Integrand size = 146, antiderivative size = 32 \[ \int \frac {4 e^{2-x}+4 x+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x}-2 e^{2-x} x-x^2\right )+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x} x-2 e^{2-x} x^2-x^3\right ) \log (x)+\left (-x+e^{2-x} x\right ) \log \left (x^4\right )}{e^{4-2 x} x+2 e^{2-x} x^2+x^3} \, dx=-e^{5+x+\frac {4}{\log (3)}} \log (x)+\frac {\log \left (x^4\right )}{e^{2-x}+x} \] Output:
ln(x^4)/(x+exp(2-x))-exp(x+5+4/ln(3))*ln(x)
Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {4 e^{2-x}+4 x+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x}-2 e^{2-x} x-x^2\right )+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x} x-2 e^{2-x} x^2-x^3\right ) \log (x)+\left (-x+e^{2-x} x\right ) \log \left (x^4\right )}{e^{4-2 x} x+2 e^{2-x} x^2+x^3} \, dx=-e^{5+x+\frac {4}{\log (3)}} \log (x)+\frac {e^x \log \left (x^4\right )}{e^2+e^x x} \] Input:
Integrate[(4*E^(2 - x) + 4*x + E^((4 + (5 + x)*Log[3])/Log[3])*(-E^(4 - 2* x) - 2*E^(2 - x)*x - x^2) + E^((4 + (5 + x)*Log[3])/Log[3])*(-(E^(4 - 2*x) *x) - 2*E^(2 - x)*x^2 - x^3)*Log[x] + (-x + E^(2 - x)*x)*Log[x^4])/(E^(4 - 2*x)*x + 2*E^(2 - x)*x^2 + x^3),x]
Output:
-(E^(5 + x + 4/Log[3])*Log[x]) + (E^x*Log[x^4])/(E^2 + E^x*x)
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.
| \(\displaystyle \int \frac {\left (e^{2-x} x-x\right ) \log \left (x^4\right )+\left (-x^2-2 e^{2-x} x-e^{4-2 x}\right ) e^{\frac {(x+5) \log (3)+4}{\log (3)}}+\left (-x^3-2 e^{2-x} x^2-e^{4-2 x} x\right ) e^{\frac {(x+5) \log (3)+4}{\log (3)}} \log (x)+4 x+4 e^{2-x}}{x^3+2 e^{2-x} x^2+e^{4-2 x} x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int e^x \left (-\frac {\left (e^x-e^2\right ) x \log \left (x^4\right )+\left (e^x x+e^2\right ) \left (x e^{x+5+\frac {4}{\log (3)}}-4+e^{7+\frac {4}{\log (3)}}\right )}{x \left (e^x x+e^2\right )^2}-e^{5+\frac {4}{\log (3)}} \log (x)\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^x \left (-e^x x \log \left (x^4\right )+e^2 x \log \left (x^4\right )+x^2 \left (-e^{2 x+5+\frac {4}{\log (3)}}\right )+4 e^x x \left (1-\frac {1}{2} e^{7+\frac {4}{\log (3)}}\right )+4 e^2 \left (1-\frac {1}{4} e^{7+\frac {4}{\log (3)}}\right )\right )}{x \left (e^x x+e^2\right )^2}-e^{x+5+\frac {4}{\log (3)}} \log (x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\log \left (x^4\right ) \int \frac {e^{2 x}}{\left (e^x x+e^2\right )^2}dx+\log \left (x^4\right ) \int \frac {e^{x+2}}{\left (e^x x+e^2\right )^2}dx+4 \int \frac {\int \frac {e^{2 x}}{\left (e^x x+e^2\right )^2}dx}{x}dx-4 \int \frac {\int \frac {e^{x+2}}{\left (e^x x+e^2\right )^2}dx}{x}dx+2 \left (2-e^{7+\frac {4}{\log (3)}}\right ) \int \frac {e^{2 x}}{\left (e^x x+e^2\right )^2}dx+\left (4-e^{7+\frac {4}{\log (3)}}\right ) \int \frac {e^{x+2}}{x \left (e^x x+e^2\right )^2}dx-\int \frac {e^{3 x+\frac {4}{\log (3)}+5} x}{\left (e^x x+e^2\right )^2}dx+e^{5+\frac {4}{\log (3)}} \operatorname {ExpIntegralEi}(x)-e^{x+5+\frac {4}{\log (3)}} \log (x)\) |
Input:
Int[(4*E^(2 - x) + 4*x + E^((4 + (5 + x)*Log[3])/Log[3])*(-E^(4 - 2*x) - 2 *E^(2 - x)*x - x^2) + E^((4 + (5 + x)*Log[3])/Log[3])*(-(E^(4 - 2*x)*x) - 2*E^(2 - x)*x^2 - x^3)*Log[x] + (-x + E^(2 - x)*x)*Log[x^4])/(E^(4 - 2*x)* x + 2*E^(2 - x)*x^2 + x^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(30)=60\).
