Integrand size = 148, antiderivative size = 17 \[ \int \frac {8 e^4 x+e^5 \left (4-16 x^3\right )}{-x^{10}+e \left (-5 x^9+5 x^{12}\right )+e^2 \left (-10 x^8+20 x^{11}-10 x^{14}\right )+e^3 \left (-10 x^7+30 x^{10}-30 x^{13}+10 x^{16}\right )+e^4 \left (-5 x^6+20 x^9-30 x^{12}+20 x^{15}-5 x^{18}\right )+e^5 \left (-x^5+5 x^8-10 x^{11}+10 x^{14}-5 x^{17}+x^{20}\right )} \, dx=\frac {1}{\left (x+x^2 \left (\frac {1}{e}-x^2\right )\right )^4} \] Output:
1/(x^2*(1/exp(1)-x^2)+x)^4
Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {8 e^4 x+e^5 \left (4-16 x^3\right )}{-x^{10}+e \left (-5 x^9+5 x^{12}\right )+e^2 \left (-10 x^8+20 x^{11}-10 x^{14}\right )+e^3 \left (-10 x^7+30 x^{10}-30 x^{13}+10 x^{16}\right )+e^4 \left (-5 x^6+20 x^9-30 x^{12}+20 x^{15}-5 x^{18}\right )+e^5 \left (-x^5+5 x^8-10 x^{11}+10 x^{14}-5 x^{17}+x^{20}\right )} \, dx=\frac {e^4}{x^4 \left (e+x-e x^3\right )^4} \] Input:
Integrate[(8*E^4*x + E^5*(4 - 16*x^3))/(-x^10 + E*(-5*x^9 + 5*x^12) + E^2* (-10*x^8 + 20*x^11 - 10*x^14) + E^3*(-10*x^7 + 30*x^10 - 30*x^13 + 10*x^16 ) + E^4*(-5*x^6 + 20*x^9 - 30*x^12 + 20*x^15 - 5*x^18) + E^5*(-x^5 + 5*x^8 - 10*x^11 + 10*x^14 - 5*x^17 + x^20)),x]
Output:
E^4/(x^4*(E + x - E*x^3)^4)
Time = 11.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {2026, 2462, 7239, 27, 25, 2023}
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.
| \(\displaystyle \int \frac {e^5 \left (4-16 x^3\right )+8 e^4 x}{-x^{10}+e \left (5 x^{12}-5 x^9\right )+e^2 \left (-10 x^{14}+20 x^{11}-10 x^8\right )+e^3 \left (10 x^{16}-30 x^{13}+30 x^{10}-10 x^7\right )+e^4 \left (-5 x^{18}+20 x^{15}-30 x^{12}+20 x^9-5 x^6\right )+e^5 \left (x^{20}-5 x^{17}+10 x^{14}-10 x^{11}+5 x^8-x^5\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^5 \left (4-16 x^3\right )+8 e^4 x}{x^5 \left (e^5 x^{15}-5 e^4 x^{13}-5 e^5 x^{12}+10 e^3 x^{11}+20 e^4 x^{10}-10 e^2 \left (1-e^3\right ) x^9-30 e^3 x^8+5 e \left (1-6 e^3\right ) x^7+10 e^2 \left (2-e^3\right ) x^6-\left (1-30 e^3\right ) x^5-5 e \left (1-4 e^3\right ) x^4-5 e^2 \left (2-e^3\right ) x^3-10 e^3 x^2-5 e^4 x-e^5\right )}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (-\frac {4}{x^5}+\frac {12}{e x^4}+\frac {4 \left (\left (5-e^3\right ) x-5 e\right )}{e^2 \left (-e x^3+x+e\right )}-\frac {20}{e^2 x^3}-\frac {4 \left (e^3-5\right )}{e^3 x^2}+\frac {4 \left (e \left (5-4 e^3\right ) x^2+2 e^3-5\right )}{e^3 \left (-e x^3+x+e\right )^2}+\frac {4 \left (e \left (5-6 e^3\right ) x^2-e^2 \left (2-e^3\right ) x+6 e^3-5\right )}{e^2 \left (-e x^3+x+e\right )^3}+\frac {4 \left (3 e \left (1-2 e^3\right ) x^2-2 e^2 \left (1-e^3\right ) x+7 e^3-3\right )}{e \left (-e x^3+x+e\right )^4}+\frac {4 \left (e \left (1-4 e^3\right ) x^2-e^2 \left (1-3 e^3\right ) x+5 e^3-1\right )}{\left (-e x^3+x+e\right )^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 e^4 \left (4 e x^3-2 x-e\right )}{x^5 \left (-e x^3+x+e\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 e^4 \int -\frac {-4 e x^3+2 x+e}{x^5 \left (-e x^3+x+e\right )^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 e^4 \int \frac {-4 e x^3+2 x+e}{x^5 \left (-e x^3+x+e\right )^5}dx\) |
