Integrand size = 109, antiderivative size = 29 \[ \int \frac {-15 e^x+6 e^{2 x} x}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx=4+\log \left (2+\frac {3}{-3+e^2-5 e^{-x}-x^2+\log (\log (4))}\right ) \] Output:
ln(3/(exp(2)+ln(2*ln(2))-5/exp(x)-x^2-3)+2)+4
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(29)=58\).
Time = 2.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.48 \[ \int \frac {-15 e^x+6 e^{2 x} x}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx=3 \left (\frac {1}{3} \log \left (10+3 e^x-2 e^{2+x}+2 e^x x^2-2 e^x \log (\log (4))\right )-\frac {1}{3} \log \left (5+3 e^x-e^{2+x}+e^x x^2-e^x \log (\log (4))\right )\right ) \] Input:
Integrate[(-15*E^x + 6*E^(2*x)*x)/(50 + E^x*(45 - 20*E^2 + 20*x^2) + E^(2* x)*(9 + 2*E^4 + 9*x^2 + 2*x^4 + E^2*(-9 - 4*x^2)) + (-20*E^x + E^(2*x)*(-9 + 4*E^2 - 4*x^2))*Log[Log[4]] + 2*E^(2*x)*Log[Log[4]]^2),x]
Output:
3*(Log[10 + 3*E^x - 2*E^(2 + x) + 2*E^x*x^2 - 2*E^x*Log[Log[4]]]/3 - Log[5 + 3*E^x - E^(2 + x) + E^x*x^2 - E^x*Log[Log[4]]]/3)
Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(29)=58\).
Time = 2.61 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {7292, 27, 25, 7293, 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.
| \(\displaystyle \int \frac {6 e^{2 x} x-15 e^x}{e^x \left (20 x^2-20 e^2+45\right )+\left (e^{2 x} \left (-4 x^2+4 e^2-9\right )-20 e^x\right ) \log (\log (4))+e^{2 x} \left (2 x^4+9 x^2+e^2 \left (-4 x^2-9\right )+2 e^4+9\right )+2 e^{2 x} \log ^2(\log (4))+50} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {3 e^x \left (2 e^x x-5\right )}{e^x \left (20 x^2-20 e^2+45\right )+\left (e^{2 x} \left (-4 x^2+4 e^2-9\right )-20 e^x\right ) \log (\log (4))+e^{2 x} \left (2 x^4+9 x^2+e^2 \left (-4 x^2-9\right )+2 e^4+9\right )+2 e^{2 x} \log ^2(\log (4))+50}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \int -\frac {e^x \left (5-2 e^x x\right )}{5 e^x \left (4 x^2-4 e^2+9\right )+e^{2 x} \left (2 x^4+9 x^2-e^2 \left (4 x^2+9\right )+2 e^4+9\right )+2 e^{2 x} \log ^2(\log (4))-\left (e^{2 x} \left (4 x^2-4 e^2+9\right )+20 e^x\right ) \log (\log (4))+50}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -3 \int \frac {e^x \left (5-2 e^x x\right )}{5 e^x \left (4 x^2-4 e^2+9\right )+e^{2 x} \left (2 x^4+9 x^2-e^2 \left (4 x^2+9\right )+2 e^4+9\right )+2 e^{2 x} \log ^2(\log (4))-\left (e^{2 x} \left (4 x^2-4 e^2+9\right )+20 e^x\right ) \log (\log (4))+50}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -3 \int \left (\frac {e^x \left (x^2+2 x-\log (\log (4))-e^2+3\right )}{3 \left (e^x x^2+3 e^x \left (1+\frac {1}{3} \left (-e^2-\log (\log (4))\right )\right )+5\right )}+\frac {e^x \left (-2 x^2-4 x+2 \log (\log (4))+2 e^2-3\right )}{3 \left (2 e^x x^2+3 e^x \left (1-\frac {2}{3} \left (e^2+\log (\log (4))\right )\right )+10\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {1}{3} \log \left (e^x x^2+e^x \left (3-e^2-\log (\log (4))\right )+5\right )-\frac {1}{3} \log \left (2 e^x x^2+e^x \left (3-2 \left (e^2+\log (\log (4))\right )\right )+10\right )\right )\) |
Input:
Int[(-15*E^x + 6*E^(2*x)*x)/(50 + E^x*(45 - 20*E^2 + 20*x^2) + E^(2*x)*(9 + 2*E^4 + 9*x^2 + 2*x^4 + E^2*(-9 - 4*x^2)) + (-20*E^x + E^(2*x)*(-9 + 4*E ^2 - 4*x^2))*Log[Log[4]] + 2*E^(2*x)*Log[Log[4]]^2),x]
Output:
-3*(Log[5 + E^x*x^2 + E^x*(3 - E^2 - Log[Log[4]])]/3 - Log[10 + 2*E^x*x^2 + E^x*(3 - 2*(E^2 + Log[Log[4]]))]/3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(29)=58\).
