Integrand size = 181, antiderivative size = 29 \[ \int \frac {e^{81 x^2+216 x^2 \log \left (-e^{-4+x}+x^2-\log (x)\right )+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )} \left (216 x-594 x^3+e^{-4+x} \left (162 x+216 x^2\right )+162 x \log (x)+\left (288 x-1008 x^3+e^{-4+x} \left (432 x+288 x^2\right )+432 x \log (x)\right ) \log \left (-e^{-4+x}+x^2-\log (x)\right )+\left (288 e^{-4+x} x-288 x^3+288 x \log (x)\right ) \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )\right )}{e^{-4+x}-x^2+\log (x)} \, dx=e^{9 x^2 \left (3+4 \log \left (-e^{-4+x}+x^2-\log (x)\right )\right )^2} \]
Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {e^{81 x^2+216 x^2 \log \left (-e^{-4+x}+x^2-\log (x)\right )+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )} \left (216 x-594 x^3+e^{-4+x} \left (162 x+216 x^2\right )+162 x \log (x)+\left (288 x-1008 x^3+e^{-4+x} \left (432 x+288 x^2\right )+432 x \log (x)\right ) \log \left (-e^{-4+x}+x^2-\log (x)\right )+\left (288 e^{-4+x} x-288 x^3+288 x \log (x)\right ) \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )\right )}{e^{-4+x}-x^2+\log (x)} \, dx=e^{81 x^2+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )} \left (-e^{-4+x}+x^2-\log (x)\right )^{216 x^2} \]
Integrate[(E^(81*x^2 + 216*x^2*Log[-E^(-4 + x) + x^2 - Log[x]] + 144*x^2*L og[-E^(-4 + x) + x^2 - Log[x]]^2)*(216*x - 594*x^3 + E^(-4 + x)*(162*x + 2 16*x^2) + 162*x*Log[x] + (288*x - 1008*x^3 + E^(-4 + x)*(432*x + 288*x^2) + 432*x*Log[x])*Log[-E^(-4 + x) + x^2 - Log[x]] + (288*E^(-4 + x)*x - 288* x^3 + 288*x*Log[x])*Log[-E^(-4 + x) + x^2 - Log[x]]^2))/(E^(-4 + x) - x^2 + Log[x]),x]
E^(81*x^2 + 144*x^2*Log[-E^(-4 + x) + x^2 - Log[x]]^2)*(-E^(-4 + x) + x^2 - Log[x])^(216*x^2)
Time = 6.89 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, 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.
\(\displaystyle \int \frac {\left (-594 x^3+e^{x-4} \left (216 x^2+162 x\right )+\left (-288 x^3+288 e^{x-4} x+288 x \log (x)\right ) \log ^2\left (x^2-e^{x-4}-\log (x)\right )+\left (-1008 x^3+e^{x-4} \left (288 x^2+432 x\right )+288 x+432 x \log (x)\right ) \log \left (x^2-e^{x-4}-\log (x)\right )+216 x+162 x \log (x)\right ) \exp \left (81 x^2+144 x^2 \log ^2\left (x^2-e^{x-4}-\log (x)\right )+216 x^2 \log \left (x^2-e^{x-4}-\log (x)\right )\right )}{-x^2+e^{x-4}+\log (x)} \, dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle e^{81 x^2+144 x^2 \log ^2\left (x^2-e^{x-4}-\log (x)\right )} \left (x^2-e^{x-4}-\log (x)\right )^{216 x^2}\) |
Int[(E^(81*x^2 + 216*x^2*Log[-E^(-4 + x) + x^2 - Log[x]] + 144*x^2*Log[-E^ (-4 + x) + x^2 - Log[x]]^2)*(216*x - 594*x^3 + E^(-4 + x)*(162*x + 216*x^2 ) + 162*x*Log[x] + (288*x - 1008*x^3 + E^(-4 + x)*(432*x + 288*x^2) + 432* x*Log[x])*Log[-E^(-4 + x) + x^2 - Log[x]] + (288*E^(-4 + x)*x - 288*x^3 + 288*x*Log[x])*Log[-E^(-4 + x) + x^2 - Log[x]]^2))/(E^(-4 + x) - x^2 + Log[ x]),x]
