Integrand size = 127, antiderivative size = 27 \[ \int \frac {e^{2-e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )}-x} \left (-x \log (x)+e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )} \left (-4 x+\left (-6 x-4 x^2\right ) \log (x)+(-4+(-4-4 x) \log (x)) \log \left (\frac {3}{2} x \log (x)\right )\right )\right )}{x \log (x)} \, dx=e^{2-e^{2 \left (x+\left (x+\log \left (\frac {3}{2} x \log (x)\right )\right )^2\right )}-x} \]
Time = 0.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {e^{2-e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )}-x} \left (-x \log (x)+e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )} \left (-4 x+\left (-6 x-4 x^2\right ) \log (x)+(-4+(-4-4 x) \log (x)) \log \left (\frac {3}{2} x \log (x)\right )\right )\right )}{x \log (x)} \, dx=e^{2-e^{2 \left (x+x^2+2 x \log \left (\frac {3}{2} x \log (x)\right )+\log ^2\left (\frac {3}{2} x \log (x)\right )\right )}-x} \]
Integrate[(E^(2 - E^(2*x + 2*x^2 + 4*x*Log[(3*x*Log[x])/2] + 2*Log[(3*x*Lo g[x])/2]^2) - x)*(-(x*Log[x]) + E^(2*x + 2*x^2 + 4*x*Log[(3*x*Log[x])/2] + 2*Log[(3*x*Log[x])/2]^2)*(-4*x + (-6*x - 4*x^2)*Log[x] + (-4 + (-4 - 4*x) *Log[x])*Log[(3*x*Log[x])/2])))/(x*Log[x]),x]
Time = 4.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {7257}
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.
\(\displaystyle \int \frac {\exp \left (-\exp \left (2 x^2+2 x+2 \log ^2\left (\frac {3}{2} x \log (x)\right )+4 x \log \left (\frac {3}{2} x \log (x)\right )\right )-x+2\right ) \left (\left (\left (-4 x^2-6 x\right ) \log (x)-4 x+((-4 x-4) \log (x)-4) \log \left (\frac {3}{2} x \log (x)\right )\right ) \exp \left (2 x^2+2 x+2 \log ^2\left (\frac {3}{2} x \log (x)\right )+4 x \log \left (\frac {3}{2} x \log (x)\right )\right )-x \log (x)\right )}{x \log (x)} \, dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle \exp \left (-\left (\frac {3}{2}\right )^{4 x} e^{2 x^2+2 x+2 \log ^2\left (\frac {3}{2} x \log (x)\right )} (x \log (x))^{4 x}-x+2\right )\) |
Int[(E^(2 - E^(2*x + 2*x^2 + 4*x*Log[(3*x*Log[x])/2] + 2*Log[(3*x*Log[x])/ 2]^2) - x)*(-(x*Log[x]) + E^(2*x + 2*x^2 + 4*x*Log[(3*x*Log[x])/2] + 2*Log [(3*x*Log[x])/2]^2)*(-4*x + (-6*x - 4*x^2)*Log[x] + (-4 + (-4 - 4*x)*Log[x ])*Log[(3*x*Log[x])/2])))/(x*Log[x]),x]
3.22.66.3.1 Defintions of rubi rules used
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 1.74 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41
method | result | size |
parallelrisch | \({\mathrm e}^{-{\mathrm e}^{2 \ln \left (\frac {3 x \ln \left (x \right )}{2}\right )^{2}+4 x \ln \left (\frac {3 x \ln \left (x \right )}{2}\right )+2 x^{2}+2 x}+2-x}\) | \(38\) |
risch | \(\text {Expression too large to display}\) | \(631\) |
int(((((-4-4*x)*ln(x)-4)*ln(3/2*x*ln(x))+(-4*x^2-6*x)*ln(x)-4*x)*exp(2*ln( 3/2*x*ln(x))^2+4*x*ln(3/2*x*ln(x))+2*x^2+2*x)-x*ln(x))*exp(-exp(2*ln(3/2*x *ln(x))^2+4*x*ln(3/2*x*ln(x))+2*x^2+2*x)+2-x)/x/ln(x),x,method=_RETURNVERB OSE)
Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {e^{2-e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )}-x} \left (-x \log (x)+e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )} \left (-4 x+\left (-6 x-4 x^2\right ) \log (x)+(-4+(-4-4 x) \log (x)) \log \left (\frac {3}{2} x \log (x)\right )\right )\right )}{x \log (x)} \, dx=e^{\left (-x - e^{\left (2 \, x^{2} + 4 \, x \log \left (\frac {3}{2} \, x \log \left (x\right )\right ) + 2 \, \log \left (\frac {3}{2} \, x \log \left (x\right )\right )^{2} + 2 \, x\right )} + 2\right )} \]
integrate(((((-4-4*x)*log(x)-4)*log(3/2*x*log(x))+(-4*x^2-6*x)*log(x)-4*x) *exp(2*log(3/2*x*log(x))^2+4*x*log(3/2*x*log(x))+2*x^2+2*x)-x*log(x))*exp( -exp(2*log(3/2*x*log(x))^2+4*x*log(3/2*x*log(x))+2*x^2+2*x)+2-x)/x/log(x), x, algorithm=\
Time = 13.78 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{2-e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )}-x} \left (-x \log (x)+e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )} \left (-4 x+\left (-6 x-4 x^2\right ) \log (x)+(-4+(-4-4 x) \log (x)) \log \left (\frac {3}{2} x \log (x)\right )\right )\right )}{x \log (x)} \, dx=e^{- x - e^{2 x^{2} + 4 x \log {\left (\frac {3 x \log {\left (x \right )}}{2} \right )} + 2 x + 2 \log {\left (\frac {3 x \log {\left (x \right )}}{2} \right )}^{2}} + 2} \]
integrate(((((-4-4*x)*ln(x)-4)*ln(3/2*x*ln(x))+(-4*x**2-6*x)*ln(x)-4*x)*ex p(2*ln(3/2*x*ln(x))**2+4*x*ln(3/2*x*ln(x))+2*x**2+2*x)-x*ln(x))*exp(-exp(2 *ln(3/2*x*ln(x))**2+4*x*ln(3/2*x*ln(x))+2*x**2+2*x)+2-x)/x/ln(x),x)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (25) = 50\).
