Integrand size = 257, antiderivative size = 22 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=11+x^2-\frac {3-2 x}{(x-e (1+x))^4} \] Output:
11+x^2-(3-2*x)/(x-(1+x)*exp(1))^4
Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=-2 x+(1+x)^2+\frac {3-5 e}{(-1+e) (e-x+e x)^4}+\frac {2}{(-1+e) (e-x+e x)^3} \] Input:
Integrate[(-12 + 6*x - 2*x^6 + E^2*(-20*x^4 - 40*x^5 - 20*x^6) + E^4*(-10* x^2 - 40*x^3 - 60*x^4 - 40*x^5 - 10*x^6) + E^5*(2*x + 10*x^2 + 20*x^3 + 20 *x^4 + 10*x^5 + 2*x^6) + E*(14 - 6*x + 10*x^5 + 10*x^6) + E^3*(20*x^3 + 60 *x^4 + 60*x^5 + 20*x^6))/(-x^5 + E^2*(-10*x^3 - 20*x^4 - 10*x^5) + E^4*(-5 *x - 20*x^2 - 30*x^3 - 20*x^4 - 5*x^5) + E^5*(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5) + E*(5*x^4 + 5*x^5) + E^3*(10*x^2 + 30*x^3 + 30*x^4 + 10*x^5) ),x]
Output:
-2*x + (1 + x)^2 + (3 - 5*E)/((-1 + E)*(E - x + E*x)^4) + 2/((-1 + E)*(E - x + E*x)^3)
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).
Time = 0.42 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2007, 2389, 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.
| \(\displaystyle \int \frac {-2 x^6+e \left (10 x^6+10 x^5-6 x+14\right )+e^2 \left (-20 x^6-40 x^5-20 x^4\right )+e^3 \left (20 x^6+60 x^5+60 x^4+20 x^3\right )+e^4 \left (-10 x^6-40 x^5-60 x^4-40 x^3-10 x^2\right )+e^5 \left (2 x^6+10 x^5+20 x^4+20 x^3+10 x^2+2 x\right )+6 x-12}{-x^5+e \left (5 x^5+5 x^4\right )+e^2 \left (-10 x^5-20 x^4-10 x^3\right )+e^4 \left (-5 x^5-20 x^4-30 x^3-20 x^2-5 x\right )+e^5 \left (x^5+5 x^4+10 x^3+10 x^2+5 x+1\right )+e^3 \left (10 x^5+30 x^4+30 x^3+10 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {-2 x^6+e \left (10 x^6+10 x^5-6 x+14\right )+e^2 \left (-20 x^6-40 x^5-20 x^4\right )+e^3 \left (20 x^6+60 x^5+60 x^4+20 x^3\right )+e^4 \left (-10 x^6-40 x^5-60 x^4-40 x^3-10 x^2\right )+e^5 \left (2 x^6+10 x^5+20 x^4+20 x^3+10 x^2+2 x\right )+6 x-12}{((e-1) x+e)^5}dx\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (2 x-\frac {6}{(e-(1-e) x)^4}+\frac {4 (5 e-3)}{(e-(1-e) x)^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^2-\frac {2}{(1-e) (e-(1-e) x)^3}-\frac {3-5 e}{(1-e) (e-(1-e) x)^4}\) |
Input:
Int[(-12 + 6*x - 2*x^6 + E^2*(-20*x^4 - 40*x^5 - 20*x^6) + E^4*(-10*x^2 - 40*x^3 - 60*x^4 - 40*x^5 - 10*x^6) + E^5*(2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6) + E*(14 - 6*x + 10*x^5 + 10*x^6) + E^3*(20*x^3 + 60*x^4 + 60*x^5 + 20*x^6))/(-x^5 + E^2*(-10*x^3 - 20*x^4 - 10*x^5) + E^4*(-5*x - 2 0*x^2 - 30*x^3 - 20*x^4 - 5*x^5) + E^5*(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5) + E*(5*x^4 + 5*x^5) + E^3*(10*x^2 + 30*x^3 + 30*x^4 + 10*x^5)),x]
Output:
x^2 - (3 - 5*E)/((1 - E)*(E - (1 - E)*x)^4) - 2/((1 - E)*(E - (1 - E)*x)^3 )
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[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(23)=46\).