E^(81*x^2 + 144*x^2*Log[-E^(-4 + x) + x^2 - Log[x]]^2)*(-E^(-4 + x) + x^2 - Log[x])^(216*x^2)
3.6.47.3.1 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 = 24.54 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55
method | result | size |
parallelrisch | \({\mathrm e}^{9 x^{2} \left (16 \ln \left (-\ln \left (x \right )-{\mathrm e}^{x -4}+x^{2}\right )^{2}+24 \ln \left (-\ln \left (x \right )-{\mathrm e}^{x -4}+x^{2}\right )+9\right )}\) | \(45\) |
risch | \(\left (-\ln \left (x \right )-{\mathrm e}^{x -4}+x^{2}\right )^{216 x^{2}} {\mathrm e}^{9 x^{2} \left (16 \ln \left (-\ln \left (x \right )-{\mathrm e}^{x -4}+x^{2}\right )^{2}+9\right )}\) | \(49\) |
int(((288*x*ln(x)+288*x*exp(x-4)-288*x^3)*ln(-ln(x)-exp(x-4)+x^2)^2+(432*x *ln(x)+(288*x^2+432*x)*exp(x-4)-1008*x^3+288*x)*ln(-ln(x)-exp(x-4)+x^2)+16 2*x*ln(x)+(216*x^2+162*x)*exp(x-4)-594*x^3+216*x)*exp(144*x^2*ln(-ln(x)-ex p(x-4)+x^2)^2+216*x^2*ln(-ln(x)-exp(x-4)+x^2)+81*x^2)/(ln(x)+exp(x-4)-x^2) ,x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {e^{81 x^2+216 x^2 \log \left (-e^{-4+x}+x^2-\log (x)\right )+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )} \left (216 x-594 x^3+e^{-4+x} \left (162 x+216 x^2\right )+162 x \log (x)+\left (288 x-1008 x^3+e^{-4+x} \left (432 x+288 x^2\right )+432 x \log (x)\right ) \log \left (-e^{-4+x}+x^2-\log (x)\right )+\left (288 e^{-4+x} x-288 x^3+288 x \log (x)\right ) \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )\right )}{e^{-4+x}-x^2+\log (x)} \, dx=e^{\left (144 \, x^{2} \log \left (x^{2} - e^{\left (x - 4\right )} - \log \left (x\right )\right )^{2} + 216 \, x^{2} \log \left (x^{2} - e^{\left (x - 4\right )} - \log \left (x\right )\right ) + 81 \, x^{2}\right )} \]
integrate(((288*x*log(x)+288*x*exp(x-4)-288*x^3)*log(-log(x)-exp(x-4)+x^2) ^2+(432*x*log(x)+(288*x^2+432*x)*exp(x-4)-1008*x^3+288*x)*log(-log(x)-exp( x-4)+x^2)+162*x*log(x)+(216*x^2+162*x)*exp(x-4)-594*x^3+216*x)*exp(144*x^2 *log(-log(x)-exp(x-4)+x^2)^2+216*x^2*log(-log(x)-exp(x-4)+x^2)+81*x^2)/(lo g(x)+exp(x-4)-x^2),x, algorithm=\
Timed out. \[ \int \frac {e^{81 x^2+216 x^2 \log \left (-e^{-4+x}+x^2-\log (x)\right )+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )} \left (216 x-594 x^3+e^{-4+x} \left (162 x+216 x^2\right )+162 x \log (x)+\left (288 x-1008 x^3+e^{-4+x} \left (432 x+288 x^2\right )+432 x \log (x)\right ) \log \left (-e^{-4+x}+x^2-\log (x)\right )+\left (288 e^{-4+x} x-288 x^3+288 x \log (x)\right ) \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )\right )}{e^{-4+x}-x^2+\log (x)} \, dx=\text {Timed out} \]
integrate(((288*x*ln(x)+288*x*exp(x-4)-288*x**3)*ln(-ln(x)-exp(x-4)+x**2)* *2+(432*x*ln(x)+(288*x**2+432*x)*exp(x-4)-1008*x**3+288*x)*ln(-ln(x)-exp(x -4)+x**2)+162*x*ln(x)+(216*x**2+162*x)*exp(x-4)-594*x**3+216*x)*exp(144*x* *2*ln(-ln(x)-exp(x-4)+x**2)**2+216*x**2*ln(-ln(x)-exp(x-4)+x**2)+81*x**2)/ (ln(x)+exp(x-4)-x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).
Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {e^{81 x^2+216 x^2 \log \left (-e^{-4+x}+x^2-\log (x)\right )+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )} \left (216 x-594 x^3+e^{-4+x} \left (162 x+216 x^2\right )+162 x \log (x)+\left (288 x-1008 x^3+e^{-4+x} \left (432 x+288 x^2\right )+432 x \log (x)\right ) \log \left (-e^{-4+x}+x^2-\log (x)\right )+\left (288 e^{-4+x} x-288 x^3+288 x \log (x)\right ) \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )\right )}{e^{-4+x}-x^2+\log (x)} \, dx=e^{\left (144 \, x^{2} \log \left (x^{2} e^{4} - e^{4} \log \left (x\right ) - e^{x}\right )^{2} - 936 \, x^{2} \log \left (x^{2} e^{4} - e^{4} \log \left (x\right ) - e^{x}\right ) + 1521 \, x^{2}\right )} \]
integrate(((288*x*log(x)+288*x*exp(x-4)-288*x^3)*log(-log(x)-exp(x-4)+x^2) ^2+(432*x*log(x)+(288*x^2+432*x)*exp(x-4)-1008*x^3+288*x)*log(-log(x)-exp( x-4)+x^2)+162*x*log(x)+(216*x^2+162*x)*exp(x-4)-594*x^3+216*x)*exp(144*x^2 *log(-log(x)-exp(x-4)+x^2)^2+216*x^2*log(-log(x)-exp(x-4)+x^2)+81*x^2)/(lo g(x)+exp(x-4)-x^2),x, algorithm=\
e^(144*x^2*log(x^2*e^4 - e^4*log(x) - e^x)^2 - 936*x^2*log(x^2*e^4 - e^4*l og(x) - e^x) + 1521*x^2)
Time = 15.79 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {e^{81 x^2+216 x^2 \log \left (-e^{-4+x}+x^2-\log (x)\right )+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )} \left (216 x-594 x^3+e^{-4+x} \left (162 x+216 x^2\right )+162 x \log (x)+\left (288 x-1008 x^3+e^{-4+x} \left (432 x+288 x^2\right )+432 x \log (x)\right ) \log \left (-e^{-4+x}+x^2-\log (x)\right )+\left (288 e^{-4+x} x-288 x^3+288 x \log (x)\right ) \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )\right )}{e^{-4+x}-x^2+\log (x)} \, dx=e^{\left (144 \, x^{2} \log \left (x^{2} - e^{\left (x - 4\right )} - \log \left (x\right )\right )^{2} + 216 \, x^{2} \log \left (x^{2} - e^{\left (x - 4\right )} - \log \left (x\right )\right ) + 81 \, x^{2}\right )} \]
integrate(((288*x*log(x)+288*x*exp(x-4)-288*x^3)*log(-log(x)-exp(x-4)+x^2) ^2+(432*x*log(x)+(288*x^2+432*x)*exp(x-4)-1008*x^3+288*x)*log(-log(x)-exp( x-4)+x^2)+162*x*log(x)+(216*x^2+162*x)*exp(x-4)-594*x^3+216*x)*exp(144*x^2 *log(-log(x)-exp(x-4)+x^2)^2+216*x^2*log(-log(x)-exp(x-4)+x^2)+81*x^2)/(lo g(x)+exp(x-4)-x^2),x, algorithm=\
Time = 9.44 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {e^{81 x^2+216 x^2 \log \left (-e^{-4+x}+x^2-\log (x)\right )+144 x^2 \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )} \left (216 x-594 x^3+e^{-4+x} \left (162 x+216 x^2\right )+162 x \log (x)+\left (288 x-1008 x^3+e^{-4+x} \left (432 x+288 x^2\right )+432 x \log (x)\right ) \log \left (-e^{-4+x}+x^2-\log (x)\right )+\left (288 e^{-4+x} x-288 x^3+288 x \log (x)\right ) \log ^2\left (-e^{-4+x}+x^2-\log (x)\right )\right )}{e^{-4+x}-x^2+\log (x)} \, dx={\mathrm {e}}^{144\,x^2\,{\ln \left (x^2-{\mathrm {e}}^{-4}\,{\mathrm {e}}^x-\ln \left (x\right )\right )}^2}\,{\mathrm {e}}^{81\,x^2}\,{\left (x^2-{\mathrm {e}}^{-4}\,{\mathrm {e}}^x-\ln \left (x\right )\right )}^{216\,x^2} \]
int((exp(144*x^2*log(x^2 - log(x) - exp(x - 4))^2 + 216*x^2*log(x^2 - log( x) - exp(x - 4)) + 81*x^2)*(216*x + exp(x - 4)*(162*x + 216*x^2) + log(x^2 - log(x) - exp(x - 4))*(288*x + exp(x - 4)*(432*x + 288*x^2) + 432*x*log( x) - 1008*x^3) + log(x^2 - log(x) - exp(x - 4))^2*(288*x*exp(x - 4) + 288* x*log(x) - 288*x^3) + 162*x*log(x) - 594*x^3))/(exp(x - 4) + log(x) - x^2) ,x)