) | \(96\) |
Input:
int(((2*x^10+11*x^9-31*x^8-135*x^7+225*x^6-40*x-160)*exp(exp(1))^2+(8*x^9+ 4*x^8-144*x^7+180*x^6-64)*exp(exp(1))+8*x^8-36*x^7+36*x^6)/((8*x^7+80*x^6+ 200*x^5)*exp(exp(1))^2+(32*x^6+160*x^5)*exp(exp(1))+32*x^5),x,method=_RETU RNVERBOSE)
Output:
1/16*x^4-3/8*x^3+9/16*x^2+exp(exp(1))/x^4/(x*exp(exp(1))+5*exp(exp(1))+2)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {2 \, x^{8} - 12 \, x^{7} + 18 \, x^{6} + {\left (x^{9} - x^{8} - 21 \, x^{7} + 45 \, x^{6} + 16\right )} e^{e}}{16 \, {\left (2 \, x^{4} + {\left (x^{5} + 5 \, x^{4}\right )} e^{e}\right )}} \] Input:
integrate(((2*x^10+11*x^9-31*x^8-135*x^7+225*x^6-40*x-160)*exp(exp(1))^2+( 8*x^9+4*x^8-144*x^7+180*x^6-64)*exp(exp(1))+8*x^8-36*x^7+36*x^6)/((8*x^7+8 0*x^6+200*x^5)*exp(exp(1))^2+(32*x^6+160*x^5)*exp(exp(1))+32*x^5),x, algor ithm="fricas")
Output:
1/16*(2*x^8 - 12*x^7 + 18*x^6 + (x^9 - x^8 - 21*x^7 + 45*x^6 + 16)*e^e)/(2 *x^4 + (x^5 + 5*x^4)*e^e)
Time = 0.76 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {x^{4}}{16} - \frac {3 x^{3}}{8} + \frac {9 x^{2}}{16} + \frac {e^{e}}{x^{5} e^{e} + x^{4} \cdot \left (2 + 5 e^{e}\right )} \] Input:
integrate(((2*x**10+11*x**9-31*x**8-135*x**7+225*x**6-40*x-160)*exp(exp(1) )**2+(8*x**9+4*x**8-144*x**7+180*x**6-64)*exp(exp(1))+8*x**8-36*x**7+36*x* *6)/((8*x**7+80*x**6+200*x**5)*exp(exp(1))**2+(32*x**6+160*x**5)*exp(exp(1 ))+32*x**5),x)
Output:
x**4/16 - 3*x**3/8 + 9*x**2/16 + exp(E)/(x**5*exp(E) + x**4*(2 + 5*exp(E)) )
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {1}{16} \, x^{4} - \frac {3}{8} \, x^{3} + \frac {9}{16} \, x^{2} + \frac {e^{e}}{x^{5} e^{e} + x^{4} {\left (5 \, e^{e} + 2\right )}} \] Input:
integrate(((2*x^10+11*x^9-31*x^8-135*x^7+225*x^6-40*x-160)*exp(exp(1))^2+( 8*x^9+4*x^8-144*x^7+180*x^6-64)*exp(exp(1))+8*x^8-36*x^7+36*x^6)/((8*x^7+8 0*x^6+200*x^5)*exp(exp(1))^2+(32*x^6+160*x^5)*exp(exp(1))+32*x^5),x, algor ithm="maxima")
Output:
1/16*x^4 - 3/8*x^3 + 9/16*x^2 + e^e/(x^5*e^e + x^4*(5*e^e + 2))
Exception generated. \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(((2*x^10+11*x^9-31*x^8-135*x^7+225*x^6-40*x-160)*exp(exp(1))^2+( 8*x^9+4*x^8-144*x^7+180*x^6-64)*exp(exp(1))+8*x^8-36*x^7+36*x^6)/((8*x^7+8 0*x^6+200*x^5)*exp(exp(1))^2+(32*x^6+160*x^5)*exp(exp(1))+32*x^5),x, algor ithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: 1/8* ((1/2*sageVARx^4*exp(2*exp(1))^4-3*sageVARx^3*exp(2*exp(1))^4+9/2*sageVARx ^2*exp(2*exp(1))^4)/exp(2*exp(1))^4+((-375000*exp(2*exp(1))^5-900000*exp(2 *exp(1))^4*exp(exp(
Time = 4.17 (sec) , antiderivative size = 314, normalized size of antiderivative = 10.83 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=x^2\,\left (\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (4\,{\mathrm {e}}^{\mathrm {e}}-31\,{\mathrm {e}}^{2\,\mathrm {e}}+8\right )}{16}-\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,{\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}^2}{8}+{\mathrm {e}}^{-\mathrm {e}}\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{2}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{8}\right )\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\right )+\frac {{\mathrm {e}}^{\mathrm {e}}}{{\mathrm {e}}^{\mathrm {e}}\,x^5+\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\,x^4}-x^3\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{6}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{24}\right )+\frac {x^4}{16}-x\,\left (\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (135\,{\mathrm {e}}^{2\,\mathrm {e}}+144\,{\mathrm {e}}^{\mathrm {e}}+36\right )}{8}+2\,{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\,\left (\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (4\,{\mathrm {e}}^{\mathrm {e}}-31\,{\mathrm {e}}^{2\,\mathrm {e}}+8\right )}{8}-\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,{\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}^2}{4}+2\,{\mathrm {e}}^{-\mathrm {e}}\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{2}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{8}\right )\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )\right )-{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}{2}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\left (11\,{\mathrm {e}}^{\mathrm {e}}+8\right )}{8}\right )\,{\left (5\,{\mathrm {e}}^{\mathrm {e}}+2\right )}^2\right ) \] Input:
int((exp(exp(1))*(180*x^6 - 144*x^7 + 4*x^8 + 8*x^9 - 64) - exp(2*exp(1))* (40*x - 225*x^6 + 135*x^7 + 31*x^8 - 11*x^9 - 2*x^10 + 160) + 36*x^6 - 36* x^7 + 8*x^8)/(exp(2*exp(1))*(200*x^5 + 80*x^6 + 8*x^7) + 32*x^5 + exp(exp( 1))*(160*x^5 + 32*x^6)),x)
Output:
x^2*((exp(-2*exp(1))*(4*exp(exp(1)) - 31*exp(2*exp(1)) + 8))/16 - (exp(-2* exp(1))*(5*exp(exp(1)) + 2)^2)/8 + exp(-exp(1))*((exp(-exp(1))*(5*exp(exp( 1)) + 2))/2 - (exp(-exp(1))*(11*exp(exp(1)) + 8))/8)*(5*exp(exp(1)) + 2)) + exp(exp(1))/(x^5*exp(exp(1)) + x^4*(5*exp(exp(1)) + 2)) - x^3*((exp(-exp (1))*(5*exp(exp(1)) + 2))/6 - (exp(-exp(1))*(11*exp(exp(1)) + 8))/24) + x^ 4/16 - x*((exp(-2*exp(1))*(135*exp(2*exp(1)) + 144*exp(exp(1)) + 36))/8 + 2*exp(-exp(1))*(5*exp(exp(1)) + 2)*((exp(-2*exp(1))*(4*exp(exp(1)) - 31*ex p(2*exp(1)) + 8))/8 - (exp(-2*exp(1))*(5*exp(exp(1)) + 2)^2)/4 + 2*exp(-ex p(1))*((exp(-exp(1))*(5*exp(exp(1)) + 2))/2 - (exp(-exp(1))*(11*exp(exp(1) ) + 8))/8)*(5*exp(exp(1)) + 2)) - exp(-2*exp(1))*((exp(-exp(1))*(5*exp(exp (1)) + 2))/2 - (exp(-exp(1))*(11*exp(exp(1)) + 8))/8)*(5*exp(exp(1)) + 2)^ 2)
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {36 x^6-36 x^7+8 x^8+e^e \left (-64+180 x^6-144 x^7+4 x^8+8 x^9\right )+e^{2 e} \left (-160-40 x+225 x^6-135 x^7-31 x^8+11 x^9+2 x^{10}\right )}{32 x^5+e^e \left (160 x^5+32 x^6\right )+e^{2 e} \left (200 x^5+80 x^6+8 x^7\right )} \, dx=\frac {e^{e} x^{9}-e^{e} x^{8}-21 e^{e} x^{7}+45 e^{e} x^{6}+16 e^{e}+2 x^{8}-12 x^{7}+18 x^{6}}{16 x^{4} \left (e^{e} x +5 e^{e}+2\right )} \] Input:
int(((2*x^10+11*x^9-31*x^8-135*x^7+225*x^6-40*x-160)*exp(exp(1))^2+(8*x^9+ 4*x^8-144*x^7+180*x^6-64)*exp(exp(1))+8*x^8-36*x^7+36*x^6)/((8*x^7+80*x^6+ 200*x^5)*exp(exp(1))^2+(32*x^6+160*x^5)*exp(exp(1))+32*x^5),x)
Output:
(e**e*x**9 - e**e*x**8 - 21*e**e*x**7 + 45*e**e*x**6 + 16*e**e + 2*x**8 - 12*x**7 + 18*x**6)/(16*x**4*(e**e*x + 5*e**e + 2))