Integrand size = 108, antiderivative size = 28 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\frac {4}{9+x}+e^{x^3} \left (3-x-e^{-x} \log (x)\right ) \] Output:
4/(x+9)+(3-x-ln(x)/exp(x))*exp(x^3)
Time = 1.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-e^{x^3} (-3+x)+\frac {4}{9+x}-e^{-x+x^3} \log (x) \] Input:
Integrate[(-4*E^x*x + E^x^3*(-81 - 18*x - x^2 + E^x*(-81*x - 18*x^2 + 728* x^3 - 81*x^4 - 45*x^5 - 3*x^6)) + E^x^3*(81*x + 18*x^2 - 242*x^3 - 54*x^4 - 3*x^5)*Log[x])/(E^x*(81*x + 18*x^2 + x^3)),x]
Output:
-(E^x^3*(-3 + x)) + 4/(9 + x) - E^(-x + x^3)*Log[x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.45 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2026, 2007, 7239, 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.
\(\displaystyle \int \frac {e^{-x} \left (e^{x^3} \left (-3 x^5-54 x^4-242 x^3+18 x^2+81 x\right ) \log (x)+e^{x^3} \left (-x^2+e^x \left (-3 x^6-45 x^5-81 x^4+728 x^3-18 x^2-81 x\right )-18 x-81\right )-4 e^x x\right )}{x^3+18 x^2+81 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{-x} \left (e^{x^3} \left (-3 x^5-54 x^4-242 x^3+18 x^2+81 x\right ) \log (x)+e^{x^3} \left (-x^2+e^x \left (-3 x^6-45 x^5-81 x^4+728 x^3-18 x^2-81 x\right )-18 x-81\right )-4 e^x x\right )}{x \left (x^2+18 x+81\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{-x} \left (e^{x^3} \left (-3 x^5-54 x^4-242 x^3+18 x^2+81 x\right ) \log (x)+e^{x^3} \left (-x^2+e^x \left (-3 x^6-45 x^5-81 x^4+728 x^3-18 x^2-81 x\right )-18 x-81\right )-4 e^x x\right )}{x (x+9)^2}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (-\frac {e^{x^3-x}}{x}+e^{x^3} \left (-3 x^3+9 x^2-1\right )+e^{x^3-x} \left (1-3 x^2\right ) \log (x)-\frac {4}{(x+9)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 e^{x^3}-e^{x^3-x} \log (x)+\frac {x \Gamma \left (\frac {1}{3},-x^3\right )}{3 \sqrt [3]{-x^3}}+\frac {x^4 \Gamma \left (\frac {4}{3},-x^3\right )}{\left (-x^3\right )^{4/3}}+\frac {4}{x+9}\) |
Input:
Int[(-4*E^x*x + E^x^3*(-81 - 18*x - x^2 + E^x*(-81*x - 18*x^2 + 728*x^3 - 81*x^4 - 45*x^5 - 3*x^6)) + E^x^3*(81*x + 18*x^2 - 242*x^3 - 54*x^4 - 3*x^ 5)*Log[x])/(E^x*(81*x + 18*x^2 + x^3)),x]
Output:
3*E^x^3 + 4/(9 + x) + (x*Gamma[1/3, -x^3])/(3*(-x^3)^(1/3)) + (x^4*Gamma[4 /3, -x^3])/(-x^3)^(4/3) - E^(-x + x^3)*Log[x]
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[(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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 3.96 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61
method | result | size |
risch | \(-\ln \left (x \right ) {\mathrm e}^{x \left (-1+x \right ) \left (1+x \right )}-\frac {x^{2} {\mathrm e}^{x^{3}}+6 \,{\mathrm e}^{x^{3}} x -27 \,{\mathrm e}^{x^{3}}-4}{x +9}\) | \(45\) |
parallelrisch | \(-\frac {\left (9 \,{\mathrm e}^{x} {\mathrm e}^{x^{3}} x^{2}+9 \ln \left (x \right ) {\mathrm e}^{x^{3}} x +54 \,{\mathrm e}^{x} {\mathrm e}^{x^{3}} x +81 \,{\mathrm e}^{x^{3}} \ln \left (x \right )-243 \,{\mathrm e}^{x} {\mathrm e}^{x^{3}}-36 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{9 \left (x +9\right )}\) | \(62\) |
orering | \(\text {Expression too large to display}\) | \(6474\) |
Input:
int(((-3*x^5-54*x^4-242*x^3+18*x^2+81*x)*exp(x^3)*ln(x)+((-3*x^6-45*x^5-81 *x^4+728*x^3-18*x^2-81*x)*exp(x)-x^2-18*x-81)*exp(x^3)-4*exp(x)*x)/(x^3+18 *x^2+81*x)/exp(x),x,method=_RETURNVERBOSE)
Output:
-ln(x)*exp(x*(-1+x)*(1+x))-(x^2*exp(x^3)+6*exp(x^3)*x-27*exp(x^3)-4)/(x+9)
