Integrand size = 80, antiderivative size = 29 \[ \int \frac {2 e^{e^4+e^e}+e^{e^4} (-20-8 x)}{300 x^2+120 x^3+12 x^4+e^{2 e^e} \left (75+30 x+3 x^2\right )+e^{e^e} \left (-300 x-120 x^2-12 x^3\right )} \, dx=2+\frac {2 e^{e^4}}{3 (5+x) \left (-e^{e^e}+2 x\right )} \]
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {2 e^{e^4+e^e}+e^{e^4} (-20-8 x)}{300 x^2+120 x^3+12 x^4+e^{2 e^e} \left (75+30 x+3 x^2\right )+e^{e^e} \left (-300 x-120 x^2-12 x^3\right )} \, dx=-\frac {2 e^{e^4}}{3 \left (e^{e^e}-2 x\right ) (5+x)} \]
Integrate[(2*E^(E^4 + E^E) + E^E^4*(-20 - 8*x))/(300*x^2 + 120*x^3 + 12*x^ 4 + E^(2*E^E)*(75 + 30*x + 3*x^2) + E^E^E*(-300*x - 120*x^2 - 12*x^3)),x]
Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2459, 27, 27, 1380, 27, 241}
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.
\(\displaystyle \int \frac {e^{e^4} (-8 x-20)+2 e^{e^4+e^e}}{12 x^4+120 x^3+300 x^2+e^{2 e^e} \left (3 x^2+30 x+75\right )+e^{e^e} \left (-12 x^3-120 x^2-300 x\right )} \, dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int -\frac {8 e^{e^4} \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )}{12 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )^4-\frac {3}{2} \left (10+e^{e^e}\right )^2 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )^2+\frac {3}{64} \left (10+e^{e^e}\right )^4}d\left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -8 e^{e^4} \int \frac {64 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )}{3 \left (256 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )^4-32 \left (10+e^{e^e}\right )^2 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )^2+\left (10+e^{e^e}\right )^4\right )}d\left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {512}{3} e^{e^4} \int \frac {x+\frac {1}{48} \left (120-12 e^{e^e}\right )}{256 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )^4-32 \left (10+e^{e^e}\right )^2 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )^2+\left (10+e^{e^e}\right )^4}d\left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle -\frac {131072}{3} e^{e^4} \int \frac {x+\frac {1}{48} \left (120-12 e^{e^e}\right )}{256 \left (\left (10+e^{e^e}\right )^2-16 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )^2\right )^2}d\left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {512}{3} e^{e^4} \int \frac {x+\frac {1}{48} \left (120-12 e^{e^e}\right )}{\left (\left (10+e^{e^e}\right )^2-16 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )^2\right )^2}d\left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )\) |
\(\Big \downarrow \) 241 |
\(\displaystyle -\frac {16 e^{e^4}}{3 \left (\left (10+e^{e^e}\right )^2-16 \left (x+\frac {1}{48} \left (120-12 e^{e^e}\right )\right )^2\right )}\) |
Int[(2*E^(E^4 + E^E) + E^E^4*(-20 - 8*x))/(300*x^2 + 120*x^3 + 12*x^4 + E^ (2*E^E)*(75 + 30*x + 3*x^2) + E^E^E*(-300*x - 120*x^2 - 12*x^3)),x]
3.28.6.3.1 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[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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[(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 = 0.37 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72
method | result | size |
norman | \(-\frac {2 \,{\mathrm e}^{{\mathrm e}^{4}}}{3 \left (5+x \right ) \left (-2 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}\right )}\) | \(21\) |
gosper | \(-\frac {2 \,{\mathrm e}^{{\mathrm e}^{4}}}{3 \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}}} x -2 x^{2}+5 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-10 x \right )}\) | \(29\) |
risch | \(-\frac {2 \,{\mathrm e}^{{\mathrm e}^{4}}}{3 \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}}} x -2 x^{2}+5 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-10 x \right )}\) | \(29\) |