| \(\Big \downarrow \) 2023 |
| \(\displaystyle \frac {e^4}{x^4 \left (-e x^3+x+e\right )^4}\) |
Input:
Int[(8*E^4*x + E^5*(4 - 16*x^3))/(-x^10 + E*(-5*x^9 + 5*x^12) + E^2*(-10*x ^8 + 20*x^11 - 10*x^14) + E^3*(-10*x^7 + 30*x^10 - 30*x^13 + 10*x^16) + E^ 4*(-5*x^6 + 20*x^9 - 30*x^12 + 20*x^15 - 5*x^18) + E^5*(-x^5 + 5*x^8 - 10* x^11 + 10*x^14 - 5*x^17 + x^20)),x]
Output:
E^4/(x^4*(E + x - E*x^3)^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, Simp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq ^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r])) , x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q ]*Coeff[Rr, x, r]*Pp, Coeff[Pp, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n}, x] && P olyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]
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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.51 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.47
| method | result | size |
| norman | \(\frac {{\mathrm e}^{4}}{x^{4} \left (x^{3} {\mathrm e}-{\mathrm e}-x \right )^{4}}\) | \(25\) |
| risch | \(\frac {{\mathrm e}^{4}}{x^{4} \left ({\mathrm e}^{4} x^{12}-4 x^{9} {\mathrm e}^{4}-4 \,{\mathrm e}^{3} x^{10}+6 x^{6} {\mathrm e}^{4}+12 \,{\mathrm e}^{3} x^{7}+6 x^{8} {\mathrm e}^{2}-4 x^{3} {\mathrm e}^{4}-12 x^{4} {\mathrm e}^{3}-12 \,{\mathrm e}^{2} x^{5}-4 x^{6} {\mathrm e}+{\mathrm e}^{4}+4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}+4 x^{3} {\mathrm e}+x^{4}\right )}\) | \(103\) |
| gosper | \(\frac {{\mathrm e}^{4}}{x^{4} \left ({\mathrm e}^{4} x^{12}-4 x^{9} {\mathrm e}^{4}-4 \,{\mathrm e}^{3} x^{10}+6 x^{6} {\mathrm e}^{4}+12 \,{\mathrm e}^{3} x^{7}+6 x^{8} {\mathrm e}^{2}-4 x^{3} {\mathrm e}^{4}-12 x^{4} {\mathrm e}^{3}-12 \,{\mathrm e}^{2} x^{5}-4 x^{6} {\mathrm e}+{\mathrm e}^{4}+4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}+4 x^{3} {\mathrm e}+x^{4}\right )}\) | \(129\) |
| parallelrisch | \(\frac {\left (6 \,{\mathrm e}^{8}-4 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-1}}{2 \left (3 \,{\mathrm e}^{3}-2\right ) x^{4} \left ({\mathrm e}^{4} x^{12}-4 x^{9} {\mathrm e}^{4}-4 \,{\mathrm e}^{3} x^{10}+6 x^{6} {\mathrm e}^{4}+12 \,{\mathrm e}^{3} x^{7}+6 x^{8} {\mathrm e}^{2}-4 x^{3} {\mathrm e}^{4}-12 x^{4} {\mathrm e}^{3}-12 \,{\mathrm e}^{2} x^{5}-4 x^{6} {\mathrm e}+{\mathrm e}^{4}+4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}+4 x^{3} {\mathrm e}+x^{4}\right )}\) | \(153\) |