Time = 1.55 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07
| method | result | size |
| norman | \(-\ln \left (-{\mathrm e}^{x} x^{2}+{\mathrm e}^{2} {\mathrm e}^{x}+{\mathrm e}^{x} \ln \left (2 \ln \left (2\right )\right )-3 \,{\mathrm e}^{x}-5\right )+\ln \left (-2 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{2} {\mathrm e}^{x}+2 \,{\mathrm e}^{x} \ln \left (2 \ln \left (2\right )\right )-3 \,{\mathrm e}^{x}-10\right )\) | \(60\) |
| parallelrisch | \(-\ln \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{2} {\mathrm e}^{x}-{\mathrm e}^{x} \ln \left (2 \ln \left (2\right )\right )+3 \,{\mathrm e}^{x}+5\right )+\ln \left ({\mathrm e}^{x} x^{2}-{\mathrm e}^{2} {\mathrm e}^{x}-{\mathrm e}^{x} \ln \left (2 \ln \left (2\right )\right )+\frac {3 \,{\mathrm e}^{x}}{2}+5\right )\) | \(60\) |
| risch | \(-\ln \left (x^{2}-\ln \left (2\right )-{\mathrm e}^{2}-\ln \left (\ln \left (2\right )\right )+3\right )+\ln \left (-2 x^{2}+2 \ln \left (2\right )+2 \,{\mathrm e}^{2}+2 \ln \left (\ln \left (2\right )\right )-3\right )+\ln \left ({\mathrm e}^{x}-\frac {10}{-2 x^{2}+2 \ln \left (2\right )+2 \,{\mathrm e}^{2}+2 \ln \left (\ln \left (2\right )\right )-3}\right )-\ln \left ({\mathrm e}^{x}-\frac {5}{-x^{2}+\ln \left (2\right )+{\mathrm e}^{2}+\ln \left (\ln \left (2\right )\right )-3}\right )\) | \(96\) |
Input:
int((6*x*exp(x)^2-15*exp(x))/(2*exp(x)^2*ln(2*ln(2))^2+((4*exp(2)-4*x^2-9) *exp(x)^2-20*exp(x))*ln(2*ln(2))+(2*exp(2)^2+(-4*x^2-9)*exp(2)+2*x^4+9*x^2 +9)*exp(x)^2+(-20*exp(2)+20*x^2+45)*exp(x)+50),x,method=_RETURNVERBOSE)
Output:
-ln(-exp(x)*x^2+exp(2)*exp(x)+exp(x)*ln(2*ln(2))-3*exp(x)-5)+ln(-2*exp(x)* x^2+2*exp(2)*exp(x)+2*exp(x)*ln(2*ln(2))-3*exp(x)-10)
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (31) = 62\).
Time = 0.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.59 \[ \int \frac {-15 e^x+6 e^{2 x} x}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx=\log \left (2 \, x^{2} - 2 \, e^{2} - 2 \, \log \left (2 \, \log \left (2\right )\right ) + 3\right ) - \log \left (x^{2} - e^{2} - \log \left (2 \, \log \left (2\right )\right ) + 3\right ) + \log \left (-\frac {{\left (2 \, x^{2} - 2 \, e^{2} + 3\right )} e^{x} - 2 \, e^{x} \log \left (2 \, \log \left (2\right )\right ) + 10}{2 \, x^{2} - 2 \, e^{2} - 2 \, \log \left (2 \, \log \left (2\right )\right ) + 3}\right ) - \log \left (-\frac {{\left (x^{2} - e^{2} + 3\right )} e^{x} - e^{x} \log \left (2 \, \log \left (2\right )\right ) + 5}{x^{2} - e^{2} - \log \left (2 \, \log \left (2\right )\right ) + 3}\right ) \] Input:
integrate((6*x*exp(x)^2-15*exp(x))/(2*exp(x)^2*log(2*log(2))^2+((4*exp(2)- 4*x^2-9)*exp(x)^2-20*exp(x))*log(2*log(2))+(2*exp(2)^2+(-4*x^2-9)*exp(2)+2 *x^4+9*x^2+9)*exp(x)^2+(-20*exp(2)+20*x^2+45)*exp(x)+50),x, algorithm="fri cas")
Output:
log(2*x^2 - 2*e^2 - 2*log(2*log(2)) + 3) - log(x^2 - e^2 - log(2*log(2)) + 3) + log(-((2*x^2 - 2*e^2 + 3)*e^x - 2*e^x*log(2*log(2)) + 10)/(2*x^2 - 2 *e^2 - 2*log(2*log(2)) + 3)) - log(-((x^2 - e^2 + 3)*e^x - e^x*log(2*log(2 )) + 5)/(x^2 - e^2 - log(2*log(2)) + 3))
Timed out. \[ \int \frac {-15 e^x+6 e^{2 x} x}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx=\text {Timed out} \] Input:
integrate((6*x*exp(x)**2-15*exp(x))/(2*exp(x)**2*ln(2*ln(2))**2+((4*exp(2) -4*x**2-9)*exp(x)**2-20*exp(x))*ln(2*ln(2))+(2*exp(2)**2+(-4*x**2-9)*exp(2 )+2*x**4+9*x**2+9)*exp(x)**2+(-20*exp(2)+20*x**2+45)*exp(x)+50),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (31) = 62\).