parallelrisch | \(-\frac {2 \,{\mathrm e}^{{\mathrm e}^{4}}}{3 \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}}} x -2 x^{2}+5 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}}}-10 x \right )}\) | \(29\) |
int((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x^2+30*x+75 )*exp(exp(exp(1)))^2+(-12*x^3-120*x^2-300*x)*exp(exp(exp(1)))+12*x^4+120*x ^3+300*x^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {2 e^{e^4+e^e}+e^{e^4} (-20-8 x)}{300 x^2+120 x^3+12 x^4+e^{2 e^e} \left (75+30 x+3 x^2\right )+e^{e^e} \left (-300 x-120 x^2-12 x^3\right )} \, dx=-\frac {2 \, e^{\left (2 \, e^{4}\right )}}{3 \, {\left ({\left (x + 5\right )} e^{\left (e^{4} + e^{e}\right )} - 2 \, {\left (x^{2} + 5 \, x\right )} e^{\left (e^{4}\right )}\right )}} \]
integrate((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x^2+3 0*x+75)*exp(exp(exp(1)))^2+(-12*x^3-120*x^2-300*x)*exp(exp(exp(1)))+12*x^4 +120*x^3+300*x^2),x, algorithm=\
Time = 0.46 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {2 e^{e^4+e^e}+e^{e^4} (-20-8 x)}{300 x^2+120 x^3+12 x^4+e^{2 e^e} \left (75+30 x+3 x^2\right )+e^{e^e} \left (-300 x-120 x^2-12 x^3\right )} \, dx=\frac {2 e^{e^{4}}}{6 x^{2} + x \left (30 - 3 e^{e^{e}}\right ) - 15 e^{e^{e}}} \]
integrate((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x**2+ 30*x+75)*exp(exp(exp(1)))**2+(-12*x**3-120*x**2-300*x)*exp(exp(exp(1)))+12 *x**4+120*x**3+300*x**2),x)
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {2 e^{e^4+e^e}+e^{e^4} (-20-8 x)}{300 x^2+120 x^3+12 x^4+e^{2 e^e} \left (75+30 x+3 x^2\right )+e^{e^e} \left (-300 x-120 x^2-12 x^3\right )} \, dx=\frac {2 \, e^{\left (e^{4}\right )}}{3 \, {\left (2 \, x^{2} - x {\left (e^{\left (e^{e}\right )} - 10\right )} - 5 \, e^{\left (e^{e}\right )}\right )}} \]
integrate((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x^2+3 0*x+75)*exp(exp(exp(1)))^2+(-12*x^3-120*x^2-300*x)*exp(exp(exp(1)))+12*x^4 +120*x^3+300*x^2),x, algorithm=\
Exception generated. \[ \int \frac {2 e^{e^4+e^e}+e^{e^4} (-20-8 x)}{300 x^2+120 x^3+12 x^4+e^{2 e^e} \left (75+30 x+3 x^2\right )+e^{e^e} \left (-300 x-120 x^2-12 x^3\right )} \, dx=\text {Exception raised: NotImplementedError} \]
integrate((2*exp(exp(4))*exp(exp(exp(1)))+(-8*x-20)*exp(exp(4)))/((3*x^2+3 0*x+75)*exp(exp(exp(1)))^2+(-12*x^3-120*x^2-300*x)*exp(exp(exp(1)))+12*x^4 +120*x^3+300*x^2),x, algorithm=\
Exception raised: NotImplementedError >> unable to parse Giac output: -2/3 *((-2*exp(2*exp(exp(1)))*exp(exp(4))-20*exp(exp(exp(1)))*exp(exp(4))+2*exp (exp(exp(1)))*exp(exp(4)+exp(exp(1)))+20*exp(exp(4)+exp(exp(1))))/(exp(2*e xp(exp(1)))^2+40*ex
Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.97 \[ \int \frac {2 e^{e^4+e^e}+e^{e^4} (-20-8 x)}{300 x^2+120 x^3+12 x^4+e^{2 e^e} \left (75+30 x+3 x^2\right )+e^{e^e} \left (-300 x-120 x^2-12 x^3\right )} \, dx=\frac {4\,\left ({\mathrm {e}}^{{\mathrm {e}}^4+2\,{\mathrm {e}}^{\mathrm {e}}}+100\,{\mathrm {e}}^{{\mathrm {e}}^4}+20\,{\mathrm {e}}^{{\mathrm {e}}^4+{\mathrm {e}}^{\mathrm {e}}}\right )}{3\,{\left ({\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}+10\right )}^3\,\left (2\,x-{\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}\right )}-\frac {2\,\left ({\mathrm {e}}^{{\mathrm {e}}^4+2\,{\mathrm {e}}^{\mathrm {e}}}+100\,{\mathrm {e}}^{{\mathrm {e}}^4}+20\,{\mathrm {e}}^{{\mathrm {e}}^4+{\mathrm {e}}^{\mathrm {e}}}\right )}{3\,{\left ({\mathrm {e}}^{{\mathrm {e}}^{\mathrm {e}}}+10\right )}^3\,\left (x+5\right )} \]
int((2*exp(exp(exp(1)))*exp(exp(4)) - exp(exp(4))*(8*x + 20))/(exp(2*exp(e xp(1)))*(30*x + 3*x^2 + 75) - exp(exp(exp(1)))*(300*x + 120*x^2 + 12*x^3) + 300*x^2 + 120*x^3 + 12*x^4),x)