| default | \(4 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-25} {\mathrm e}^{20}}{x^{3}}+\frac {{\mathrm e}^{-25} \left ({\mathrm e}^{21}-5 \,{\mathrm e}^{18}\right )}{x}+\frac {5 \,{\mathrm e}^{-25} {\mathrm e}^{19}}{2 x^{2}}+\frac {{\mathrm e}^{21} {\mathrm e}^{-25}}{4 x^{4}}-\frac {{\mathrm e}^{-25} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left ({\mathrm e}^{5} \textit {\_Z}^{15}-5 \,{\mathrm e}^{4} \textit {\_Z}^{13}-5 \,{\mathrm e}^{5} \textit {\_Z}^{12}+10 \,{\mathrm e}^{3} \textit {\_Z}^{11}+20 \,{\mathrm e}^{4} \textit {\_Z}^{10}+\left (10 \,{\mathrm e}^{5}-10 \,{\mathrm e}^{2}\right ) \textit {\_Z}^{9}-30 \,{\mathrm e}^{3} \textit {\_Z}^{8}+\left (5 \,{\mathrm e}-30 \,{\mathrm e}^{4}\right ) \textit {\_Z}^{7}+\left (-10 \,{\mathrm e}^{5}+20 \,{\mathrm e}^{2}\right ) \textit {\_Z}^{6}+\left (30 \,{\mathrm e}^{3}-1\right ) \textit {\_Z}^{5}+\left (-5 \,{\mathrm e}+20 \,{\mathrm e}^{4}\right ) \textit {\_Z}^{4}+\left (5 \,{\mathrm e}^{5}-10 \,{\mathrm e}^{2}\right ) \textit {\_Z}^{3}-10 \textit {\_Z}^{2} {\mathrm e}^{3}-5 \textit {\_Z} \,{\mathrm e}^{4}-{\mathrm e}^{5}\right )}{\sum }\frac {\left (\left (5 \,{\mathrm e}^{23}-{\mathrm e}^{26}\right ) \textit {\_R}^{13}-5 \,{\mathrm e}^{24} \textit {\_R}^{12}+\left (-25 \,{\mathrm e}^{22}+8 \,{\mathrm e}^{25}\right ) \textit {\_R}^{11}+4 \textit {\_R}^{10} {\mathrm e}^{26}+50 \textit {\_R}^{9} {\mathrm e}^{21}+10 \left (5 \,{\mathrm e}^{22}-3 \,{\mathrm e}^{25}\right ) \textit {\_R}^{8}+5 \left (-2 \,{\mathrm e}^{23}-{\mathrm e}^{26}-10 \,{\mathrm e}^{20}\right ) \textit {\_R}^{7}+10 \left (-10 \,{\mathrm e}^{21}+3 \,{\mathrm e}^{24}\right ) \textit {\_R}^{6}+5 \left (-7 \,{\mathrm e}^{22}+8 \,{\mathrm e}^{25}+5 \,{\mathrm e}^{19}\right ) \textit {\_R}^{5}+75 \,{\mathrm e}^{20} \textit {\_R}^{4}+\left (66 \,{\mathrm e}^{21}-40 \,{\mathrm e}^{24}-5 \,{\mathrm e}^{18}\right ) \textit {\_R}^{3}+10 \left ({\mathrm e}^{22}-2 \,{\mathrm e}^{25}-2 \,{\mathrm e}^{19}\right ) \textit {\_R}^{2}+\left (5 \,{\mathrm e}^{23}+5 \,{\mathrm e}^{26}-28 \,{\mathrm e}^{20}\right ) \textit {\_R} -14 \,{\mathrm e}^{21}+15 \,{\mathrm e}^{24}\right ) \ln \left (x -\textit {\_R} \right )}{3 \,{\mathrm e}^{5} \textit {\_R}^{14}-12 \,{\mathrm e}^{5} \textit {\_R}^{11}-13 \,{\mathrm e}^{4} \textit {\_R}^{12}+18 \textit {\_R}^{8} {\mathrm e}^{5}+40 \textit {\_R}^{9} {\mathrm e}^{4}+22 \,{\mathrm e}^{3} \textit {\_R}^{10}-12 \textit {\_R}^{5} {\mathrm e}^{5}-42 \textit {\_R}^{6} {\mathrm e}^{4}-48 \,{\mathrm e}^{3} \textit {\_R}^{7}-18 \textit {\_R}^{8} {\mathrm e}^{2}+3 \textit {\_R}^{2} {\mathrm e}^{5}+16 \textit {\_R}^{3} {\mathrm e}^{4}+30 \textit {\_R}^{4} {\mathrm e}^{3}+24 \,{\mathrm e}^{2} \textit {\_R}^{5}+7 \textit {\_R}^{6} {\mathrm e}-{\mathrm e}^{4}-4 \textit {\_R} \,{\mathrm e}^{3}-6 \textit {\_R}^{2} {\mathrm e}^{2}-4 \textit {\_R}^{3} {\mathrm e}-\textit {\_R}^{4}}\right )}{5}\right )\) | \(507\) |
Input:
int(((-16*x^3+4)*exp(1)^5+8*x*exp(1)^4)/((x^20-5*x^17+10*x^14-10*x^11+5*x^ 8-x^5)*exp(1)^5+(-5*x^18+20*x^15-30*x^12+20*x^9-5*x^6)*exp(1)^4+(10*x^16-3 0*x^13+30*x^10-10*x^7)*exp(1)^3+(-10*x^14+20*x^11-10*x^8)*exp(1)^2+(5*x^12 -5*x^9)*exp(1)-x^10),x,method=_RETURNVERBOSE)
Output:
exp(1)^4/x^4/(x^3*exp(1)-exp(1)-x)^4
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (18) = 36\).
Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 5.06 \[ \int \frac {8 e^4 x+e^5 \left (4-16 x^3\right )}{-x^{10}+e \left (-5 x^9+5 x^{12}\right )+e^2 \left (-10 x^8+20 x^{11}-10 x^{14}\right )+e^3 \left (-10 x^7+30 x^{10}-30 x^{13}+10 x^{16}\right )+e^4 \left (-5 x^6+20 x^9-30 x^{12}+20 x^{15}-5 x^{18}\right )+e^5 \left (-x^5+5 x^8-10 x^{11}+10 x^{14}-5 x^{17}+x^{20}\right )} \, dx=\frac {e^{4}}{x^{8} + {\left (x^{16} - 4 \, x^{13} + 6 \, x^{10} - 4 \, x^{7} + x^{4}\right )} e^{4} - 4 \, {\left (x^{14} - 3 \, x^{11} + 3 \, x^{8} - x^{5}\right )} e^{3} + 6 \, {\left (x^{12} - 2 \, x^{9} + x^{6}\right )} e^{2} - 4 \, {\left (x^{10} - x^{7}\right )} e} \] Input:
integrate(((-16*x^3+4)*exp(1)^5+8*x*exp(1)^4)/((x^20-5*x^17+10*x^14-10*x^1 1+5*x^8-x^5)*exp(1)^5+(-5*x^18+20*x^15-30*x^12+20*x^9-5*x^6)*exp(1)^4+(10* x^16-30*x^13+30*x^10-10*x^7)*exp(1)^3+(-10*x^14+20*x^11-10*x^8)*exp(1)^2+( 5*x^12-5*x^9)*exp(1)-x^10),x, algorithm="fricas")
Output:
e^4/(x^8 + (x^16 - 4*x^13 + 6*x^10 - 4*x^7 + x^4)*e^4 - 4*(x^14 - 3*x^11 + 3*x^8 - x^5)*e^3 + 6*(x^12 - 2*x^9 + x^6)*e^2 - 4*(x^10 - x^7)*e)
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (15) = 30\).
Time = 2.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 6.59 \[ \int \frac {8 e^4 x+e^5 \left (4-16 x^3\right )}{-x^{10}+e \left (-5 x^9+5 x^{12}\right )+e^2 \left (-10 x^8+20 x^{11}-10 x^{14}\right )+e^3 \left (-10 x^7+30 x^{10}-30 x^{13}+10 x^{16}\right )+e^4 \left (-5 x^6+20 x^9-30 x^{12}+20 x^{15}-5 x^{18}\right )+e^5 \left (-x^5+5 x^8-10 x^{11}+10 x^{14}-5 x^{17}+x^{20}\right )} \, dx=\frac {e^{4}}{x^{16} e^{4} - 4 x^{14} e^{3} - 4 x^{13} e^{4} + 6 x^{12} e^{2} + 12 x^{11} e^{3} + x^{10} \left (- 4 e + 6 e^{4}\right ) - 12 x^{9} e^{2} + x^{8} \cdot \left (1 - 12 e^{3}\right ) + x^{7} \left (- 4 e^{4} + 4 e\right ) + 6 x^{6} e^{2} + 4 x^{5} e^{3} + x^{4} e^{4}} \] Input:
integrate(((-16*x**3+4)*exp(1)**5+8*x*exp(1)**4)/((x**20-5*x**17+10*x**14- 10*x**11+5*x**8-x**5)*exp(1)**5+(-5*x**18+20*x**15-30*x**12+20*x**9-5*x**6 )*exp(1)**4+(10*x**16-30*x**13+30*x**10-10*x**7)*exp(1)**3+(-10*x**14+20*x **11-10*x**8)*exp(1)**2+(5*x**12-5*x**9)*exp(1)-x**10),x)
Output:
exp(4)/(x**16*exp(4) - 4*x**14*exp(3) - 4*x**13*exp(4) + 6*x**12*exp(2) + 12*x**11*exp(3) + x**10*(-4*E + 6*exp(4)) - 12*x**9*exp(2) + x**8*(1 - 12* exp(3)) + x**7*(-4*exp(4) + 4*E) + 6*x**6*exp(2) + 4*x**5*exp(3) + x**4*ex p(4))
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (18) = 36\).