Time = 0.62 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.89 \[ \int \frac {e^{2-e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )}-x} \left (-x \log (x)+e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )} \left (-4 x+\left (-6 x-4 x^2\right ) \log (x)+(-4+(-4-4 x) \log (x)) \log \left (\frac {3}{2} x \log (x)\right )\right )\right )}{x \log (x)} \, dx=e^{\left (-x - \frac {e^{\left (2 \, x^{2} + 4 \, x \log \left (3\right ) + 2 \, \log \left (3\right )^{2} - 4 \, x \log \left (2\right ) + 2 \, \log \left (2\right )^{2} + 4 \, x \log \left (x\right ) + 4 \, \log \left (3\right ) \log \left (x\right ) - 4 \, \log \left (2\right ) \log \left (x\right ) + 2 \, \log \left (x\right )^{2} + 4 \, x \log \left (\log \left (x\right )\right ) + 4 \, \log \left (3\right ) \log \left (\log \left (x\right )\right ) - 4 \, \log \left (2\right ) \log \left (\log \left (x\right )\right ) + 4 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 2 \, \log \left (\log \left (x\right )\right )^{2} + 2 \, x\right )}}{2^{4 \, \log \left (3\right )}} + 2\right )} \]
integrate(((((-4-4*x)*log(x)-4)*log(3/2*x*log(x))+(-4*x^2-6*x)*log(x)-4*x) *exp(2*log(3/2*x*log(x))^2+4*x*log(3/2*x*log(x))+2*x^2+2*x)-x*log(x))*exp( -exp(2*log(3/2*x*log(x))^2+4*x*log(3/2*x*log(x))+2*x^2+2*x)+2-x)/x/log(x), x, algorithm=\
e^(-x - e^(2*x^2 + 4*x*log(3) + 2*log(3)^2 - 4*x*log(2) + 2*log(2)^2 + 4*x *log(x) + 4*log(3)*log(x) - 4*log(2)*log(x) + 2*log(x)^2 + 4*x*log(log(x)) + 4*log(3)*log(log(x)) - 4*log(2)*log(log(x)) + 4*log(x)*log(log(x)) + 2* log(log(x))^2 + 2*x)/2^(4*log(3)) + 2)
Time = 19.94 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {e^{2-e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )}-x} \left (-x \log (x)+e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )} \left (-4 x+\left (-6 x-4 x^2\right ) \log (x)+(-4+(-4-4 x) \log (x)) \log \left (\frac {3}{2} x \log (x)\right )\right )\right )}{x \log (x)} \, dx=e^{\left (-x - e^{\left (2 \, x^{2} + 4 \, x \log \left (\frac {3}{2} \, x \log \left (x\right )\right ) + 2 \, \log \left (\frac {3}{2} \, x \log \left (x\right )\right )^{2} + 2 \, x\right )} + 2\right )} \]
integrate(((((-4-4*x)*log(x)-4)*log(3/2*x*log(x))+(-4*x^2-6*x)*log(x)-4*x) *exp(2*log(3/2*x*log(x))^2+4*x*log(3/2*x*log(x))+2*x^2+2*x)-x*log(x))*exp( -exp(2*log(3/2*x*log(x))^2+4*x*log(3/2*x*log(x))+2*x^2+2*x)+2-x)/x/log(x), x, algorithm=\
Time = 9.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81 \[ \int \frac {e^{2-e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )}-x} \left (-x \log (x)+e^{2 x+2 x^2+4 x \log \left (\frac {3}{2} x \log (x)\right )+2 \log ^2\left (\frac {3}{2} x \log (x)\right )} \left (-4 x+\left (-6 x-4 x^2\right ) \log (x)+(-4+(-4-4 x) \log (x)) \log \left (\frac {3}{2} x \log (x)\right )\right )\right )}{x \log (x)} \, dx={\mathrm {e}}^{-\frac {1}{2^{4\,x+4\,\ln \left (3\right )}}\,{81}^x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{2\,{\ln \left (x\,\ln \left (x\right )\right )}^2}\,{\mathrm {e}}^{2\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{2\,{\ln \left (3\right )}^2}\,{\mathrm {e}}^{2\,x^2}\,{\left (x\,\ln \left (x\right )\right )}^{4\,x-4\,\ln \left (2\right )+4\,\ln \left (3\right )}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^2 \]
int(-(exp(2 - exp(2*x + 2*log((3*x*log(x))/2)^2 + 2*x^2 + 4*x*log((3*x*log (x))/2)) - x)*(exp(2*x + 2*log((3*x*log(x))/2)^2 + 2*x^2 + 4*x*log((3*x*lo g(x))/2))*(4*x + log((3*x*log(x))/2)*(log(x)*(4*x + 4) + 4) + log(x)*(6*x + 4*x^2)) + x*log(x)))/(x*log(x)),x)