Integrand size = 80, antiderivative size = 25 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4 e^{\frac {4 e^{-2 e^x}}{\left (-30+\sqrt {e}\right )^2}}}{x} \] Output:
4*exp(4/exp(exp(x))^2/(exp(1/4)^2-30)^2)/x
Time = 0.90 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4 e^{\frac {4 e^{-2 e^x}}{\left (-30+\sqrt {e}\right )^2}}}{x} \] Input:
Integrate[(E^(-2*E^x + 4/(E^(2*E^x)*(900 - 60*Sqrt[E] + E)))*((-3600 + 240 *Sqrt[E] - 4*E)*E^(2*E^x) - 32*E^x*x))/(900*x^2 - 60*Sqrt[E]*x^2 + E*x^2), x]
Output:
(4*E^(4/((-30 + Sqrt[E])^2*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.
| \(\displaystyle \int \frac {e^{\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}-2 e^x} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{e x^2-60 \sqrt {e} x^2+900 x^2} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^{\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}-2 e^x} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{e x^2+\left (900-60 \sqrt {e}\right ) x^2}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^{\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}-2 e^x} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{\left (900-60 \sqrt {e}+e\right ) x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {4 e^{\frac {4 e^{-2 e^x}}{\left (30-\sqrt {e}\right )^2}-2 e^x} \left (8 e^x x+\left (30-\sqrt {e}\right )^2 e^{2 e^x}\right )}{x^2}dx}{\left (30-\sqrt {e}\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \int \frac {e^{\frac {4 e^{-2 e^x}}{\left (30-\sqrt {e}\right )^2}-2 e^x} \left (8 e^x x+\left (30-\sqrt {e}\right )^2 e^{2 e^x}\right )}{x^2}dx}{\left (30-\sqrt {e}\right )^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {4 \int \left (\frac {8 e^{x+\frac {4 e^{-2 e^x}}{\left (30-\sqrt {e}\right )^2}-2 e^x}}{x}+\frac {\left (-30+\sqrt {e}\right )^2 e^{\frac {4 e^{-2 e^x}}{\left (30-\sqrt {e}\right )^2}}}{x^2}\right )dx}{\left (30-\sqrt {e}\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (\left (30-\sqrt {e}\right )^2 \int \frac {e^{\frac {4 e^{-2 e^x}}{\left (30-\sqrt {e}\right )^2}}}{x^2}dx+8 \int \frac {e^{x+\frac {4 e^{-2 e^x}}{\left (30-\sqrt {e}\right )^2}-2 e^x}}{x}dx\right )}{\left (30-\sqrt {e}\right )^2}\) |
Input:
Int[(E^(-2*E^x + 4/(E^(2*E^x)*(900 - 60*Sqrt[E] + E)))*((-3600 + 240*Sqrt[ E] - 4*E)*E^(2*E^x) - 32*E^x*x))/(900*x^2 - 60*Sqrt[E]*x^2 + E*x^2),x]
Output:
$Aborted
Time = 2.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(\frac {4 \,{\mathrm e}^{\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}}}{{\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900}}}{x}\) | \(24\) |
