Integrand size = 91, antiderivative size = 23 \[ \int \frac {3 x^2+8 x^3+x^4+e^2 \left (5+2 x-x^2\right )+e^6 \left (e^2 x^2+2 x^3\right )}{1+8 x+18 x^2+e^{12} x^2+8 x^3+x^4+e^6 \left (2 x+8 x^2+2 x^3\right )} \, dx=\frac {x^2+e^2 (5+x)}{4+e^6+\frac {1}{x}+x} \] Output:
((5+x)*exp(2)+x^2)/(exp(3)^2+4+x+1/x)
Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(23)=46\).
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {3 x^2+8 x^3+x^4+e^2 \left (5+2 x-x^2\right )+e^6 \left (e^2 x^2+2 x^3\right )}{1+8 x+18 x^2+e^{12} x^2+8 x^3+x^4+e^6 \left (2 x+8 x^2+2 x^3\right )} \, dx=x+\frac {4+e^2 (-1+x)+15 x-e^8 x+e^{12} x+e^6 (1+8 x)}{1+\left (4+e^6\right ) x+x^2} \] Input:
Integrate[(3*x^2 + 8*x^3 + x^4 + E^2*(5 + 2*x - x^2) + E^6*(E^2*x^2 + 2*x^ 3))/(1 + 8*x + 18*x^2 + E^12*x^2 + 8*x^3 + x^4 + E^6*(2*x + 8*x^2 + 2*x^3) ),x]
Output:
x + (4 + E^2*(-1 + x) + 15*x - E^8*x + E^12*x + E^6*(1 + 8*x))/(1 + (4 + E ^6)*x + x^2)
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(23)=46\).
Time = 0.49 (sec) , antiderivative size = 141, normalized size of antiderivative = 6.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {6, 2459, 1380, 25, 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.
\(\displaystyle \int \frac {x^4+8 x^3+3 x^2+e^2 \left (-x^2+2 x+5\right )+e^6 \left (2 x^3+e^2 x^2\right )}{x^4+8 x^3+e^{12} x^2+18 x^2+e^6 \left (2 x^3+8 x^2+2 x\right )+8 x+1} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {x^4+8 x^3+3 x^2+e^2 \left (-x^2+2 x+5\right )+e^6 \left (2 x^3+e^2 x^2\right )}{x^4+8 x^3+\left (18+e^{12}\right ) x^2+e^6 \left (2 x^3+8 x^2+2 x\right )+8 x+1}dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int \frac {\left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^4-\frac {1}{2} \left (42+2 e^2+24 e^6-2 e^8+3 e^{12}\right ) \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^2+\left (52+6 e^2+45 e^6-3 e^8+12 e^{12}-e^{14}+e^{18}\right ) \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )-\frac {1}{16} \left (2+e^6\right ) \left (6+e^6\right ) \left (48+4 e^2+24 e^6-4 e^8+3 e^{12}\right )}{\left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^4-\frac {1}{2} \left (12+8 e^6+e^{12}\right ) \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^2+\frac {1}{16} \left (12+8 e^6+e^{12}\right )^2}d\left (x+\frac {1}{4} \left (8+2 e^6\right )\right )\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int -\frac {-16 \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^4+8 \left (42+2 e^2+24 e^6-2 e^8+3 e^{12}\right ) \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^2-16 \left (52+6 e^2+45 e^6-3 e^8+12 e^{12}-e^{14}+e^{18}\right ) \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )+\left (2+e^6\right ) \left (6+e^6\right ) \left (48+4 e^2+24 e^6-4 e^8+3 e^{12}\right )}{\left (-4 \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^2+e^{12}+8 e^6+12\right )^2}d\left (x+\frac {1}{4} \left (8+2 e^6\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {-16 \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^4+8 \left (42+2 e^2+24 e^6-2 e^8+3 e^{12}\right ) \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^2-16 \left (52+6 e^2+45 e^6-3 e^8+12 e^{12}-e^{14}+e^{18}\right ) \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )+\left (2+e^6\right ) \left (6+e^6\right ) \left (48+4 e^2+24 e^6-4 e^8+3 e^{12}\right )}{\left (-4 \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^2+e^{12}+8 e^6+12\right )^2}d\left (x+\frac {1}{4} \left (8+2 e^6\right )\right )\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {\int 2 \left (12+8 e^6+e^{12}\right )d\left (x+\frac {1}{4} \left (8+2 e^6\right )\right )}{2 \left (12+8 e^6+e^{12}\right )}+\frac {2 \left (\left (12+8 e^6+e^{12}\right ) \left (52+6 e^2+45 e^6-3 e^8+12 e^{12}-e^{14}+e^{18}\right )-2 \left (2+e^6\right ) \left (6+e^6\right ) \left (15+e^2+8 e^6-e^8+e^{12}\right ) \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )\right )}{\left (12+8 e^6+e^{12}\right ) \left (-4 \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^2+e^{12}+8 e^6+12\right )}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle x+\frac {2 \left (\left (12+8 e^6+e^{12}\right ) \left (52+6 e^2+45 e^6-3 e^8+12 e^{12}-e^{14}+e^{18}\right )-2 \left (2+e^6\right ) \left (6+e^6\right ) \left (15+e^2+8 e^6-e^8+e^{12}\right ) \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )\right )}{\left (12+8 e^6+e^{12}\right ) \left (-4 \left (x+\frac {1}{4} \left (8+2 e^6\right )\right )^2+e^{12}+8 e^6+12\right )}+\frac {1}{4} \left (8+2 e^6\right )\) |
Input:
Int[(3*x^2 + 8*x^3 + x^4 + E^2*(5 + 2*x - x^2) + E^6*(E^2*x^2 + 2*x^3))/(1 + 8*x + 18*x^2 + E^12*x^2 + 8*x^3 + x^4 + E^6*(2*x + 8*x^2 + 2*x^3)),x]
Output:
(8 + 2*E^6)/4 + x + (2*((12 + 8*E^6 + E^12)*(52 + 6*E^2 + 45*E^6 - 3*E^8 + 12*E^12 - E^14 + E^18) - 2*(2 + E^6)*(6 + E^6)*(15 + E^2 + 8*E^6 - E^8 + E^12)*((8 + 2*E^6)/4 + x)))/((12 + 8*E^6 + E^12)*(12 + 8*E^6 + E^12 - 4*(( 8 + 2*E^6)/4 + 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[(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[(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 = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70
method | result | size |
norman | \(\frac {x^{3}+\left (-{\mathrm e}^{6} {\mathrm e}^{2}+{\mathrm e}^{2}\right ) x -{\mathrm e}^{2}}{1+x \,{\mathrm e}^{6}+x^{2}+4 x}\) | \(39\) |
gosper | \(-\frac {{\mathrm e}^{6} {\mathrm e}^{2} x -x^{3}-{\mathrm e}^{2} x +{\mathrm e}^{2}}{1+x \,{\mathrm e}^{6}+x^{2}+4 x}\) | \(40\) |