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-\frac {{\left ({\left (x + 9\right )} e^{\left (x^{3}\right )} \log \left (x\right ) + {\left (x^{2} + 6 \, x - 27\right )} e^{\left (x^{3} + x\right )} - 4 \, e^{x}\right )} e^{\left (-x\right )}}{x + 9} \] Input:
integrate(((-3*x^5-54*x^4-242*x^3+18*x^2+81*x)*exp(x^3)*log(x)+((-3*x^6-45 *x^5-81*x^4+728*x^3-18*x^2-81*x)*exp(x)-x^2-18*x-81)*exp(x^3)-4*exp(x)*x)/ (x^3+18*x^2+81*x)/exp(x),x, algorithm="fricas")
Output:
-((x + 9)*e^(x^3)*log(x) + (x^2 + 6*x - 27)*e^(x^3 + x) - 4*e^x)*e^(-x)/(x + 9)
Time = 109.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\left (- x + 3 - e^{- x} \log {\left (x \right )}\right ) e^{x^{3}} + \frac {4}{x + 9} \] Input:
integrate(((-3*x**5-54*x**4-242*x**3+18*x**2+81*x)*exp(x**3)*ln(x)+((-3*x* *6-45*x**5-81*x**4+728*x**3-18*x**2-81*x)*exp(x)-x**2-18*x-81)*exp(x**3)-4 *exp(x)*x)/(x**3+18*x**2+81*x)/exp(x),x)
Output:
(-x + 3 - exp(-x)*log(x))*exp(x**3) + 4/(x + 9)
Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-{\left ({\left (x - 3\right )} e^{x} + \log \left (x\right )\right )} e^{\left (x^{3} - x\right )} + \frac {4}{x + 9} \] Input:
integrate(((-3*x^5-54*x^4-242*x^3+18*x^2+81*x)*exp(x^3)*log(x)+((-3*x^6-45 *x^5-81*x^4+728*x^3-18*x^2-81*x)*exp(x)-x^2-18*x-81)*exp(x^3)-4*exp(x)*x)/ (x^3+18*x^2+81*x)/exp(x),x, algorithm="maxima")
Output:
-((x - 3)*e^x + log(x))*e^(x^3 - x) + 4/(x + 9)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=-\frac {x^{2} e^{\left (x^{3}\right )} + x e^{\left (x^{3} - x\right )} \log \left (x\right ) + 6 \, x e^{\left (x^{3}\right )} + 9 \, e^{\left (x^{3} - x\right )} \log \left (x\right ) - 27 \, e^{\left (x^{3}\right )} - 4}{x + 9} \] Input:
integrate(((-3*x^5-54*x^4-242*x^3+18*x^2+81*x)*exp(x^3)*log(x)+((-3*x^6-45 *x^5-81*x^4+728*x^3-18*x^2-81*x)*exp(x)-x^2-18*x-81)*exp(x^3)-4*exp(x)*x)/ (x^3+18*x^2+81*x)/exp(x),x, algorithm="giac")
Output:
-(x^2*e^(x^3) + x*e^(x^3 - x)*log(x) + 6*x*e^(x^3) + 9*e^(x^3 - x)*log(x) - 27*e^(x^3) - 4)/(x + 9)
Timed out. \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\int -\frac {{\mathrm {e}}^{-x}\,\left ({\mathrm {e}}^{x^3}\,\left (18\,x+{\mathrm {e}}^x\,\left (3\,x^6+45\,x^5+81\,x^4-728\,x^3+18\,x^2+81\,x\right )+x^2+81\right )+4\,x\,{\mathrm {e}}^x+{\mathrm {e}}^{x^3}\,\ln \left (x\right )\,\left (3\,x^5+54\,x^4+242\,x^3-18\,x^2-81\,x\right )\right )}{x^3+18\,x^2+81\,x} \,d x \] Input:
int(-(exp(-x)*(exp(x^3)*(18*x + exp(x)*(81*x + 18*x^2 - 728*x^3 + 81*x^4 + 45*x^5 + 3*x^6) + x^2 + 81) + 4*x*exp(x) + exp(x^3)*log(x)*(242*x^3 - 18* x^2 - 81*x + 54*x^4 + 3*x^5)))/(81*x + 18*x^2 + x^3),x)
Output:
int(-(exp(-x)*(exp(x^3)*(18*x + exp(x)*(81*x + 18*x^2 - 728*x^3 + 81*x^4 + 45*x^5 + 3*x^6) + x^2 + 81) + 4*x*exp(x) + exp(x^3)*log(x)*(242*x^3 - 18* x^2 - 81*x + 54*x^4 + 3*x^5)))/(81*x + 18*x^2 + x^3), x)
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {e^{-x} \left (-4 e^x x+e^{x^3} \left (-81-18 x-x^2+e^x \left (-81 x-18 x^2+728 x^3-81 x^4-45 x^5-3 x^6\right )\right )+e^{x^3} \left (81 x+18 x^2-242 x^3-54 x^4-3 x^5\right ) \log (x)\right )}{81 x+18 x^2+x^3} \, dx=\frac {-9 e^{x^{3}+x} x^{2}-54 e^{x^{3}+x} x +243 e^{x^{3}+x}-9 e^{x^{3}} \mathrm {log}\left (x \right ) x -81 e^{x^{3}} \mathrm {log}\left (x \right )-4 e^{x} x}{9 e^{x} \left (x +9\right )} \] Input:
int(((-3*x^5-54*x^4-242*x^3+18*x^2+81*x)*exp(x^3)*log(x)+((-3*x^6-45*x^5-8 1*x^4+728*x^3-18*x^2-81*x)*exp(x)-x^2-18*x-81)*exp(x^3)-4*exp(x)*x)/(x^3+1 8*x^2+81*x)/exp(x),x)
Output:
( - 9*e**(x**3 + x)*x**2 - 54*e**(x**3 + x)*x + 243*e**(x**3 + x) - 9*e**( x**3)*log(x)*x - 81*e**(x**3)*log(x) - 4*e**x*x)/(9*e**x*(x + 9))