| parallelrisch | \(\frac {4 \,{\mathrm e} \,{\mathrm e}^{\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}}}{{\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900}}-240 \,{\mathrm e}^{\frac {1}{2}} {\mathrm e}^{\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}}}{{\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900}}+3600 \,{\mathrm e}^{\frac {4 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}}}{{\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900}}}{\left ({\mathrm e}-60 \,{\mathrm e}^{\frac {1}{2}}+900\right ) x}\) | \(100\) |
Input:
int(((-4*exp(1/4)^4+240*exp(1/4)^2-3600)*exp(exp(x))^2-32*exp(x)*x)*exp(4/ (exp(1/4)^4-60*exp(1/4)^2+900)/exp(exp(x))^2)/(x^2*exp(1/4)^4-60*x^2*exp(1 /4)^2+900*x^2)/exp(exp(x))^2,x,method=_RETURNVERBOSE)
Output:
4/x*exp(4/(exp(1)-60*exp(1/2)+900)*exp(-2*exp(x)))
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4 \, e^{\left (-\frac {2 \, {\left ({\left (e - 60 \, e^{\frac {1}{2}} + 900\right )} e^{\left (x + 2 \, e^{x}\right )} - 2\right )} e^{\left (-2 \, e^{x}\right )}}{e - 60 \, e^{\frac {1}{2}} + 900} + 2 \, e^{x}\right )}}{x} \] Input:
integrate(((-4*exp(1/4)^4+240*exp(1/4)^2-3600)*exp(exp(x))^2-32*exp(x)*x)* exp(4/(exp(1/4)^4-60*exp(1/4)^2+900)/exp(exp(x))^2)/(x^2*exp(1/4)^4-60*x^2 *exp(1/4)^2+900*x^2)/exp(exp(x))^2,x, algorithm="fricas")
Output:
4*e^(-2*((e - 60*e^(1/2) + 900)*e^(x + 2*e^x) - 2)*e^(-2*e^x)/(e - 60*e^(1 /2) + 900) + 2*e^x)/x
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4 e^{\frac {4 e^{- 2 e^{x}}}{- 60 e^{\frac {1}{2}} + e + 900}}}{x} \] Input:
integrate(((-4*exp(1/4)**4+240*exp(1/4)**2-3600)*exp(exp(x))**2-32*exp(x)* x)*exp(4/(exp(1/4)**4-60*exp(1/4)**2+900)/exp(exp(x))**2)/(x**2*exp(1/4)** 4-60*x**2*exp(1/4)**2+900*x**2)/exp(exp(x))**2,x)
Output:
4*exp(4*exp(-2*exp(x))/(-60*exp(1/2) + E + 900))/x
\[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\int { -\frac {4 \, {\left (8 \, x e^{x} + {\left (e - 60 \, e^{\frac {1}{2}} + 900\right )} e^{\left (2 \, e^{x}\right )}\right )} e^{\left (\frac {4 \, e^{\left (-2 \, e^{x}\right )}}{e - 60 \, e^{\frac {1}{2}} + 900} - 2 \, e^{x}\right )}}{x^{2} e - 60 \, x^{2} e^{\frac {1}{2}} + 900 \, x^{2}} \,d x } \] Input:
integrate(((-4*exp(1/4)^4+240*exp(1/4)^2-3600)*exp(exp(x))^2-32*exp(x)*x)* exp(4/(exp(1/4)^4-60*exp(1/4)^2+900)/exp(exp(x))^2)/(x^2*exp(1/4)^4-60*x^2 *exp(1/4)^2+900*x^2)/exp(exp(x))^2,x, algorithm="maxima")
Output:
-4*integrate((8*x*e^x + (e - 60*e^(1/2) + 900)*e^(2*e^x))*e^(4*e^(-2*e^x)/ (e - 60*e^(1/2) + 900) - 2*e^x)/(x^2*e - 60*x^2*e^(1/2) + 900*x^2), x)
\[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\int { -\frac {4 \, {\left (8 \, x e^{x} + {\left (e - 60 \, e^{\frac {1}{2}} + 900\right )} e^{\left (2 \, e^{x}\right )}\right )} e^{\left (\frac {4 \, e^{\left (-2 \, e^{x}\right )}}{e - 60 \, e^{\frac {1}{2}} + 900} - 2 \, e^{x}\right )}}{x^{2} e - 60 \, x^{2} e^{\frac {1}{2}} + 900 \, x^{2}} \,d x } \] Input:
integrate(((-4*exp(1/4)^4+240*exp(1/4)^2-3600)*exp(exp(x))^2-32*exp(x)*x)* exp(4/(exp(1/4)^4-60*exp(1/4)^2+900)/exp(exp(x))^2)/(x^2*exp(1/4)^4-60*x^2 *exp(1/4)^2+900*x^2)/exp(exp(x))^2,x, algorithm="giac")
Output:
integrate(-4*(8*x*e^x + (e - 60*e^(1/2) + 900)*e^(2*e^x))*e^(4*e^(-2*e^x)/ (e - 60*e^(1/2) + 900) - 2*e^x)/(x^2*e - 60*x^2*e^(1/2) + 900*x^2), x)
Time = 3.44 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{-2\,{\mathrm {e}}^x}}{\mathrm {e}-60\,\sqrt {\mathrm {e}}+900}}}{x} \] Input:
int(-(exp((4*exp(-2*exp(x)))/(exp(1) - 60*exp(1/2) + 900))*exp(-2*exp(x))* (32*x*exp(x) + exp(2*exp(x))*(4*exp(1) - 240*exp(1/2) + 3600)))/(x^2*exp(1 ) - 60*x^2*exp(1/2) + 900*x^2),x)
Output:
(4*exp((4*exp(-2*exp(x)))/(exp(1) - 60*exp(1/2) + 900)))/x
\[ \int \frac {e^{-2 e^x+\frac {4 e^{-2 e^x}}{900-60 \sqrt {e}+e}} \left (\left (-3600+240 \sqrt {e}-4 e\right ) e^{2 e^x}-32 e^x x\right )}{900 x^2-60 \sqrt {e} x^2+e x^2} \, dx=\frac {4 e^{\frac {e +900}{120 \sqrt {e}-2 e -1800}} \left (-60 \sqrt {e}\, \left (\int \frac {1}{e^{\frac {4}{60 e^{2 e^{x}+\frac {1}{2}}-e^{2 e^{x}} e -900 e^{2 e^{x}}}} x^{2}}d x \right )+8 \left (\int \frac {e^{x}}{e^{\frac {120 e^{2 e^{x}+x +\frac {1}{2}}-2 e^{2 e^{x}+x} e -1800 e^{2 e^{x}+x}+4}{60 e^{2 e^{x}+\frac {1}{2}}-e^{2 e^{x}} e -900 e^{2 e^{x}}}} x}d x \right )+\left (\int \frac {1}{e^{\frac {4}{60 e^{2 e^{x}+\frac {1}{2}}-e^{2 e^{x}} e -900 e^{2 e^{x}}}} x^{2}}d x \right ) e +900 \left (\int \frac {1}{e^{\frac {4}{60 e^{2 e^{x}+\frac {1}{2}}-e^{2 e^{x}} e -900 e^{2 e^{x}}}} x^{2}}d x \right )\right )}{60 e^{\frac {30 \sqrt {e}}{60 \sqrt {e}-e -900}}-e^{\frac {e +900}{120 \sqrt {e}-2 e -1800}} e -900 e^{\frac {e +900}{120 \sqrt {e}-2 e -1800}}} \] Input:
int(((-4*exp(1/4)^4+240*exp(1/4)^2-3600)*exp(exp(x))^2-32*exp(x)*x)*exp(4/ (exp(1/4)^4-60*exp(1/4)^2+900)/exp(exp(x))^2)/(x^2*exp(1/4)^4-60*x^2*exp(1 /4)^2+900*x^2)/exp(exp(x))^2,x)
Output:
(4*e**((e + 900)/(120*sqrt(e) - 2*e - 1800))*( - 60*sqrt(e)*int(1/(e**(4/( 60*e**((4*e**x + 1)/2) - e**(2*e**x)*e - 900*e**(2*e**x)))*x**2),x) + 8*in t(e**x/(e**((120*e**((4*e**x + 2*x + 1)/2) - 2*e**(2*e**x + x)*e - 1800*e* *(2*e**x + x) + 4)/(60*e**((4*e**x + 1)/2) - e**(2*e**x)*e - 900*e**(2*e** x)))*x),x) + int(1/(e**(4/(60*e**((4*e**x + 1)/2) - e**(2*e**x)*e - 900*e* *(2*e**x)))*x**2),x)*e + 900*int(1/(e**(4/(60*e**((4*e**x + 1)/2) - e**(2* e**x)*e - 900*e**(2*e**x)))*x**2),x)))/(60*e**((30*sqrt(e))/(60*sqrt(e) - e - 900)) - e**((e + 900)/(120*sqrt(e) - 2*e - 1800))*e - 900*e**((e + 900 )/(120*sqrt(e) - 2*e - 1800)))