Time = 0.19 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.79 \[ \int \frac {-15 e^x+6 e^{2 x} x}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx=\log \left (2 \, x^{2} - 2 \, e^{2} - 2 \, \log \left (2\right ) - 2 \, \log \left (\log \left (2\right )\right ) + 3\right ) - \log \left (x^{2} - e^{2} - \log \left (2\right ) - \log \left (\log \left (2\right )\right ) + 3\right ) + \log \left (\frac {{\left (2 \, x^{2} - 2 \, e^{2} - 2 \, \log \left (2\right ) - 2 \, \log \left (\log \left (2\right )\right ) + 3\right )} e^{x} + 10}{2 \, x^{2} - 2 \, e^{2} - 2 \, \log \left (2\right ) - 2 \, \log \left (\log \left (2\right )\right ) + 3}\right ) - \log \left (\frac {{\left (x^{2} - e^{2} - \log \left (2\right ) - \log \left (\log \left (2\right )\right ) + 3\right )} e^{x} + 5}{x^{2} - e^{2} - \log \left (2\right ) - \log \left (\log \left (2\right )\right ) + 3}\right ) \] Input:
integrate((6*x*exp(x)^2-15*exp(x))/(2*exp(x)^2*log(2*log(2))^2+((4*exp(2)- 4*x^2-9)*exp(x)^2-20*exp(x))*log(2*log(2))+(2*exp(2)^2+(-4*x^2-9)*exp(2)+2 *x^4+9*x^2+9)*exp(x)^2+(-20*exp(2)+20*x^2+45)*exp(x)+50),x, algorithm="max ima")
Output:
log(2*x^2 - 2*e^2 - 2*log(2) - 2*log(log(2)) + 3) - log(x^2 - e^2 - log(2) - log(log(2)) + 3) + log(((2*x^2 - 2*e^2 - 2*log(2) - 2*log(log(2)) + 3)* e^x + 10)/(2*x^2 - 2*e^2 - 2*log(2) - 2*log(log(2)) + 3)) - log(((x^2 - e^ 2 - log(2) - log(log(2)) + 3)*e^x + 5)/(x^2 - e^2 - log(2) - log(log(2)) + 3))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (31) = 62\).
Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.34 \[ \int \frac {-15 e^x+6 e^{2 x} x}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx=\log \left (2 \, x^{2} e^{x} - 2 \, e^{x} \log \left (2\right ) - 2 \, e^{x} \log \left (\log \left (2\right )\right ) - 2 \, e^{\left (x + 2\right )} + 3 \, e^{x} + 10\right ) - \log \left (x^{2} e^{x} - e^{x} \log \left (2\right ) - e^{x} \log \left (\log \left (2\right )\right ) - e^{\left (x + 2\right )} + 3 \, e^{x} + 5\right ) \] Input:
integrate((6*x*exp(x)^2-15*exp(x))/(2*exp(x)^2*log(2*log(2))^2+((4*exp(2)- 4*x^2-9)*exp(x)^2-20*exp(x))*log(2*log(2))+(2*exp(2)^2+(-4*x^2-9)*exp(2)+2 *x^4+9*x^2+9)*exp(x)^2+(-20*exp(2)+20*x^2+45)*exp(x)+50),x, algorithm="gia c")
Output:
log(2*x^2*e^x - 2*e^x*log(2) - 2*e^x*log(log(2)) - 2*e^(x + 2) + 3*e^x + 1 0) - log(x^2*e^x - e^x*log(2) - e^x*log(log(2)) - e^(x + 2) + 3*e^x + 5)
Timed out. \[ \int \frac {-15 e^x+6 e^{2 x} x}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx=\int -\frac {15\,{\mathrm {e}}^x-6\,x\,{\mathrm {e}}^{2\,x}}{2\,{\ln \left (2\,\ln \left (2\right )\right )}^2\,{\mathrm {e}}^{2\,x}-\ln \left (2\,\ln \left (2\right )\right )\,\left (20\,{\mathrm {e}}^x+{\mathrm {e}}^{2\,x}\,\left (4\,x^2-4\,{\mathrm {e}}^2+9\right )\right )+{\mathrm {e}}^{2\,x}\,\left (2\,{\mathrm {e}}^4-{\mathrm {e}}^2\,\left (4\,x^2+9\right )+9\,x^2+2\,x^4+9\right )+{\mathrm {e}}^x\,\left (20\,x^2-20\,{\mathrm {e}}^2+45\right )+50} \,d x \] Input:
int(-(15*exp(x) - 6*x*exp(2*x))/(2*log(2*log(2))^2*exp(2*x) - log(2*log(2) )*(20*exp(x) + exp(2*x)*(4*x^2 - 4*exp(2) + 9)) + exp(2*x)*(2*exp(4) - exp (2)*(4*x^2 + 9) + 9*x^2 + 2*x^4 + 9) + exp(x)*(20*x^2 - 20*exp(2) + 45) + 50),x)
Output:
int(-(15*exp(x) - 6*x*exp(2*x))/(2*log(2*log(2))^2*exp(2*x) - log(2*log(2) )*(20*exp(x) + exp(2*x)*(4*x^2 - 4*exp(2) + 9)) + exp(2*x)*(2*exp(4) - exp (2)*(4*x^2 + 9) + 9*x^2 + 2*x^4 + 9) + exp(x)*(20*x^2 - 20*exp(2) + 45) + 50), x)
\[ \int \frac {-15 e^x+6 e^{2 x} x}{50+e^x \left (45-20 e^2+20 x^2\right )+e^{2 x} \left (9+2 e^4+9 x^2+2 x^4+e^2 \left (-9-4 x^2\right )\right )+\left (-20 e^x+e^{2 x} \left (-9+4 e^2-4 x^2\right )\right ) \log (\log (4))+2 e^{2 x} \log ^2(\log (4))} \, dx=-15 \left (\int \frac {e^{x}}{2 e^{2 x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2}+4 e^{2 x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{2}-4 e^{2 x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) x^{2}-9 e^{2 x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )+2 e^{2 x} e^{4}-4 e^{2 x} e^{2} x^{2}-9 e^{2 x} e^{2}+2 e^{2 x} x^{4}+9 e^{2 x} x^{2}+9 e^{2 x}-20 e^{x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )-20 e^{x} e^{2}+20 e^{x} x^{2}+45 e^{x}+50}d x \right )+6 \left (\int \frac {e^{2 x} x}{2 e^{2 x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )^{2}+4 e^{2 x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) e^{2}-4 e^{2 x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) x^{2}-9 e^{2 x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )+2 e^{2 x} e^{4}-4 e^{2 x} e^{2} x^{2}-9 e^{2 x} e^{2}+2 e^{2 x} x^{4}+9 e^{2 x} x^{2}+9 e^{2 x}-20 e^{x} \mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )-20 e^{x} e^{2}+20 e^{x} x^{2}+45 e^{x}+50}d x \right ) \] Input:
int((6*x*exp(x)^2-15*exp(x))/(2*exp(x)^2*log(2*log(2))^2+((4*exp(2)-4*x^2- 9)*exp(x)^2-20*exp(x))*log(2*log(2))+(2*exp(2)^2+(-4*x^2-9)*exp(2)+2*x^4+9 *x^2+9)*exp(x)^2+(-20*exp(2)+20*x^2+45)*exp(x)+50),x)
Output:
3*( - 5*int(e**x/(2*e**(2*x)*log(2*log(2))**2 + 4*e**(2*x)*log(2*log(2))*e **2 - 4*e**(2*x)*log(2*log(2))*x**2 - 9*e**(2*x)*log(2*log(2)) + 2*e**(2*x )*e**4 - 4*e**(2*x)*e**2*x**2 - 9*e**(2*x)*e**2 + 2*e**(2*x)*x**4 + 9*e**( 2*x)*x**2 + 9*e**(2*x) - 20*e**x*log(2*log(2)) - 20*e**x*e**2 + 20*e**x*x* *2 + 45*e**x + 50),x) + 2*int((e**(2*x)*x)/(2*e**(2*x)*log(2*log(2))**2 + 4*e**(2*x)*log(2*log(2))*e**2 - 4*e**(2*x)*log(2*log(2))*x**2 - 9*e**(2*x) *log(2*log(2)) + 2*e**(2*x)*e**4 - 4*e**(2*x)*e**2*x**2 - 9*e**(2*x)*e**2 + 2*e**(2*x)*x**4 + 9*e**(2*x)*x**2 + 9*e**(2*x) - 20*e**x*log(2*log(2)) - 20*e**x*e**2 + 20*e**x*x**2 + 45*e**x + 50),x))