Integrand size = 59, antiderivative size = 26 \[ \int \frac {1}{3} e^{-x-e^{-x} \left (75 x-15 x^2+2 e^x x^2\right )} \left (-75 x+105 x^2-15 x^3+e^x \left (1-4 x^2\right )\right ) \, dx=\frac {1}{3} e^{-15 e^{-x} (5-x) x-2 x^2} x \] Output:
1/3*x/exp(15*x*(5-x)/exp(x)+2*x^2)
Time = 0.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{3} e^{-x-e^{-x} \left (75 x-15 x^2+2 e^x x^2\right )} \left (-75 x+105 x^2-15 x^3+e^x \left (1-4 x^2\right )\right ) \, dx=\frac {1}{3} e^{15 e^{-x} (-5+x) x-2 x^2} x \] Input:
Integrate[(E^(-x - (75*x - 15*x^2 + 2*E^x*x^2)/E^x)*(-75*x + 105*x^2 - 15* x^3 + E^x*(1 - 4*x^2)))/3,x]
Output:
(E^((15*(-5 + x)*x)/E^x - 2*x^2)*x)/3
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.
\(\displaystyle \int \frac {1}{3} e^{-e^{-x} \left (2 e^x x^2-15 x^2+75 x\right )-x} \left (-15 x^3+105 x^2+e^x \left (1-4 x^2\right )-75 x\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -e^{-x-e^{-x} \left (2 e^x x^2-15 x^2+75 x\right )} \left (15 x^3-105 x^2+75 x-e^x \left (1-4 x^2\right )\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int e^{-x-e^{-x} \left (2 e^x x^2-15 x^2+75 x\right )} \left (15 x^3-105 x^2+75 x-e^x \left (1-4 x^2\right )\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\frac {1}{3} \int e^{-e^{-x} x \left (2 e^x x-15 x+e^x+75\right )} \left (15 x^3-105 x^2+75 x-e^x \left (1-4 x^2\right )\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (15 e^{-e^{-x} x \left (2 e^x x-15 x+e^x+75\right )} x^3-105 e^{-e^{-x} x \left (2 e^x x-15 x+e^x+75\right )} x^2+75 e^{-e^{-x} x \left (2 e^x x-15 x+e^x+75\right )} x+e^{x-e^{-x} x \left (2 e^x x-15 x+e^x+75\right )} \left (4 x^2-1\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-15 \int e^{-e^{-x} x \left (2 e^x x-15 x+e^x+75\right )} x^3dx-4 \int e^{-e^{-x} x \left (2 e^x x-15 x+75\right )} x^2dx+105 \int e^{-e^{-x} x \left (2 e^x x-15 x+e^x+75\right )} x^2dx+\int e^{-e^{-x} x \left (2 e^x x-15 x+75\right )}dx-75 \int e^{-e^{-x} x \left (2 e^x x-15 x+e^x+75\right )} xdx\right )\) |
Input:
Int[(E^(-x - (75*x - 15*x^2 + 2*E^x*x^2)/E^x)*(-75*x + 105*x^2 - 15*x^3 + E^x*(1 - 4*x^2)))/3,x]
Output:
$Aborted
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {x \,{\mathrm e}^{-x \left (2 \,{\mathrm e}^{x} x -15 x +75\right ) {\mathrm e}^{-x}}}{3}\) | \(22\) |
parallelrisch | \(\frac {x \,{\mathrm e}^{-x \left (2 \,{\mathrm e}^{x} x -15 x +75\right ) {\mathrm e}^{-x}}}{3}\) | \(23\) |
norman | \(\frac {x \,{\mathrm e}^{-\left (2 \,{\mathrm e}^{x} x^{2}-15 x^{2}+75 x \right ) {\mathrm e}^{-x}}}{3}\) | \(28\) |
Input:
int(1/3*((-4*x^2+1)*exp(x)-15*x^3+105*x^2-75*x)/exp(x)/exp((2*exp(x)*x^2-1 5*x^2+75*x)/exp(x)),x,method=_RETURNVERBOSE)
Output:
1/3*x*exp(-x*(2*exp(x)*x-15*x+75)*exp(-x))