Time = 0.05 (sec) , antiderivative size = 104, normalized size of antiderivative = 6.12 \[ \int \frac {8 e^4 x+e^5 \left (4-16 x^3\right )}{-x^{10}+e \left (-5 x^9+5 x^{12}\right )+e^2 \left (-10 x^8+20 x^{11}-10 x^{14}\right )+e^3 \left (-10 x^7+30 x^{10}-30 x^{13}+10 x^{16}\right )+e^4 \left (-5 x^6+20 x^9-30 x^{12}+20 x^{15}-5 x^{18}\right )+e^5 \left (-x^5+5 x^8-10 x^{11}+10 x^{14}-5 x^{17}+x^{20}\right )} \, dx=\frac {e^{4}}{x^{16} e^{4} - 4 \, x^{14} e^{3} - 4 \, x^{13} e^{4} + 6 \, x^{12} e^{2} + 12 \, x^{11} e^{3} + 2 \, x^{10} {\left (3 \, e^{4} - 2 \, e\right )} - 12 \, x^{9} e^{2} - x^{8} {\left (12 \, e^{3} - 1\right )} - 4 \, x^{7} {\left (e^{4} - e\right )} + 6 \, x^{6} e^{2} + 4 \, x^{5} e^{3} + x^{4} e^{4}} \] Input:
integrate(((-16*x^3+4)*exp(1)^5+8*x*exp(1)^4)/((x^20-5*x^17+10*x^14-10*x^1 1+5*x^8-x^5)*exp(1)^5+(-5*x^18+20*x^15-30*x^12+20*x^9-5*x^6)*exp(1)^4+(10* x^16-30*x^13+30*x^10-10*x^7)*exp(1)^3+(-10*x^14+20*x^11-10*x^8)*exp(1)^2+( 5*x^12-5*x^9)*exp(1)-x^10),x, algorithm="maxima")
Output:
e^4/(x^16*e^4 - 4*x^14*e^3 - 4*x^13*e^4 + 6*x^12*e^2 + 12*x^11*e^3 + 2*x^1 0*(3*e^4 - 2*e) - 12*x^9*e^2 - x^8*(12*e^3 - 1) - 4*x^7*(e^4 - e) + 6*x^6* e^2 + 4*x^5*e^3 + x^4*e^4)
Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.35 \[ \int \frac {8 e^4 x+e^5 \left (4-16 x^3\right )}{-x^{10}+e \left (-5 x^9+5 x^{12}\right )+e^2 \left (-10 x^8+20 x^{11}-10 x^{14}\right )+e^3 \left (-10 x^7+30 x^{10}-30 x^{13}+10 x^{16}\right )+e^4 \left (-5 x^6+20 x^9-30 x^{12}+20 x^{15}-5 x^{18}\right )+e^5 \left (-x^5+5 x^8-10 x^{11}+10 x^{14}-5 x^{17}+x^{20}\right )} \, dx=\frac {e^{20}}{{\left (x^{2} e^{4} - {\left (x^{4} - x\right )} e^{5}\right )}^{4}} \] Input:
integrate(((-16*x^3+4)*exp(1)^5+8*x*exp(1)^4)/((x^20-5*x^17+10*x^14-10*x^1 1+5*x^8-x^5)*exp(1)^5+(-5*x^18+20*x^15-30*x^12+20*x^9-5*x^6)*exp(1)^4+(10* x^16-30*x^13+30*x^10-10*x^7)*exp(1)^3+(-10*x^14+20*x^11-10*x^8)*exp(1)^2+( 5*x^12-5*x^9)*exp(1)-x^10),x, algorithm="giac")
Output:
e^20/(x^2*e^4 - (x^4 - x)*e^5)^4