parallelrisch | \(-\frac {{\mathrm e}^{6} {\mathrm e}^{2} x -x^{3}-{\mathrm e}^{2} x +{\mathrm e}^{2}}{1+x \,{\mathrm e}^{6}+x^{2}+4 x}\) | \(40\) |
risch | \(x +\frac {\left (8 \,{\mathrm e}^{6}+{\mathrm e}^{12}+15-{\mathrm e}^{8}+{\mathrm e}^{2}\right ) x +{\mathrm e}^{6}-{\mathrm e}^{2}+4}{1+x \,{\mathrm e}^{6}+x^{2}+4 x}\) | \(42\) |
default | \(x -\frac {-\frac {\left (8 \,{\mathrm e}^{18}+\left ({\mathrm e}^{12}\right )^{2}-{\mathrm e}^{8} {\mathrm e}^{12}+{\mathrm e}^{12} {\mathrm e}^{2}+91 \,{\mathrm e}^{12}+8 \,{\mathrm e}^{6} {\mathrm e}^{12}-8 \,{\mathrm e}^{6} {\mathrm e}^{8}+10 \,{\mathrm e}^{6} {\mathrm e}^{2}+216 \,{\mathrm e}^{6}-14 \,{\mathrm e}^{8}+12 \,{\mathrm e}^{2}+180\right ) x}{{\mathrm e}^{12}+8 \,{\mathrm e}^{6}+12}-\frac {12 \,{\mathrm e}^{12}+{\mathrm e}^{6} {\mathrm e}^{12}-{\mathrm e}^{6} {\mathrm e}^{8}-4 \,{\mathrm e}^{6} {\mathrm e}^{2}+44 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{8}-12 \,{\mathrm e}^{2}+48}{{\mathrm e}^{12}+8 \,{\mathrm e}^{6}+12}}{1+x \,{\mathrm e}^{6}+x^{2}+4 x}+\frac {4 \left (-{\mathrm e}^{6} {\mathrm e}^{2}+{\mathrm e}^{8}\right ) \operatorname {arctanh}\left (\frac {{\mathrm e}^{6}+2 x +4}{\sqrt {{\mathrm e}^{12}+8 \,{\mathrm e}^{6}+12}}\right )}{\left ({\mathrm e}^{12}+8 \,{\mathrm e}^{6}+12\right )^{\frac {3}{2}}}\) | \(212\) |
Input:
int(((x^2*exp(2)+2*x^3)*exp(3)^2+(-x^2+2*x+5)*exp(2)+x^4+8*x^3+3*x^2)/(x^2 *exp(3)^4+(2*x^3+8*x^2+2*x)*exp(3)^2+x^4+8*x^3+18*x^2+8*x+1),x,method=_RET URNVERBOSE)
Output:
(x^3+(-exp(2)*exp(3)^2+exp(2))*x-exp(2))/(x*exp(3)^2+x^2+4*x+1)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {3 x^2+8 x^3+x^4+e^2 \left (5+2 x-x^2\right )+e^6 \left (e^2 x^2+2 x^3\right )}{1+8 x+18 x^2+e^{12} x^2+8 x^3+x^4+e^6 \left (2 x+8 x^2+2 x^3\right )} \, dx=\frac {x^{3} + 4 \, x^{2} + x e^{12} - x e^{8} + {\left (x^{2} + 8 \, x + 1\right )} e^{6} + {\left (x - 1\right )} e^{2} + 16 \, x + 4}{x^{2} + x e^{6} + 4 \, x + 1} \] Input:
integrate(((x^2*exp(2)+2*x^3)*exp(3)^2+(-x^2+2*x+5)*exp(2)+x^4+8*x^3+3*x^2 )/(x^2*exp(3)^4+(2*x^3+8*x^2+2*x)*exp(3)^2+x^4+8*x^3+18*x^2+8*x+1),x, algo rithm="fricas")
Output:
(x^3 + 4*x^2 + x*e^12 - x*e^8 + (x^2 + 8*x + 1)*e^6 + (x - 1)*e^2 + 16*x + 4)/(x^2 + x*e^6 + 4*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.95 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {3 x^2+8 x^3+x^4+e^2 \left (5+2 x-x^2\right )+e^6 \left (e^2 x^2+2 x^3\right )}{1+8 x+18 x^2+e^{12} x^2+8 x^3+x^4+e^6 \left (2 x+8 x^2+2 x^3\right )} \, dx=x + \frac {x \left (- e^{8} + e^{2} + 15 + 8 e^{6} + e^{12}\right ) - e^{2} + 4 + e^{6}}{x^{2} + x \left (4 + e^{6}\right ) + 1} \] Input:
integrate(((x**2*exp(2)+2*x**3)*exp(3)**2+(-x**2+2*x+5)*exp(2)+x**4+8*x**3 +3*x**2)/(x**2*exp(3)**4+(2*x**3+8*x**2+2*x)*exp(3)**2+x**4+8*x**3+18*x**2 +8*x+1),x)
Output:
x + (x*(-exp(8) + exp(2) + 15 + 8*exp(6) + exp(12)) - exp(2) + 4 + exp(6)) /(x**2 + x*(4 + exp(6)) + 1)
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {3 x^2+8 x^3+x^4+e^2 \left (5+2 x-x^2\right )+e^6 \left (e^2 x^2+2 x^3\right )}{1+8 x+18 x^2+e^{12} x^2+8 x^3+x^4+e^6 \left (2 x+8 x^2+2 x^3\right )} \, dx=x + \frac {x {\left (e^{12} - e^{8} + 8 \, e^{6} + e^{2} + 15\right )} + e^{6} - e^{2} + 4}{x^{2} + x {\left (e^{6} + 4\right )} + 1} \] Input:
integrate(((x^2*exp(2)+2*x^3)*exp(3)^2+(-x^2+2*x+5)*exp(2)+x^4+8*x^3+3*x^2 )/(x^2*exp(3)^4+(2*x^3+8*x^2+2*x)*exp(3)^2+x^4+8*x^3+18*x^2+8*x+1),x, algo rithm="maxima")
Output:
x + (x*(e^12 - e^8 + 8*e^6 + e^2 + 15) + e^6 - e^2 + 4)/(x^2 + x*(e^6 + 4) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {3 x^2+8 x^3+x^4+e^2 \left (5+2 x-x^2\right )+e^6 \left (e^2 x^2+2 x^3\right )}{1+8 x+18 x^2+e^{12} x^2+8 x^3+x^4+e^6 \left (2 x+8 x^2+2 x^3\right )} \, dx=x + \frac {x e^{12} - x e^{8} + 8 \, x e^{6} + x e^{2} + 15 \, x + e^{6} - e^{2} + 4}{x^{2} + x e^{6} + 4 \, x + 1} \] Input:
integrate(((x^2*exp(2)+2*x^3)*exp(3)^2+(-x^2+2*x+5)*exp(2)+x^4+8*x^3+3*x^2 )/(x^2*exp(3)^4+(2*x^3+8*x^2+2*x)*exp(3)^2+x^4+8*x^3+18*x^2+8*x+1),x, algo rithm="giac")
Output:
x + (x*e^12 - x*e^8 + 8*x*e^6 + x*e^2 + 15*x + e^6 - e^2 + 4)/(x^2 + x*e^6 + 4*x + 1)
Time = 0.10 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {3 x^2+8 x^3+x^4+e^2 \left (5+2 x-x^2\right )+e^6 \left (e^2 x^2+2 x^3\right )}{1+8 x+18 x^2+e^{12} x^2+8 x^3+x^4+e^6 \left (2 x+8 x^2+2 x^3\right )} \, dx=x+\frac {{\mathrm {e}}^6-{\mathrm {e}}^2+x\,\left ({\mathrm {e}}^2+8\,{\mathrm {e}}^6-{\mathrm {e}}^8+{\mathrm {e}}^{12}+15\right )+4}{x^2+\left ({\mathrm {e}}^6+4\right )\,x+1} \] Input:
int((exp(2)*(2*x - x^2 + 5) + 3*x^2 + 8*x^3 + x^4 + exp(6)*(x^2*exp(2) + 2 *x^3))/(8*x + exp(6)*(2*x + 8*x^2 + 2*x^3) + x^2*exp(12) + 18*x^2 + 8*x^3 + x^4 + 1),x)
Output:
x + (exp(6) - exp(2) + x*(exp(2) + 8*exp(6) - exp(8) + exp(12) + 15) + 4)/ (x*(exp(6) + 4) + x^2 + 1)
Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int \frac {3 x^2+8 x^3+x^4+e^2 \left (5+2 x-x^2\right )+e^6 \left (e^2 x^2+2 x^3\right )}{1+8 x+18 x^2+e^{12} x^2+8 x^3+x^4+e^6 \left (2 x+8 x^2+2 x^3\right )} \, dx=\frac {e^{8} x^{2}+e^{6} x^{3}-e^{2} x^{2}+4 x^{3}-5 e^{2}}{e^{12} x +e^{6} x^{2}+8 e^{6} x +e^{6}+4 x^{2}+16 x +4} \] Input:
int(((x^2*exp(2)+2*x^3)*exp(3)^2+(-x^2+2*x+5)*exp(2)+x^4+8*x^3+3*x^2)/(x^2 *exp(3)^4+(2*x^3+8*x^2+2*x)*exp(3)^2+x^4+8*x^3+18*x^2+8*x+1),x)
Output:
(e**8*x**2 + e**6*x**3 - e**2*x**2 - 5*e**2 + 4*x**3)/(e**12*x + e**6*x**2 + 8*e**6*x + e**6 + 4*x**2 + 16*x + 4)