Integrand size = 137, antiderivative size = 35 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {x+e^{-\frac {-3+\frac {1}{x}-\frac {3}{\log (x)}}{x}} x}{x \left (e^5+2 x\right )} \]
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {1+e^{\frac {3 x+(-1+3 x) \log (x)}{x^2 \log (x)}}}{e^5+2 x} \]
Integrate[(-3*E^5*x - 6*x^2 + (-3*E^5*x - 6*x^2)*Log[x] - 2*E^((-3*x + (1 - 3*x)*Log[x])/(x^2*Log[x]))*x^3*Log[x]^2 + (E^5*(2 - 3*x) + 4*x - 6*x^2 - 2*x^3)*Log[x]^2)/(E^((-3*x + (1 - 3*x)*Log[x])/(x^2*Log[x]))*(E^10*x^3 + 4*E^5*x^4 + 4*x^5)*Log[x]^2),x]
Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(35)=70\).
Time = 16.69 (sec) , antiderivative size = 180, normalized size of antiderivative = 5.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2026, 2007, 7293, 7292, 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.
\(\displaystyle \int \frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-6 x^2+\left (-6 x^2-3 e^5 x\right ) \log (x)-2 x^3 e^{\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \log ^2(x)+\left (-2 x^3-6 x^2+4 x+e^5 (2-3 x)\right ) \log ^2(x)-3 e^5 x\right )}{\left (4 x^5+4 e^5 x^4+e^{10} x^3\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-6 x^2+\left (-6 x^2-3 e^5 x\right ) \log (x)-2 x^3 e^{\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \log ^2(x)+\left (-2 x^3-6 x^2+4 x+e^5 (2-3 x)\right ) \log ^2(x)-3 e^5 x\right )}{x^3 \left (4 x^2+4 e^5 x+e^{10}\right ) \log ^2(x)}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-6 x^2+\left (-6 x^2-3 e^5 x\right ) \log (x)-2 x^3 e^{\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \log ^2(x)+\left (-2 x^3-6 x^2+4 x+e^5 (2-3 x)\right ) \log ^2(x)-3 e^5 x\right )}{x^3 \left (2 x+e^5\right )^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-2 x^3 \log ^2(x)-6 x^2-6 x^2 \log ^2(x)-6 x^2 \log (x)-3 e^5 x+4 \left (1-\frac {3 e^5}{4}\right ) x \log ^2(x)+2 e^5 \log ^2(x)-3 e^5 x \log (x)\right )}{x^3 \left (2 x+e^5\right )^2 \log ^2(x)}-\frac {2 \exp \left (\frac {-3 x-3 x \log (x)+\log (x)}{x^2 \log (x)}-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}\right )}{\left (2 x+e^5\right )^2}\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \left (\frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-2 x^3 \log ^2(x)-6 x^2-6 x^2 \log ^2(x)-6 x^2 \log (x)-3 e^5 x+4 \left (1-\frac {3 e^5}{4}\right ) x \log ^2(x)+2 e^5 \log ^2(x)-3 e^5 x \log (x)\right )}{x^3 \left (2 x+e^5\right )^2 \log ^2(x)}-\frac {2}{\left (2 x+e^5\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2 x+e^5}-\frac {e^{\frac {3}{x \log (x)}-\frac {1-3 x}{x^2}} \left (6 x^2+6 x^2 \log ^2(x)+6 x^2 \log (x)+3 e^5 x-\left (4-3 e^5\right ) x \log ^2(x)-2 e^5 \log ^2(x)+3 e^5 x \log (x)\right )}{x^3 \left (2 x+e^5\right )^2 \log ^2(x) \left (-\frac {3 x-(1-3 x) \log (x)}{x^3 \log ^2(x)}-\frac {2 (3 x-(1-3 x) \log (x))}{x^3 \log (x)}+\frac {-\frac {1-3 x}{x}+3 \log (x)+3}{x^2 \log (x)}\right )}\) |
Int[(-3*E^5*x - 6*x^2 + (-3*E^5*x - 6*x^2)*Log[x] - 2*E^((-3*x + (1 - 3*x) *Log[x])/(x^2*Log[x]))*x^3*Log[x]^2 + (E^5*(2 - 3*x) + 4*x - 6*x^2 - 2*x^3 )*Log[x]^2)/(E^((-3*x + (1 - 3*x)*Log[x])/(x^2*Log[x]))*(E^10*x^3 + 4*E^5* x^4 + 4*x^5)*Log[x]^2),x]
