\(\int \frac {e^{-x+\frac {e^{-x} (45 x^3+e^x (15-10 x-x^3))}{-15+x^3}} (-2025 x^2+675 x^3-45 x^6+e^x (150+20 x^3))}{225-30 x^3+x^6} \, dx\) [910]

Optimal result

Integrand size = 79, antiderivative size = 29 \[ \int \frac {e^{-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}} \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{225-30 x^3+x^6} \, dx=e^{-1+\frac {x \left (-2+9 e^{-x} x^2\right )}{-3+\frac {x^3}{5}}} \] Output:

exp((9*x^2/exp(x)-2)*x/(1/5*x^3-3)-1)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 4.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}} \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{225-30 x^3+x^6} \, dx=e^{-1-\frac {10 x}{-15+x^3}+\frac {45 e^{-x} x^3}{-15+x^3}} \] Input:

Integrate[(E^(-x + (45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))*(-2 
025*x^2 + 675*x^3 - 45*x^6 + E^x*(150 + 20*x^3)))/(225 - 30*x^3 + x^6),x]
 

Output:

E^(-1 - (10*x)/(-15 + x^3) + (45*x^3)/(E^x*(-15 + x^3)))
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (-45 x^6+675 x^3+e^x \left (20 x^3+150\right )-2025 x^2\right ) \exp \left (\frac {e^{-x} \left (45 x^3+e^x \left (-x^3-10 x+15\right )\right )}{x^3-15}-x\right )}{x^6-30 x^3+225} \, dx\)

\(\Big \downarrow \) 1380

\(\displaystyle \int -\frac {5 \left (9 x^6-135 x^3-2 e^x \left (2 x^3+15\right )+405 x^2\right ) \exp \left (-\frac {e^{-x} \left (45 x^3+e^x \left (-x^3-10 x+15\right )\right )}{15-x^3}-x\right )}{\left (15-x^3\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -5 \int \frac {\exp \left (-x-\frac {e^{-x} \left (45 x^3+e^x \left (-x^3-10 x+15\right )\right )}{15-x^3}\right ) \left (9 x^6-135 x^3+405 x^2-2 e^x \left (2 x^3+15\right )\right )}{\left (15-x^3\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (\frac {9 \exp \left (-x-\frac {e^{-x} \left (45 x^3+e^x \left (-x^3-10 x+15\right )\right )}{15-x^3}\right ) x^2 \left (x^4-15 x+45\right )}{\left (x^3-15\right )^2}-\frac {2 \exp \left (-\frac {e^{-x} \left (45 x^3+e^x \left (-x^3-10 x+15\right )\right )}{15-x^3}\right ) \left (2 x^3+15\right )}{\left (x^3-15\right )^2}\right )dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -5 \int \left (\frac {9 \exp \left (-\frac {e^{-x} \left (e^x x^4+e^x x^3-45 x^3-5 e^x x-15 e^x\right )}{x^3-15}\right ) \left (x^4-15 x+45\right ) x^2}{\left (15-x^3\right )^2}+\frac {2 \exp \left (-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}\right ) \left (-2 x^3-15\right )}{\left (15-x^3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -5 \left (9 \int e^{-\frac {e^{-x} \left (e^x x^4+e^x x^3-45 x^3-5 e^x x-15 e^x\right )}{x^3-15}}dx+\frac {4 \int \frac {e^{-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}}}{\sqrt [3]{15}-x}dx}{3\ 15^{2/3}}-3 \sqrt [3]{15} \int \frac {e^{-\frac {e^{-x} \left (e^x x^4+e^x x^3-45 x^3-5 e^x x-15 e^x\right )}{x^3-15}}}{\sqrt [3]{15}-x}dx-\frac {2 \int \frac {e^{-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}}}{\left (\sqrt [3]{-1} x+\sqrt [3]{15}\right )^2}dx}{3 \sqrt [3]{15}}+\frac {4 \int \frac {e^{-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}}}{\sqrt [3]{-1} x+\sqrt [3]{15}}dx}{3\ 15^{2/3}}-3 \sqrt [3]{15} \int \frac {e^{-\frac {e^{-x} \left (e^x x^4+e^x x^3-45 x^3-5 e^x x-15 e^x\right )}{x^3-15}}}{\sqrt [3]{-1} x+\sqrt [3]{15}}dx+\frac {4 \int \frac {e^{-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}}}{\sqrt [3]{15}-(-1)^{2/3} x}dx}{3\ 15^{2/3}}-3 \sqrt [3]{15} \int \frac {e^{-\frac {e^{-x} \left (e^x x^4+e^x x^3-45 x^3-5 e^x x-15 e^x\right )}{x^3-15}}}{\sqrt [3]{15}-(-1)^{2/3} x}dx-\frac {8 \sqrt [3]{5} \int \frac {e^{-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}}}{\left (2\ 15^{2/3}-2 \sqrt [3]{15} x\right )^2}dx}{3^{2/3}}+\frac {4}{3} \int \frac {e^{-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}}}{15^{2/3} x-15}dx-\frac {2\ 3^{2/3} \int \frac {e^{-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}}}{\left (\sqrt [3]{15} \left (1-\sqrt [3]{-1}\right )-\left (1+(-1)^{2/3}\right ) x\right )^2}dx}{\sqrt [3]{5} \left (1+\sqrt [3]{-1}\right )^4}-\frac {8}{3} \int \frac {e^{-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}}}{15^{2/3} \left (1-i \sqrt {3}\right ) x+30}dx-\frac {8}{3} \int \frac {e^{-\frac {e^{-x} \left (e^x x^3-45 x^3+10 e^x x-15 e^x\right )}{x^3-15}}}{15^{2/3} \left (1+i \sqrt {3}\right ) x+30}dx+405 \int \frac {e^{-\frac {e^{-x} \left (e^x x^4+e^x x^3-45 x^3-5 e^x x-15 e^x\right )}{x^3-15}} x^2}{\left (x^3-15\right )^2}dx\right )\)

