Integrand size = 91, antiderivative size = 22 \[ \int \frac {-8 x-6 e x+2 e^4 x-4 e^2 x^2}{16+9 e^2+e^8-8 e^6 x-32 x^2+16 x^4+e \left (24-24 x^2\right )+e^4 \left (-8-6 e+24 x^2\right )+e^2 \left (32 x+24 e x-32 x^3\right )} \, dx=\frac {x^2}{-4-3 e+\left (-e^2+2 x\right )^2} \] Output:
x^2/((2*x-exp(2))^2-4-3*exp(1))
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {-8 x-6 e x+2 e^4 x-4 e^2 x^2}{16+9 e^2+e^8-8 e^6 x-32 x^2+16 x^4+e \left (24-24 x^2\right )+e^4 \left (-8-6 e+24 x^2\right )+e^2 \left (32 x+24 e x-32 x^3\right )} \, dx=\frac {4+3 e-e^4+4 e^2 x}{4 \left (-4-3 e+e^4-4 e^2 x+4 x^2\right )} \] Input:
Integrate[(-8*x - 6*E*x + 2*E^4*x - 4*E^2*x^2)/(16 + 9*E^2 + E^8 - 8*E^6*x - 32*x^2 + 16*x^4 + E*(24 - 24*x^2) + E^4*(-8 - 6*E + 24*x^2) + E^2*(32*x + 24*E*x - 32*x^3)),x]
Output:
(4 + 3*E - E^4 + 4*E^2*x)/(4*(-4 - 3*E + E^4 - 4*E^2*x + 4*x^2))
Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(22)=44\).
Time = 0.41 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {6, 6, 2027, 2459, 1380, 27, 2345, 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.
\(\displaystyle \int \frac {-4 e^2 x^2+2 e^4 x-6 e x-8 x}{16 x^4+e^2 \left (-32 x^3+24 e x+32 x\right )-32 x^2+e \left (24-24 x^2\right )+e^4 \left (24 x^2-6 e-8\right )-8 e^6 x+e^8+9 e^2+16} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-4 e^2 x^2+2 e^4 x+(-8-6 e) x}{16 x^4+e^2 \left (-32 x^3+24 e x+32 x\right )-32 x^2+e \left (24-24 x^2\right )+e^4 \left (24 x^2-6 e-8\right )-8 e^6 x+e^8+9 e^2+16}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (-8-6 e+2 e^4\right ) x-4 e^2 x^2}{16 x^4+e^2 \left (-32 x^3+24 e x+32 x\right )-32 x^2+e \left (24-24 x^2\right )+e^4 \left (24 x^2-6 e-8\right )-8 e^6 x+e^8+9 e^2+16}dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x \left (-4 e^2 x+2 e^4-6 e-8\right )}{16 x^4+e^2 \left (-32 x^3+24 e x+32 x\right )-32 x^2+e \left (24-24 x^2\right )+e^4 \left (24 x^2-6 e-8\right )-8 e^6 x+e^8+9 e^2+16}dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int \frac {-4 e^2 \left (x-\frac {e^2}{2}\right )^2-2 \left (4+3 e+e^4\right ) \left (x-\frac {e^2}{2}\right )-e^2 (4+3 e)}{16 \left (x-\frac {e^2}{2}\right )^4-8 (4+3 e) \left (x-\frac {e^2}{2}\right )^2+(4+3 e)^2}d\left (x-\frac {e^2}{2}\right )\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle 16 \int -\frac {4 e^2 \left (x-\frac {e^2}{2}\right )^2+2 \left (4+3 e+e^4\right ) \left (x-\frac {e^2}{2}\right )+e^2 (4+3 e)}{16 \left (-4 \left (x-\frac {e^2}{2}\right )^2+3 e+4\right )^2}d\left (x-\frac {e^2}{2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {4 e^2 \left (x-\frac {e^2}{2}\right )^2+2 \left (4+3 e+e^4\right ) \left (x-\frac {e^2}{2}\right )+e^2 (4+3 e)}{\left (-4 \left (x-\frac {e^2}{2}\right )^2+3 e+4\right )^2}d\left (x-\frac {e^2}{2}\right )\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {\int 0d\left (x-\frac {e^2}{2}\right )}{2 (4+3 e)}-\frac {4 e^2 \left (x-\frac {e^2}{2}\right )+e^4+3 e+4}{4 \left (-4 \left (x-\frac {e^2}{2}\right )^2+3 e+4\right )}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {4 e^2 \left (x-\frac {e^2}{2}\right )+e^4+3 e+4}{4 \left (-4 \left (x-\frac {e^2}{2}\right )^2+3 e+4\right )}\) |