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {1}{3} e^{-x-e^{-x} \left (75 x-15 x^2+2 e^x x^2\right )} \left (-75 x+105 x^2-15 x^3+e^x \left (1-4 x^2\right )\right ) \, dx=\frac {1}{3} \, x e^{\left ({\left (15 \, x^{2} - {\left (2 \, x^{2} + x\right )} e^{x} - 75 \, x\right )} e^{\left (-x\right )} + x\right )} \] Input:
integrate(1/3*((-4*x^2+1)*exp(x)-15*x^3+105*x^2-75*x)/exp(x)/exp((2*exp(x) *x^2-15*x^2+75*x)/exp(x)),x, algorithm="fricas")
Output:
1/3*x*e^((15*x^2 - (2*x^2 + x)*e^x - 75*x)*e^(-x) + x)
Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{3} e^{-x-e^{-x} \left (75 x-15 x^2+2 e^x x^2\right )} \left (-75 x+105 x^2-15 x^3+e^x \left (1-4 x^2\right )\right ) \, dx=\frac {x e^{- \left (2 x^{2} e^{x} - 15 x^{2} + 75 x\right ) e^{- x}}}{3} \] Input:
integrate(1/3*((-4*x**2+1)*exp(x)-15*x**3+105*x**2-75*x)/exp(x)/exp((2*exp (x)*x**2-15*x**2+75*x)/exp(x)),x)
Output:
x*exp(-(2*x**2*exp(x) - 15*x**2 + 75*x)*exp(-x))/3
Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{3} e^{-x-e^{-x} \left (75 x-15 x^2+2 e^x x^2\right )} \left (-75 x+105 x^2-15 x^3+e^x \left (1-4 x^2\right )\right ) \, dx=\frac {1}{3} \, x e^{\left (15 \, x^{2} e^{\left (-x\right )} - 2 \, x^{2} - 75 \, x e^{\left (-x\right )}\right )} \] Input:
integrate(1/3*((-4*x^2+1)*exp(x)-15*x^3+105*x^2-75*x)/exp(x)/exp((2*exp(x) *x^2-15*x^2+75*x)/exp(x)),x, algorithm="maxima")
Output:
1/3*x*e^(15*x^2*e^(-x) - 2*x^2 - 75*x*e^(-x))
\[ \int \frac {1}{3} e^{-x-e^{-x} \left (75 x-15 x^2+2 e^x x^2\right )} \left (-75 x+105 x^2-15 x^3+e^x \left (1-4 x^2\right )\right ) \, dx=\int { -\frac {1}{3} \, {\left (15 \, x^{3} - 105 \, x^{2} + {\left (4 \, x^{2} - 1\right )} e^{x} + 75 \, x\right )} e^{\left (-{\left (2 \, x^{2} e^{x} - 15 \, x^{2} + 75 \, x\right )} e^{\left (-x\right )} - x\right )} \,d x } \] Input:
integrate(1/3*((-4*x^2+1)*exp(x)-15*x^3+105*x^2-75*x)/exp(x)/exp((2*exp(x) *x^2-15*x^2+75*x)/exp(x)),x, algorithm="giac")
Output:
integrate(-1/3*(15*x^3 - 105*x^2 + (4*x^2 - 1)*e^x + 75*x)*e^(-(2*x^2*e^x - 15*x^2 + 75*x)*e^(-x) - x), x)
Time = 4.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {1}{3} e^{-x-e^{-x} \left (75 x-15 x^2+2 e^x x^2\right )} \left (-75 x+105 x^2-15 x^3+e^x \left (1-4 x^2\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^{-75\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^{15\,x^2\,{\mathrm {e}}^{-x}}}{3} \] Input:
int(-exp(-x)*exp(-exp(-x)*(75*x + 2*x^2*exp(x) - 15*x^2))*(25*x + (exp(x)* (4*x^2 - 1))/3 - 35*x^2 + 5*x^3),x)
Output:
(x*exp(-75*x*exp(-x))*exp(-2*x^2)*exp(15*x^2*exp(-x)))/3
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {1}{3} e^{-x-e^{-x} \left (75 x-15 x^2+2 e^x x^2\right )} \left (-75 x+105 x^2-15 x^3+e^x \left (1-4 x^2\right )\right ) \, dx=\frac {e^{\frac {15 x^{2}}{e^{x}}} x}{3 e^{\frac {2 e^{x} x^{2}+75 x}{e^{x}}}} \] Input:
int(1/3*((-4*x^2+1)*exp(x)-15*x^3+105*x^2-75*x)/exp(x)/exp((2*exp(x)*x^2-1 5*x^2+75*x)/exp(x)),x)
Output:
(e**((15*x**2)/e**x)*x)/(3*e**((2*e**x*x**2 + 75*x)/e**x))