Time = 1.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.77
| method | result | size |
| risch | \(x^{2}+\frac {-3+2 x}{x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{3}+6 x^{2} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3}+6 x^{4} {\mathrm e}^{2}+4 x \,{\mathrm e}^{4}-12 x^{2} {\mathrm e}^{3}+12 x^{3} {\mathrm e}^{2}-4 x^{4} {\mathrm e}+{\mathrm e}^{4}-4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}-4 x^{3} {\mathrm e}+x^{4}}\) | \(105\) |
| norman | \(\frac {\left ({\mathrm e}^{4}-4 \,{\mathrm e}^{3}+6 \,{\mathrm e}^{2}-4 \,{\mathrm e}+1\right ) x^{6}+{\mathrm e}^{-2} \left ({\mathrm e}^{6}+18 \,{\mathrm e}^{2}-36 \,{\mathrm e}+18\right ) x^{2}+\left (6 \,{\mathrm e}^{8}-12 \,{\mathrm e}^{7}+6 \,{\mathrm e}^{6}+3+3 \,{\mathrm e}^{4}-12 \,{\mathrm e}^{3}+18 \,{\mathrm e}^{2}-12 \,{\mathrm e}\right ) {\mathrm e}^{-4} x^{4}+2 \,{\mathrm e}^{-1} \left (7 \,{\mathrm e}-6\right ) x +4 \left ({\mathrm e}^{7}-{\mathrm e}^{6}+3 \,{\mathrm e}^{3}-9 \,{\mathrm e}^{2}+9 \,{\mathrm e}-3\right ) {\mathrm e}^{-3} x^{3}+4 \,{\mathrm e} \left ({\mathrm e}^{3}-3 \,{\mathrm e}^{2}+3 \,{\mathrm e}-1\right ) x^{5}}{\left (x \,{\mathrm e}+{\mathrm e}-x \right )^{4}}\) | \(188\) |
| default | \(x^{2}-\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-\left (10 \,{\mathrm e}^{2}-10 \,{\mathrm e}^{3}+5 \,{\mathrm e}^{4}-5 \,{\mathrm e}-{\mathrm e}^{5}+1\right ) \textit {\_Z}^{5}-\left (20 \,{\mathrm e}^{2}-30 \,{\mathrm e}^{3}+20 \,{\mathrm e}^{4}-5 \,{\mathrm e}-5 \,{\mathrm e}^{5}\right ) \textit {\_Z}^{4}-\left (10 \,{\mathrm e}^{2}-30 \,{\mathrm e}^{3}+30 \,{\mathrm e}^{4}-10 \,{\mathrm e}^{5}\right ) \textit {\_Z}^{3}-\left (-10 \,{\mathrm e}^{3}+20 \,{\mathrm e}^{4}-10 \,{\mathrm e}^{5}\right ) \textit {\_Z}^{2}-\left (5 \,{\mathrm e}^{4}-5 \,{\mathrm e}^{5}\right ) \textit {\_Z} +{\mathrm e}^{5}\right )}{\sum }\frac {\left (-6+3 \left (1-{\mathrm e}\right ) \textit {\_R} +7 \,{\mathrm e}\right ) \ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{4} {\mathrm e}^{5}-4 \textit {\_R}^{3} {\mathrm e}^{5}+5 \textit {\_R}^{4} {\mathrm e}^{4}-6 \textit {\_R}^{2} {\mathrm e}^{5}+16 \textit {\_R}^{3} {\mathrm e}^{4}-10 \textit {\_R}^{4} {\mathrm e}^{3}-4 \textit {\_R} \,{\mathrm e}^{5}+18 \textit {\_R}^{2} {\mathrm e}^{4}-24 \textit {\_R}^{3} {\mathrm e}^{3}+10 \textit {\_R}^{4} {\mathrm e}^{2}-{\mathrm e}^{5}+8 \textit {\_R} \,{\mathrm e}^{4}-18 \textit {\_R}^{2} {\mathrm e}^{3}+16 \textit {\_R}^{3} {\mathrm e}^{2}-5 \textit {\_R}^{4} {\mathrm e}+{\mathrm e}^{4}-4 \textit {\_R} \,{\mathrm e}^{3}+6 \textit {\_R}^{2} {\mathrm e}^{2}-4 \textit {\_R}^{3} {\mathrm e}+\textit {\_R}^{4}}\right )}{5}\) | \(266\) |