Input:
Int[(-8*x - 6*E*x + 2*E^4*x - 4*E^2*x^2)/(16 + 9*E^2 + E^8 - 8*E^6*x - 32* x^2 + 16*x^4 + E*(24 - 24*x^2) + E^4*(-8 - 6*E + 24*x^2) + E^2*(32*x + 24* E*x - 32*x^3)),x]
Output:
-1/4*(4 + 3*E + E^4 + 4*E^2*(-1/2*E^2 + x))/(4 + 3*E - 4*(-1/2*E^2 + x)^2)
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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 = 0.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64
method | result | size |
risch | \(\frac {{\mathrm e}^{2} x -\frac {{\mathrm e}^{4}}{4}+\frac {3 \,{\mathrm e}}{4}+1}{{\mathrm e}^{4}-4 \,{\mathrm e}^{2} x +4 x^{2}-3 \,{\mathrm e}-4}\) | \(36\) |
gosper | \(-\frac {{\mathrm e}^{4}-4 \,{\mathrm e}^{2} x -3 \,{\mathrm e}-4}{4 \left ({\mathrm e}^{4}-4 \,{\mathrm e}^{2} x +4 x^{2}-3 \,{\mathrm e}-4\right )}\) | \(40\) |
norman | \(\frac {{\mathrm e}^{2} x -\frac {{\mathrm e}^{4}}{4}+\frac {3 \,{\mathrm e}}{4}+1}{{\mathrm e}^{4}-4 \,{\mathrm e}^{2} x +4 x^{2}-3 \,{\mathrm e}-4}\) | \(40\) |
parallelrisch | \(\frac {-{\mathrm e}^{4}+4 \,{\mathrm e}^{2} x +3 \,{\mathrm e}+4}{4 \,{\mathrm e}^{4}-16 \,{\mathrm e}^{2} x +16 x^{2}-12 \,{\mathrm e}-16}\) | \(42\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (16 \textit {\_Z}^{4}-32 \textit {\_Z}^{3} {\mathrm e}^{2}+\left (-24 \,{\mathrm e}+24 \,{\mathrm e}^{4}-32\right ) \textit {\_Z}^{2}+\left (32 \,{\mathrm e}^{2}-8 \,{\mathrm e}^{6}+24 \,{\mathrm e}^{3}\right ) \textit {\_Z} +9 \,{\mathrm e}^{2}+24 \,{\mathrm e}+{\mathrm e}^{8}-8 \,{\mathrm e}^{4}-6 \,{\mathrm e}^{5}+16\right )}{\sum }\frac {\left (2 \textit {\_R}^{2} {\mathrm e}^{2}+\left (-{\mathrm e}^{4}+4+3 \,{\mathrm e}\right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{{\mathrm e}^{6}-6 \textit {\_R} \,{\mathrm e}^{4}+12 \textit {\_R}^{2} {\mathrm e}^{2}-8 \textit {\_R}^{3}-3 \,{\mathrm e}^{3}+6 \textit {\_R} \,{\mathrm e}-4 \,{\mathrm e}^{2}+8 \textit {\_R}}\right )}{4}\) | \(133\) |
Input:
int((2*x*exp(2)^2-4*x^2*exp(2)-6*x*exp(1)-8*x)/(exp(2)^4-8*x*exp(2)^3+(-6* exp(1)+24*x^2-8)*exp(2)^2+(24*x*exp(1)-32*x^3+32*x)*exp(2)+9*exp(1)^2+(-24 *x^2+24)*exp(1)+16*x^4-32*x^2+16),x,method=_RETURNVERBOSE)
Output:
(exp(2)*x-1/4*exp(4)+3/4*exp(1)+1)/(exp(4)-4*exp(2)*x+4*x^2-3*exp(1)-4)
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {-8 x-6 e x+2 e^4 x-4 e^2 x^2}{16+9 e^2+e^8-8 e^6 x-32 x^2+16 x^4+e \left (24-24 x^2\right )+e^4 \left (-8-6 e+24 x^2\right )+e^2 \left (32 x+24 e x-32 x^3\right )} \, dx=\frac {4 \, x e^{2} - e^{4} + 3 \, e + 4}{4 \, {\left (4 \, x^{2} - 4 \, x e^{2} + e^{4} - 3 \, e - 4\right )}} \] Input:
integrate((2*x*exp(2)^2-4*x^2*exp(2)-6*exp(1)*x-8*x)/(exp(2)^4-8*x*exp(2)^ 3+(-6*exp(1)+24*x^2-8)*exp(2)^2+(24*exp(1)*x-32*x^3+32*x)*exp(2)+9*exp(1)^ 2+(-24*x^2+24)*exp(1)+16*x^4-32*x^2+16),x, algorithm="fricas")
Output:
1/4*(4*x*e^2 - e^4 + 3*e + 4)/(4*x^2 - 4*x*e^2 + e^4 - 3*e - 4)
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (17) = 34\).