Time = 2.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.97
| method | result | size |
| parallelrisch | \(-\frac {-48 \ln \left (x^{4}\right )+48 \,{\mathrm e}^{\frac {\left (5+x \right ) \ln \left (3\right )+4}{\ln \left (3\right )}} {\mathrm e}^{2-x} \ln \left (x \right )+48 \,{\mathrm e}^{\frac {\left (5+x \right ) \ln \left (3\right )+4}{\ln \left (3\right )}} \ln \left (x \right ) x}{48 \left (x +{\mathrm e}^{2-x}\right )}\) | \(63\) |
| risch | \(-\frac {\left (x \,{\mathrm e}^{\frac {7 \ln \left (3\right )+4}{\ln \left (3\right )}}+{\mathrm e}^{-\frac {x \ln \left (3\right )-9 \ln \left (3\right )-4}{\ln \left (3\right )}}-4 \,{\mathrm e}^{2-x}\right ) {\mathrm e}^{-2+x} \ln \left (x \right )}{x +{\mathrm e}^{2-x}}-\frac {i \left (\operatorname {csgn}\left (i x^{3}\right )^{3}-\operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x \right )-\operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x^{2}\right )-\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right ) \operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x^{4}\right )^{3}-\operatorname {csgn}\left (i x^{4}\right )^{2} \operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-2 \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\operatorname {csgn}\left (i x^{2}\right )^{3}\right ) \pi }{2 \left (x +{\mathrm e}^{2-x}\right )}\) | \(241\) |
Input:
int(((x*exp(2-x)-x)*ln(x^4)+(-x*exp(2-x)^2-2*x^2*exp(2-x)-x^3)*exp(((5+x)* ln(3)+4)/ln(3))*ln(x)+(-exp(2-x)^2-2*x*exp(2-x)-x^2)*exp(((5+x)*ln(3)+4)/l n(3))+4*exp(2-x)+4*x)/(x*exp(2-x)^2+2*x^2*exp(2-x)+x^3),x,method=_RETURNVE RBOSE)
Output:
-1/48*(-48*ln(x^4)+48*exp(((5+x)*ln(3)+4)/ln(3))*exp(2-x)*ln(x)+48*exp(((5 +x)*ln(3)+4)/ln(3))*ln(x)*x)/(x+exp(2-x))
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (30) = 60\).
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {4 e^{2-x}+4 x+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x}-2 e^{2-x} x-x^2\right )+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x} x-2 e^{2-x} x^2-x^3\right ) \log (x)+\left (-x+e^{2-x} x\right ) \log \left (x^4\right )}{e^{4-2 x} x+2 e^{2-x} x^2+x^3} \, dx=-\frac {{\left (x e^{\left (\frac {2 \, {\left ({\left (x + 5\right )} \log \left (3\right ) + 4\right )}}{\log \left (3\right )}\right )} + {\left (e^{\left (\frac {7 \, \log \left (3\right ) + 4}{\log \left (3\right )}\right )} - 4\right )} e^{\left (\frac {{\left (x + 5\right )} \log \left (3\right ) + 4}{\log \left (3\right )}\right )}\right )} \log \left (x\right )}{x e^{\left (\frac {{\left (x + 5\right )} \log \left (3\right ) + 4}{\log \left (3\right )}\right )} + e^{\left (\frac {7 \, \log \left (3\right ) + 4}{\log \left (3\right )}\right )}} \] Input:
integrate(((x*exp(2-x)-x)*log(x^4)+(-x*exp(2-x)^2-2*x^2*exp(2-x)-x^3)*exp( ((5+x)*log(3)+4)/log(3))*log(x)+(-exp(2-x)^2-2*x*exp(2-x)-x^2)*exp(((5+x)* log(3)+4)/log(3))+4*exp(2-x)+4*x)/(x*exp(2-x)^2+2*x^2*exp(2-x)+x^3),x, alg orithm="fricas")
Output:
-(x*e^(2*((x + 5)*log(3) + 4)/log(3)) + (e^((7*log(3) + 4)/log(3)) - 4)*e^ (((x + 5)*log(3) + 4)/log(3)))*log(x)/(x*e^(((x + 5)*log(3) + 4)/log(3)) + e^((7*log(3) + 4)/log(3)))
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {4 e^{2-x}+4 x+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x}-2 e^{2-x} x-x^2\right )+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x} x-2 e^{2-x} x^2-x^3\right ) \log (x)+\left (-x+e^{2-x} x\right ) \log \left (x^4\right )}{e^{4-2 x} x+2 e^{2-x} x^2+x^3} \, dx=- e^{\frac {\left (x + 5\right ) \log {\left (3 \right )} + 4}{\log {\left (3 \right )}}} \log {\left (x \right )} - \frac {4 e^{7} e^{\frac {4}{\log {\left (3 \right )}}} \log {\left (x \right )}}{x^{2} e^{\frac {\left (x + 5\right ) \log {\left (3 \right )} + 4}{\log {\left (3 \right )}}} + x e^{7} e^{\frac {4}{\log {\left (3 \right )}}}} + \frac {4 \log {\left (x \right )}}{x} \] Input:
integrate(((x*exp(2-x)-x)*ln(x**4)+(-x*exp(2-x)**2-2*x**2*exp(2-x)-x**3)*e xp(((5+x)*ln(3)+4)/ln(3))*ln(x)+(-exp(2-x)**2-2*x*exp(2-x)-x**2)*exp(((5+x )*ln(3)+4)/ln(3))+4*exp(2-x)+4*x)/(x*exp(2-x)**2+2*x**2*exp(2-x)+x**3),x)
Output:
-exp(((x + 5)*log(3) + 4)/log(3))*log(x) - 4*exp(7)*exp(4/log(3))*log(x)/( x**2*exp(((x + 5)*log(3) + 4)/log(3)) + x*exp(7)*exp(4/log(3))) + 4*log(x) /x
Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {4 e^{2-x}+4 x+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x}-2 e^{2-x} x-x^2\right )+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x} x-2 e^{2-x} x^2-x^3\right ) \log (x)+\left (-x+e^{2-x} x\right ) \log \left (x^4\right )}{e^{4-2 x} x+2 e^{2-x} x^2+x^3} \, dx=-\frac {x e^{\left (2 \, x + \frac {4}{\log \left (3\right )} + 5\right )} \log \left (x\right ) + {\left (e^{\left (\frac {4}{\log \left (3\right )} + 7\right )} - 4\right )} e^{x} \log \left (x\right )}{x e^{x} + e^{2}} \] Input:
integrate(((x*exp(2-x)-x)*log(x^4)+(-x*exp(2-x)^2-2*x^2*exp(2-x)-x^3)*exp( ((5+x)*log(3)+4)/log(3))*log(x)+(-exp(2-x)^2-2*x*exp(2-x)-x^2)*exp(((5+x)* log(3)+4)/log(3))+4*exp(2-x)+4*x)/(x*exp(2-x)^2+2*x^2*exp(2-x)+x^3),x, alg orithm="maxima")
Output:
-(x*e^(2*x + 4/log(3) + 5)*log(x) + (e^(4/log(3) + 7) - 4)*e^x*log(x))/(x* e^x + e^2)
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (30) = 60\).
Time = 1.72 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.34 \[ \int \frac {4 e^{2-x}+4 x+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x}-2 e^{2-x} x-x^2\right )+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x} x-2 e^{2-x} x^2-x^3\right ) \log (x)+\left (-x+e^{2-x} x\right ) \log \left (x^4\right )}{e^{4-2 x} x+2 e^{2-x} x^2+x^3} \, dx=\frac {4 \, x^{\frac {3}{2}} e^{4} \log \left (x\right )}{x^{\frac {5}{2}} e^{4} + x^{\frac {3}{2}} e^{4} + x^{\frac {3}{2}} e^{\left (-x + 6\right )} + \sqrt {x} e^{\left (-x + 6\right )}} - e^{\left (x + \frac {5 \, \log \left (3\right ) + 4}{\log \left (3\right )}\right )} \log \left (x\right ) + \frac {4 \, \sqrt {x} e^{4} \log \left (x\right )}{x^{\frac {5}{2}} e^{4} + x^{\frac {3}{2}} e^{4} + x^{\frac {3}{2}} e^{\left (-x + 6\right )} + \sqrt {x} e^{\left (-x + 6\right )}} \] Input:
integrate(((x*exp(2-x)-x)*log(x^4)+(-x*exp(2-x)^2-2*x^2*exp(2-x)-x^3)*exp( ((5+x)*log(3)+4)/log(3))*log(x)+(-exp(2-x)^2-2*x*exp(2-x)-x^2)*exp(((5+x)* log(3)+4)/log(3))+4*exp(2-x)+4*x)/(x*exp(2-x)^2+2*x^2*exp(2-x)+x^3),x, alg orithm="giac")
Output:
4*x^(3/2)*e^4*log(x)/(x^(5/2)*e^4 + x^(3/2)*e^4 + x^(3/2)*e^(-x + 6) + sqr t(x)*e^(-x + 6)) - e^(x + (5*log(3) + 4)/log(3))*log(x) + 4*sqrt(x)*e^4*lo g(x)/(x^(5/2)*e^4 + x^(3/2)*e^4 + x^(3/2)*e^(-x + 6) + sqrt(x)*e^(-x + 6))
Timed out. \[ \int \frac {4 e^{2-x}+4 x+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x}-2 e^{2-x} x-x^2\right )+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x} x-2 e^{2-x} x^2-x^3\right ) \log (x)+\left (-x+e^{2-x} x\right ) \log \left (x^4\right )}{e^{4-2 x} x+2 e^{2-x} x^2+x^3} \, dx=\int -\frac {{\mathrm {e}}^{\frac {\ln \left (3\right )\,\left (x+5\right )+4}{\ln \left (3\right )}}\,\left ({\mathrm {e}}^{4-2\,x}+2\,x\,{\mathrm {e}}^{2-x}+x^2\right )-4\,{\mathrm {e}}^{2-x}-4\,x+\ln \left (x^4\right )\,\left (x-x\,{\mathrm {e}}^{2-x}\right )+{\mathrm {e}}^{\frac {\ln \left (3\right )\,\left (x+5\right )+4}{\ln \left (3\right )}}\,\ln \left (x\right )\,\left (x\,{\mathrm {e}}^{4-2\,x}+2\,x^2\,{\mathrm {e}}^{2-x}+x^3\right )}{x\,{\mathrm {e}}^{4-2\,x}+2\,x^2\,{\mathrm {e}}^{2-x}+x^3} \,d x \] Input:
int(-(exp((log(3)*(x + 5) + 4)/log(3))*(exp(4 - 2*x) + 2*x*exp(2 - x) + x^ 2) - 4*exp(2 - x) - 4*x + log(x^4)*(x - x*exp(2 - x)) + exp((log(3)*(x + 5 ) + 4)/log(3))*log(x)*(x*exp(4 - 2*x) + 2*x^2*exp(2 - x) + x^3))/(x*exp(4 - 2*x) + 2*x^2*exp(2 - x) + x^3),x)
Output:
int(-(exp((log(3)*(x + 5) + 4)/log(3))*(exp(4 - 2*x) + 2*x*exp(2 - x) + x^ 2) - 4*exp(2 - x) - 4*x + log(x^4)*(x - x*exp(2 - x)) + exp((log(3)*(x + 5 ) + 4)/log(3))*log(x)*(x*exp(4 - 2*x) + 2*x^2*exp(2 - x) + x^3))/(x*exp(4 - 2*x) + 2*x^2*exp(2 - x) + x^3), x)
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75 \[ \int \frac {4 e^{2-x}+4 x+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x}-2 e^{2-x} x-x^2\right )+e^{\frac {4+(5+x) \log (3)}{\log (3)}} \left (-e^{4-2 x} x-2 e^{2-x} x^2-x^3\right ) \log (x)+\left (-x+e^{2-x} x\right ) \log \left (x^4\right )}{e^{4-2 x} x+2 e^{2-x} x^2+x^3} \, dx=\frac {e^{x} \left (-e^{\frac {\mathrm {log}\left (3\right ) x +4}{\mathrm {log}\left (3\right )}} \mathrm {log}\left (x \right ) e^{5} x -e^{\frac {4}{\mathrm {log}\left (3\right )}} \mathrm {log}\left (x \right ) e^{7}+\mathrm {log}\left (x^{4}\right )\right )}{e^{x} x +e^{2}} \] Input:
int(((x*exp(2-x)-x)*log(x^4)+(-x*exp(2-x)^2-2*x^2*exp(2-x)-x^3)*exp(((5+x) *log(3)+4)/log(3))*log(x)+(-exp(2-x)^2-2*x*exp(2-x)-x^2)*exp(((5+x)*log(3) +4)/log(3))+4*exp(2-x)+4*x)/(x*exp(2-x)^2+2*x^2*exp(2-x)+x^3),x)
Output:
(e**x*( - e**((log(3)*x + 4)/log(3))*log(x)*e**5*x - e**(4/log(3))*log(x)* e**7 + log(x**4)))/(e**x*x + e**2)