| gosper | \(\frac {x \left (-12 x^{4} {\mathrm e}^{7}+x^{5} {\mathrm e}^{4}+4 x^{2} {\mathrm e}^{8}-12 x^{3} {\mathrm e}^{3}+6 x^{3} {\mathrm e}^{6}-4 x^{4} {\mathrm e}^{5}-12 x^{3} {\mathrm e}-36 x \,{\mathrm e}^{3}-36 x^{2} {\mathrm e}^{3}+3 x^{3} {\mathrm e}^{4}+12 x^{2} {\mathrm e}^{4}-12 x^{2} {\mathrm e}+36 x^{2} {\mathrm e}^{2}+18 x \,{\mathrm e}^{4}-4 \,{\mathrm e}^{7} x^{5}-12 \,{\mathrm e}^{7} x^{3}-4 x^{5} {\mathrm e}^{5}-4 x^{2} {\mathrm e}^{7}+18 x^{3} {\mathrm e}^{2}+18 \,{\mathrm e}^{2} x +14 \,{\mathrm e}^{4}+3 x^{3}-12 \,{\mathrm e}^{3}+4 x^{4} {\mathrm e}^{8}+6 x^{3} {\mathrm e}^{8}+6 \,{\mathrm e}^{6} x^{5}+12 \,{\mathrm e}^{6} x^{4}+x \,{\mathrm e}^{8}+x^{5} {\mathrm e}^{8}\right ) {\mathrm e}^{-4}}{x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{3}+6 x^{2} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3}+6 x^{4} {\mathrm e}^{2}+4 x \,{\mathrm e}^{4}-12 x^{2} {\mathrm e}^{3}+12 x^{3} {\mathrm e}^{2}-4 x^{4} {\mathrm e}+{\mathrm e}^{4}-4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}-4 x^{3} {\mathrm e}+x^{4}}\) | \(362\) |
| parallelrisch | \(\frac {\left (-12 x^{4} {\mathrm e}^{7}-12 x^{4} {\mathrm e}^{3}+x^{2} {\mathrm e}^{8}-36 x^{3} {\mathrm e}^{3}+18 x^{4} {\mathrm e}^{2}-12 x^{4} {\mathrm e}-12 x^{3} {\mathrm e}-12 x \,{\mathrm e}^{3}-36 x^{2} {\mathrm e}^{3}+12 x^{3} {\mathrm e}^{4}+18 x^{2} {\mathrm e}^{4}+18 x^{2} {\mathrm e}^{2}+14 x \,{\mathrm e}^{4}-4 \,{\mathrm e}^{7} x^{6}-12 \,{\mathrm e}^{7} x^{5}-4 \,{\mathrm e}^{7} x^{3}-4 x^{5} {\mathrm e}^{5}+3 x^{4} {\mathrm e}^{4}-4 x^{6} {\mathrm e}^{5}+36 x^{3} {\mathrm e}^{2}+3 x^{4}+6 x^{4} {\mathrm e}^{8}+4 x^{3} {\mathrm e}^{8}+12 \,{\mathrm e}^{6} x^{5}+6 \,{\mathrm e}^{6} x^{4}+6 \,{\mathrm e}^{6} x^{6}+{\mathrm e}^{4} x^{6}+4 x^{5} {\mathrm e}^{8}+{\mathrm e}^{8} x^{6}\right ) {\mathrm e}^{-4}}{x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{3}+6 x^{2} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3}+6 x^{4} {\mathrm e}^{2}+4 x \,{\mathrm e}^{4}-12 x^{2} {\mathrm e}^{3}+12 x^{3} {\mathrm e}^{2}-4 x^{4} {\mathrm e}+{\mathrm e}^{4}-4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}-4 x^{3} {\mathrm e}+x^{4}}\) | \(371\) |
Input:
int(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x ^4-40*x^3-10*x^2)*exp(1)^4+(20*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6 -40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-12)/((x^5 +5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp (1)^4+(10*x^5+30*x^4+30*x^3+10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1 )^2+(5*x^5+5*x^4)*exp(1)-x^5),x,method=_RETURNVERBOSE)
Output:
x^2+(-3+2*x)/(x^4*exp(4)+4*x^3*exp(4)-4*x^4*exp(3)+6*x^2*exp(4)-12*x^3*exp (3)+6*x^4*exp(2)+4*x*exp(4)-12*x^2*exp(3)+12*x^3*exp(2)-4*x^4*exp(1)+exp(4 )-4*x*exp(3)+6*x^2*exp(2)-4*x^3*exp(1)+x^4)
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 155, normalized size of antiderivative = 7.05 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=\frac {x^{6} + {\left (x^{6} + 4 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} + x^{2}\right )} e^{4} - 4 \, {\left (x^{6} + 3 \, x^{5} + 3 \, x^{4} + x^{3}\right )} e^{3} + 6 \, {\left (x^{6} + 2 \, x^{5} + x^{4}\right )} e^{2} - 4 \, {\left (x^{6} + x^{5}\right )} e + 2 \, x - 3}{x^{4} + {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )} e^{4} - 4 \, {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} + x\right )} e^{3} + 6 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{2} - 4 \, {\left (x^{4} + x^{3}\right )} e} \] Input:
integrate(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^ 5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(- 20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-12) /((x^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5* x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3) *exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x, algorithm="fricas")
Output:
(x^6 + (x^6 + 4*x^5 + 6*x^4 + 4*x^3 + x^2)*e^4 - 4*(x^6 + 3*x^5 + 3*x^4 + x^3)*e^3 + 6*(x^6 + 2*x^5 + x^4)*e^2 - 4*(x^6 + x^5)*e + 2*x - 3)/(x^4 + ( x^4 + 4*x^3 + 6*x^2 + 4*x + 1)*e^4 - 4*(x^4 + 3*x^3 + 3*x^2 + x)*e^3 + 6*( x^4 + 2*x^3 + x^2)*e^2 - 4*(x^4 + x^3)*e)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (19) = 38\).
Time = 3.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.00 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=x^{2} + \frac {2 x - 3}{x^{4} \left (- 4 e^{3} - 4 e + 1 + 6 e^{2} + e^{4}\right ) + x^{3} \left (- 12 e^{3} - 4 e + 12 e^{2} + 4 e^{4}\right ) + x^{2} \left (- 12 e^{3} + 6 e^{2} + 6 e^{4}\right ) + x \left (- 4 e^{3} + 4 e^{4}\right ) + e^{4}} \] Input:
integrate(((2*x**6+10*x**5+20*x**4+20*x**3+10*x**2+2*x)*exp(1)**5+(-10*x** 6-40*x**5-60*x**4-40*x**3-10*x**2)*exp(1)**4+(20*x**6+60*x**5+60*x**4+20*x **3)*exp(1)**3+(-20*x**6-40*x**5-20*x**4)*exp(1)**2+(10*x**6+10*x**5-6*x+1 4)*exp(1)-2*x**6+6*x-12)/((x**5+5*x**4+10*x**3+10*x**2+5*x+1)*exp(1)**5+(- 5*x**5-20*x**4-30*x**3-20*x**2-5*x)*exp(1)**4+(10*x**5+30*x**4+30*x**3+10* x**2)*exp(1)**3+(-10*x**5-20*x**4-10*x**3)*exp(1)**2+(5*x**5+5*x**4)*exp(1 )-x**5),x)
Output:
x**2 + (2*x - 3)/(x**4*(-4*exp(3) - 4*E + 1 + 6*exp(2) + exp(4)) + x**3*(- 12*exp(3) - 4*E + 12*exp(2) + 4*exp(4)) + x**2*(-12*exp(3) + 6*exp(2) + 6* exp(4)) + x*(-4*exp(3) + 4*exp(4)) + exp(4))
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (23) = 46\).