Time = 0.58 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {-8 x-6 e x+2 e^4 x-4 e^2 x^2}{16+9 e^2+e^8-8 e^6 x-32 x^2+16 x^4+e \left (24-24 x^2\right )+e^4 \left (-8-6 e+24 x^2\right )+e^2 \left (32 x+24 e x-32 x^3\right )} \, dx=- \frac {- 4 x e^{2} - 3 e - 4 + e^{4}}{16 x^{2} - 16 x e^{2} - 12 e - 16 + 4 e^{4}} \] Input:
integrate((2*x*exp(2)**2-4*x**2*exp(2)-6*exp(1)*x-8*x)/(exp(2)**4-8*x*exp( 2)**3+(-6*exp(1)+24*x**2-8)*exp(2)**2+(24*exp(1)*x-32*x**3+32*x)*exp(2)+9* exp(1)**2+(-24*x**2+24)*exp(1)+16*x**4-32*x**2+16),x)
Output:
-(-4*x*exp(2) - 3*E - 4 + exp(4))/(16*x**2 - 16*x*exp(2) - 12*E - 16 + 4*e xp(4))
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {-8 x-6 e x+2 e^4 x-4 e^2 x^2}{16+9 e^2+e^8-8 e^6 x-32 x^2+16 x^4+e \left (24-24 x^2\right )+e^4 \left (-8-6 e+24 x^2\right )+e^2 \left (32 x+24 e x-32 x^3\right )} \, dx=\frac {4 \, x e^{2} - e^{4} + 3 \, e + 4}{4 \, {\left (4 \, x^{2} - 4 \, x e^{2} + e^{4} - 3 \, e - 4\right )}} \] Input:
integrate((2*x*exp(2)^2-4*x^2*exp(2)-6*exp(1)*x-8*x)/(exp(2)^4-8*x*exp(2)^ 3+(-6*exp(1)+24*x^2-8)*exp(2)^2+(24*exp(1)*x-32*x^3+32*x)*exp(2)+9*exp(1)^ 2+(-24*x^2+24)*exp(1)+16*x^4-32*x^2+16),x, algorithm="maxima")
Output:
1/4*(4*x*e^2 - e^4 + 3*e + 4)/(4*x^2 - 4*x*e^2 + e^4 - 3*e - 4)
Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {-8 x-6 e x+2 e^4 x-4 e^2 x^2}{16+9 e^2+e^8-8 e^6 x-32 x^2+16 x^4+e \left (24-24 x^2\right )+e^4 \left (-8-6 e+24 x^2\right )+e^2 \left (32 x+24 e x-32 x^3\right )} \, dx=\frac {4 \, x e^{2} - e^{4} + 3 \, e + 4}{4 \, {\left (4 \, x^{2} - 4 \, x e^{2} + e^{4} - 3 \, e - 4\right )}} \] Input:
integrate((2*x*exp(2)^2-4*x^2*exp(2)-6*exp(1)*x-8*x)/(exp(2)^4-8*x*exp(2)^ 3+(-6*exp(1)+24*x^2-8)*exp(2)^2+(24*exp(1)*x-32*x^3+32*x)*exp(2)+9*exp(1)^ 2+(-24*x^2+24)*exp(1)+16*x^4-32*x^2+16),x, algorithm="giac")
Output:
1/4*(4*x*e^2 - e^4 + 3*e + 4)/(4*x^2 - 4*x*e^2 + e^4 - 3*e - 4)
Time = 0.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {-8 x-6 e x+2 e^4 x-4 e^2 x^2}{16+9 e^2+e^8-8 e^6 x-32 x^2+16 x^4+e \left (24-24 x^2\right )+e^4 \left (-8-6 e+24 x^2\right )+e^2 \left (32 x+24 e x-32 x^3\right )} \, dx=-\frac {\frac {3\,\mathrm {e}}{4}-\frac {{\mathrm {e}}^4}{4}+x\,{\mathrm {e}}^2+1}{-4\,x^2+4\,{\mathrm {e}}^2\,x+3\,\mathrm {e}-{\mathrm {e}}^4+4} \] Input:
int(-(8*x + 6*x*exp(1) - 2*x*exp(4) + 4*x^2*exp(2))/(9*exp(2) + exp(8) - e xp(4)*(6*exp(1) - 24*x^2 + 8) - 8*x*exp(6) - exp(1)*(24*x^2 - 24) + exp(2) *(32*x + 24*x*exp(1) - 32*x^3) - 32*x^2 + 16*x^4 + 16),x)
Output:
-((3*exp(1))/4 - exp(4)/4 + x*exp(2) + 1)/(3*exp(1) - exp(4) + 4*x*exp(2) - 4*x^2 + 4)
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {-8 x-6 e x+2 e^4 x-4 e^2 x^2}{16+9 e^2+e^8-8 e^6 x-32 x^2+16 x^4+e \left (24-24 x^2\right )+e^4 \left (-8-6 e+24 x^2\right )+e^2 \left (32 x+24 e x-32 x^3\right )} \, dx=\frac {x^{2}}{e^{4}-4 e^{2} x +4 x^{2}-3 e -4} \] Input:
int((2*x*exp(2)^2-4*x^2*exp(2)-6*exp(1)*x-8*x)/(exp(2)^4-8*x*exp(2)^3+(-6* exp(1)+24*x^2-8)*exp(2)^2+(24*exp(1)*x-32*x^3+32*x)*exp(2)+9*exp(1)^2+(-24 *x^2+24)*exp(1)+16*x^4-32*x^2+16),x)
Output:
x**2/(e**4 - 4*e**2*x - 3*e + 4*x**2 - 4)