Input:

Int[(E^(-x + (45*x^3 + E^x*(15 - 10*x - x^3))/(E^x*(-15 + x^3)))*(-2025*x^ 
2 + 675*x^3 - 45*x^6 + E^x*(150 + 20*x^3)))/(225 - 30*x^3 + x^6),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14

method	result	size
parallelrisch	\({\mathrm e}^{\frac {\left (\left (-x^{3}-10 x +15\right ) {\mathrm e}^{x}+45 x^{3}\right ) {\mathrm e}^{-x}}{x^{3}-15}}\)	\(33\)
risch	\({\mathrm e}^{-\frac {\left ({\mathrm e}^{x} x^{3}-45 x^{3}+10 \,{\mathrm e}^{x} x -15 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{x^{3}-15}}\)	\(36\)
norman	\(\frac {\left ({\mathrm e}^{x} x^{3} {\mathrm e}^{\frac {\left (\left (-x^{3}-10 x +15\right ) {\mathrm e}^{x}+45 x^{3}\right ) {\mathrm e}^{-x}}{x^{3}-15}}-15 \,{\mathrm e}^{x} {\mathrm e}^{\frac {\left (\left (-x^{3}-10 x +15\right ) {\mathrm e}^{x}+45 x^{3}\right ) {\mathrm e}^{-x}}{x^{3}-15}}\right ) {\mathrm e}^{-x}}{x^{3}-15}\)	\(88\)

Input:

int(((20*x^3+150)*exp(x)-45*x^6+675*x^3-2025*x^2)*exp(((-x^3-10*x+15)*exp( 
x)+45*x^3)/(x^3-15)/exp(x))/(x^6-30*x^3+225)/exp(x),x,method=_RETURNVERBOS 
E)
 