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.59 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=x^{2} + \frac {2 \, x - 3}{x^{4} {\left (e^{4} - 4 \, e^{3} + 6 \, e^{2} - 4 \, e + 1\right )} + 4 \, x^{3} {\left (e^{4} - 3 \, e^{3} + 3 \, e^{2} - e\right )} + 6 \, x^{2} {\left (e^{4} - 2 \, e^{3} + e^{2}\right )} + 4 \, x {\left (e^{4} - e^{3}\right )} + e^{4}} \] Input:
integrate(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^ 5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(- 20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-12) /((x^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5* x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3) *exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x, algorithm="maxima")
Output:
x^2 + (2*x - 3)/(x^4*(e^4 - 4*e^3 + 6*e^2 - 4*e + 1) + 4*x^3*(e^4 - 3*e^3 + 3*e^2 - e) + 6*x^2*(e^4 - 2*e^3 + e^2) + 4*x*(e^4 - e^3) + e^4)
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (23) = 46\).
Time = 0.11 (sec) , antiderivative size = 135, normalized size of antiderivative = 6.14 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=\frac {x^{2} e^{10} - 10 \, x^{2} e^{9} + 45 \, x^{2} e^{8} - 120 \, x^{2} e^{7} + 210 \, x^{2} e^{6} - 252 \, x^{2} e^{5} + 210 \, x^{2} e^{4} - 120 \, x^{2} e^{3} + 45 \, x^{2} e^{2} - 10 \, x^{2} e + x^{2}}{e^{10} - 10 \, e^{9} + 45 \, e^{8} - 120 \, e^{7} + 210 \, e^{6} - 252 \, e^{5} + 210 \, e^{4} - 120 \, e^{3} + 45 \, e^{2} - 10 \, e + 1} + \frac {2 \, x - 3}{{\left (x e - x + e\right )}^{4}} \] Input:
integrate(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^ 5-60*x^4-40*x^3-10*x^2)*exp(1)^4+(20*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(- 20*x^6-40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-12) /((x^5+5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5* x)*exp(1)^4+(10*x^5+30*x^4+30*x^3+10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3) *exp(1)^2+(5*x^5+5*x^4)*exp(1)-x^5),x, algorithm="giac")
Output:
(x^2*e^10 - 10*x^2*e^9 + 45*x^2*e^8 - 120*x^2*e^7 + 210*x^2*e^6 - 252*x^2* e^5 + 210*x^2*e^4 - 120*x^2*e^3 + 45*x^2*e^2 - 10*x^2*e + x^2)/(e^10 - 10* e^9 + 45*e^8 - 120*e^7 + 210*e^6 - 252*e^5 + 210*e^4 - 120*e^3 + 45*e^2 - 10*e + 1) + (2*x - 3)/(x*e - x + e)^4
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=\frac {2}{{\left (\mathrm {e}+x\,\left (\mathrm {e}-1\right )\right )}^3\,\left (\mathrm {e}-1\right )}+x^2-\frac {5\,\mathrm {e}-3}{{\left (\mathrm {e}+x\,\left (\mathrm {e}-1\right )\right )}^4\,\left (\mathrm {e}-1\right )} \] Input:
int((6*x + exp(5)*(2*x + 10*x^2 + 20*x^3 + 20*x^4 + 10*x^5 + 2*x^6) - exp( 4)*(10*x^2 + 40*x^3 + 60*x^4 + 40*x^5 + 10*x^6) + exp(1)*(10*x^5 - 6*x + 1 0*x^6 + 14) - exp(2)*(20*x^4 + 40*x^5 + 20*x^6) - 2*x^6 + exp(3)*(20*x^3 + 60*x^4 + 60*x^5 + 20*x^6) - 12)/(exp(5)*(5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5 + 1) + exp(1)*(5*x^4 + 5*x^5) - exp(2)*(10*x^3 + 20*x^4 + 10*x^5) - ex p(4)*(5*x + 20*x^2 + 30*x^3 + 20*x^4 + 5*x^5) - x^5 + exp(3)*(10*x^2 + 30* x^3 + 30*x^4 + 10*x^5)),x)
Output:
2/((exp(1) + x*(exp(1) - 1))^3*(exp(1) - 1)) + x^2 - (5*exp(1) - 3)/((exp( 1) + x*(exp(1) - 1))^4*(exp(1) - 1))
Time = 0.19 (sec) , antiderivative size = 371, normalized size of antiderivative = 16.86 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=\frac {e^{6} x^{6}+4 e^{6} x^{5}-6 e^{5} x^{6}+5 e^{6} x^{4}-20 e^{5} x^{5}+15 e^{4} x^{6}-20 e^{5} x^{4}+40 e^{4} x^{5}-20 e^{3} x^{6}-5 e^{6} x^{2}+30 e^{4} x^{4}-40 e^{3} x^{5}+15 e^{2} x^{6}-4 e^{6} x +10 e^{5} x^{2}-20 e^{3} x^{4}+20 e^{2} x^{5}-6 e \,x^{6}-e^{6}+4 e^{5} x -5 e^{4} x^{2}+5 e^{2} x^{4}-4 e \,x^{5}+x^{6}+2 e^{2} x -3 e^{2}-4 e x +6 e +2 x -3}{e^{6} x^{4}+4 e^{6} x^{3}-6 e^{5} x^{4}+6 e^{6} x^{2}-20 e^{5} x^{3}+15 e^{4} x^{4}+4 e^{6} x -24 e^{5} x^{2}+40 e^{4} x^{3}-20 e^{3} x^{4}+e^{6}-12 e^{5} x +36 e^{4} x^{2}-40 e^{3} x^{3}+15 e^{2} x^{4}-2 e^{5}+12 e^{4} x -24 e^{3} x^{2}+20 e^{2} x^{3}-6 e \,x^{4}+e^{4}-4 e^{3} x +6 e^{2} x^{2}-4 e \,x^{3}+x^{4}} \] Input:
int(((2*x^6+10*x^5+20*x^4+20*x^3+10*x^2+2*x)*exp(1)^5+(-10*x^6-40*x^5-60*x ^4-40*x^3-10*x^2)*exp(1)^4+(20*x^6+60*x^5+60*x^4+20*x^3)*exp(1)^3+(-20*x^6 -40*x^5-20*x^4)*exp(1)^2+(10*x^6+10*x^5-6*x+14)*exp(1)-2*x^6+6*x-12)/((x^5 +5*x^4+10*x^3+10*x^2+5*x+1)*exp(1)^5+(-5*x^5-20*x^4-30*x^3-20*x^2-5*x)*exp (1)^4+(10*x^5+30*x^4+30*x^3+10*x^2)*exp(1)^3+(-10*x^5-20*x^4-10*x^3)*exp(1 )^2+(5*x^5+5*x^4)*exp(1)-x^5),x)
Output:
(e**6*x**6 + 4*e**6*x**5 + 5*e**6*x**4 - 5*e**6*x**2 - 4*e**6*x - e**6 - 6 *e**5*x**6 - 20*e**5*x**5 - 20*e**5*x**4 + 10*e**5*x**2 + 4*e**5*x + 15*e* *4*x**6 + 40*e**4*x**5 + 30*e**4*x**4 - 5*e**4*x**2 - 20*e**3*x**6 - 40*e* *3*x**5 - 20*e**3*x**4 + 15*e**2*x**6 + 20*e**2*x**5 + 5*e**2*x**4 + 2*e** 2*x - 3*e**2 - 6*e*x**6 - 4*e*x**5 - 4*e*x + 6*e + x**6 + 2*x - 3)/(e**6*x **4 + 4*e**6*x**3 + 6*e**6*x**2 + 4*e**6*x + e**6 - 6*e**5*x**4 - 20*e**5* x**3 - 24*e**5*x**2 - 12*e**5*x - 2*e**5 + 15*e**4*x**4 + 40*e**4*x**3 + 3 6*e**4*x**2 + 12*e**4*x + e**4 - 20*e**3*x**4 - 40*e**3*x**3 - 24*e**3*x** 2 - 4*e**3*x + 15*e**2*x**4 + 20*e**2*x**3 + 6*e**2*x**2 - 6*e*x**4 - 4*e* x**3 + x**4)