Time = 2.60 (sec) , antiderivative size = 105, normalized size of antiderivative = 6.18 \[ \int \frac {8 e^4 x+e^5 \left (4-16 x^3\right )}{-x^{10}+e \left (-5 x^9+5 x^{12}\right )+e^2 \left (-10 x^8+20 x^{11}-10 x^{14}\right )+e^3 \left (-10 x^7+30 x^{10}-30 x^{13}+10 x^{16}\right )+e^4 \left (-5 x^6+20 x^9-30 x^{12}+20 x^{15}-5 x^{18}\right )+e^5 \left (-x^5+5 x^8-10 x^{11}+10 x^{14}-5 x^{17}+x^{20}\right )} \, dx=\frac {{\mathrm {e}}^4}{{\mathrm {e}}^4\,x^{16}-4\,{\mathrm {e}}^3\,x^{14}-4\,{\mathrm {e}}^4\,x^{13}+6\,{\mathrm {e}}^2\,x^{12}+12\,{\mathrm {e}}^3\,x^{11}+\left (6\,{\mathrm {e}}^4-4\,\mathrm {e}\right )\,x^{10}-12\,{\mathrm {e}}^2\,x^9+\left (1-12\,{\mathrm {e}}^3\right )\,x^8+\left (4\,\mathrm {e}-4\,{\mathrm {e}}^4\right )\,x^7+6\,{\mathrm {e}}^2\,x^6+4\,{\mathrm {e}}^3\,x^5+{\mathrm {e}}^4\,x^4} \] Input:
int(-(8*x*exp(4) - exp(5)*(16*x^3 - 4))/(exp(5)*(x^5 - 5*x^8 + 10*x^11 - 1 0*x^14 + 5*x^17 - x^20) + exp(4)*(5*x^6 - 20*x^9 + 30*x^12 - 20*x^15 + 5*x ^18) + exp(1)*(5*x^9 - 5*x^12) + exp(2)*(10*x^8 - 20*x^11 + 10*x^14) + x^1 0 + exp(3)*(10*x^7 - 30*x^10 + 30*x^13 - 10*x^16)),x)
Output:
exp(4)/(x^7*(4*exp(1) - 4*exp(4)) - x^10*(4*exp(1) - 6*exp(4)) - x^8*(12*e xp(3) - 1) + x^4*exp(4) + 4*x^5*exp(3) + 6*x^6*exp(2) - 12*x^9*exp(2) + 12 *x^11*exp(3) + 6*x^12*exp(2) - 4*x^13*exp(4) - 4*x^14*exp(3) + x^16*exp(4) )
Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 6.65 \[ \int \frac {8 e^4 x+e^5 \left (4-16 x^3\right )}{-x^{10}+e \left (-5 x^9+5 x^{12}\right )+e^2 \left (-10 x^8+20 x^{11}-10 x^{14}\right )+e^3 \left (-10 x^7+30 x^{10}-30 x^{13}+10 x^{16}\right )+e^4 \left (-5 x^6+20 x^9-30 x^{12}+20 x^{15}-5 x^{18}\right )+e^5 \left (-x^5+5 x^8-10 x^{11}+10 x^{14}-5 x^{17}+x^{20}\right )} \, dx=\frac {e^{4}}{x^{4} \left (e^{4} x^{12}-4 e^{4} x^{9}-4 e^{3} x^{10}+6 e^{4} x^{6}+12 e^{3} x^{7}+6 e^{2} x^{8}-4 e^{4} x^{3}-12 e^{3} x^{4}-12 e^{2} x^{5}-4 e \,x^{6}+e^{4}+4 e^{3} x +6 e^{2} x^{2}+4 e \,x^{3}+x^{4}\right )} \] Input:
int(((-16*x^3+4)*exp(1)^5+8*x*exp(1)^4)/((x^20-5*x^17+10*x^14-10*x^11+5*x^ 8-x^5)*exp(1)^5+(-5*x^18+20*x^15-30*x^12+20*x^9-5*x^6)*exp(1)^4+(10*x^16-3 0*x^13+30*x^10-10*x^7)*exp(1)^3+(-10*x^14+20*x^11-10*x^8)*exp(1)^2+(5*x^12 -5*x^9)*exp(1)-x^10),x)
Output:
e**4/(x**4*(e**4*x**12 - 4*e**4*x**9 + 6*e**4*x**6 - 4*e**4*x**3 + e**4 - 4*e**3*x**10 + 12*e**3*x**7 - 12*e**3*x**4 + 4*e**3*x + 6*e**2*x**8 - 12*e **2*x**5 + 6*e**2*x**2 - 4*e*x**6 + 4*e*x**3 + x**4))