(E^5 + 2*x)^(-1) - (E^(-((1 - 3*x)/x^2) + 3/(x*Log[x]))*(3*E^5*x + 6*x^2 + 3*E^5*x*Log[x] + 6*x^2*Log[x] - 2*E^5*Log[x]^2 - (4 - 3*E^5)*x*Log[x]^2 + 6*x^2*Log[x]^2))/(x^3*(E^5 + 2*x)^2*Log[x]^2*((3 - (1 - 3*x)/x + 3*Log[x] )/(x^2*Log[x]) - (3*x - (1 - 3*x)*Log[x])/(x^3*Log[x]^2) - (2*(3*x - (1 - 3*x)*Log[x]))/(x^3*Log[x])))
3.5.33.3.1 Defintions of rubi rules used
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])
Time = 8.58 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {1}{2 x +{\mathrm e}^{5}}+\frac {{\mathrm e}^{\frac {3 x \ln \left (x \right )-\ln \left (x \right )+3 x}{\ln \left (x \right ) x^{2}}}}{2 x +{\mathrm e}^{5}}\) | \(41\) |
parallelrisch | \(\frac {\left (2 \,{\mathrm e}^{\frac {\left (1-3 x \right ) \ln \left (x \right )-3 x}{x^{2} \ln \left (x \right )}} \ln \left (x \right ) x^{2}+2 x^{2} \ln \left (x \right )\right ) {\mathrm e}^{-\frac {\left (1-3 x \right ) \ln \left (x \right )-3 x}{x^{2} \ln \left (x \right )}}}{2 \ln \left (x \right ) x^{2} \left (2 x +{\mathrm e}^{5}\right )}\) | \(77\) |
int((-2*x^3*ln(x)^2*exp(((1-3*x)*ln(x)-3*x)/x^2/ln(x))+((2-3*x)*exp(5)-2*x ^3-6*x^2+4*x)*ln(x)^2+(-3*x*exp(5)-6*x^2)*ln(x)-3*x*exp(5)-6*x^2)/(x^3*exp (5)^2+4*x^4*exp(5)+4*x^5)/ln(x)^2/exp(((1-3*x)*ln(x)-3*x)/x^2/ln(x)),x,met hod=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {e^{\left (\frac {{\left (3 \, x - 1\right )} \log \left (x\right ) + 3 \, x}{x^{2} \log \left (x\right )}\right )} + 1}{2 \, x + e^{5}} \]
integrate((-2*x^3*log(x)^2*exp(((1-3*x)*log(x)-3*x)/x^2/log(x))+((2-3*x)*e xp(5)-2*x^3-6*x^2+4*x)*log(x)^2+(-3*x*exp(5)-6*x^2)*log(x)-3*x*exp(5)-6*x^ 2)/(x^3*exp(5)^2+4*x^4*exp(5)+4*x^5)/log(x)^2/exp(((1-3*x)*log(x)-3*x)/x^2 /log(x)),x, algorithm=\
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {2}{4 x + 2 e^{5}} + \frac {e^{- \frac {- 3 x + \left (1 - 3 x\right ) \log {\left (x \right )}}{x^{2} \log {\left (x \right )}}}}{2 x + e^{5}} \]
integrate((-2*x**3*ln(x)**2*exp(((1-3*x)*ln(x)-3*x)/x**2/ln(x))+((2-3*x)*e xp(5)-2*x**3-6*x**2+4*x)*ln(x)**2+(-3*x*exp(5)-6*x**2)*ln(x)-3*x*exp(5)-6* x**2)/(x**3*exp(5)**2+4*x**4*exp(5)+4*x**5)/ln(x)**2/exp(((1-3*x)*ln(x)-3* x)/x**2/ln(x)),x)
Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {{\left (e^{\left (\frac {3}{x} + \frac {3}{x \log \left (x\right )}\right )} + e^{\left (\frac {1}{x^{2}}\right )}\right )} e^{\left (-\frac {1}{x^{2}}\right )}}{2 \, x + e^{5}} \]
integrate((-2*x^3*log(x)^2*exp(((1-3*x)*log(x)-3*x)/x^2/log(x))+((2-3*x)*e xp(5)-2*x^3-6*x^2+4*x)*log(x)^2+(-3*x*exp(5)-6*x^2)*log(x)-3*x*exp(5)-6*x^ 2)/(x^3*exp(5)^2+4*x^4*exp(5)+4*x^5)/log(x)^2/exp(((1-3*x)*log(x)-3*x)/x^2 /log(x)),x, algorithm=\
Exception generated. \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]
integrate((-2*x^3*log(x)^2*exp(((1-3*x)*log(x)-3*x)/x^2/log(x))+((2-3*x)*e xp(5)-2*x^3-6*x^2+4*x)*log(x)^2+(-3*x*exp(5)-6*x^2)*log(x)-3*x*exp(5)-6*x^ 2)/(x^3*exp(5)^2+4*x^4*exp(5)+4*x^5)/log(x)^2/exp(((1-3*x)*log(x)-3*x)/x^2 /log(x)),x, algorithm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-13824,[0,25,0]%%%}+%%%{-48384,[0,24,1]%%%}+%%%{-72576,[0, 23,2]%%%}
Time = 8.68 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {1}{2\,x+{\mathrm {e}}^5}+\frac {{\mathrm {e}}^{-\frac {1}{x^2}}\,{\mathrm {e}}^{3/x}\,{\mathrm {e}}^{\frac {3}{x\,\ln \left (x\right )}}}{2\,x+{\mathrm {e}}^5} \]