Output:

exp(((-x^3-10*x+15)*exp(x)+45*x^3)/(x^3-15)/exp(x))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^{-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}} \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{225-30 x^3+x^6} \, dx=e^{\left (x + \frac {{\left (45 \, x^{3} - {\left (x^{4} + x^{3} - 5 \, x - 15\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{3} - 15}\right )} \] Input:

integrate(((20*x^3+150)*exp(x)-45*x^6+675*x^3-2025*x^2)*exp(((-x^3-10*x+15 
)*exp(x)+45*x^3)/(x^3-15)/exp(x))/(x^6-30*x^3+225)/exp(x),x, algorithm="fr 
icas")
 

Output:

e^(x + (45*x^3 - (x^4 + x^3 - 5*x - 15)*e^x)*e^(-x)/(x^3 - 15))
 

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}} \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{225-30 x^3+x^6} \, dx=e^{\frac {\left (45 x^{3} + \left (- x^{3} - 10 x + 15\right ) e^{x}\right ) e^{- x}}{x^{3} - 15}} \] Input:

integrate(((20*x**3+150)*exp(x)-45*x**6+675*x**3-2025*x**2)*exp(((-x**3-10 
*x+15)*exp(x)+45*x**3)/(x**3-15)/exp(x))/(x**6-30*x**3+225)/exp(x),x)
 

Output:

exp((45*x**3 + (-x**3 - 10*x + 15)*exp(x))*exp(-x)/(x**3 - 15))
 

Maxima [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}} \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{225-30 x^3+x^6} \, dx=e^{\left (-\frac {10 \, x}{x^{3} - 15} + \frac {675 \, e^{\left (-x\right )}}{x^{3} - 15} + 45 \, e^{\left (-x\right )} - 1\right )} \] Input:

integrate(((20*x^3+150)*exp(x)-45*x^6+675*x^3-2025*x^2)*exp(((-x^3-10*x+15 
)*exp(x)+45*x^3)/(x^3-15)/exp(x))/(x^6-30*x^3+225)/exp(x),x, algorithm="ma 
xima")
 

Output:

e^(-10*x/(x^3 - 15) + 675*e^(-x)/(x^3 - 15) + 45*e^(-x) - 1)
 

Giac [F]

\[ \int \frac {e^{-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}} \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{225-30 x^3+x^6} \, dx=\int { -\frac {5 \, {\left (9 \, x^{6} - 135 \, x^{3} + 405 \, x^{2} - 2 \, {\left (2 \, x^{3} + 15\right )} e^{x}\right )} e^{\left (-x + \frac {{\left (45 \, x^{3} - {\left (x^{3} + 10 \, x - 15\right )} e^{x}\right )} e^{\left (-x\right )}}{x^{3} - 15}\right )}}{x^{6} - 30 \, x^{3} + 225} \,d x } \] Input:

integrate(((20*x^3+150)*exp(x)-45*x^6+675*x^3-2025*x^2)*exp(((-x^3-10*x+15 
)*exp(x)+45*x^3)/(x^3-15)/exp(x))/(x^6-30*x^3+225)/exp(x),x, algorithm="gi 
ac")
 

Output:

integrate(-5*(9*x^6 - 135*x^3 + 405*x^2 - 2*(2*x^3 + 15)*e^x)*e^(-x + (45* 
x^3 - (x^3 + 10*x - 15)*e^x)*e^(-x)/(x^3 - 15))/(x^6 - 30*x^3 + 225), x)
 

Mupad [B] (verification not implemented)

Time = 2.87 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {e^{-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}} \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{225-30 x^3+x^6} \, dx={\mathrm {e}}^{-\frac {x^3}{x^3-15}}\,{\mathrm {e}}^{\frac {15}{x^3-15}}\,{\mathrm {e}}^{\frac {45\,x^3\,{\mathrm {e}}^{-x}}{x^3-15}}\,{\mathrm {e}}^{-\frac {10\,x}{x^3-15}} \] Input:

int((exp(-x)*exp(-(exp(-x)*(exp(x)*(10*x + x^3 - 15) - 45*x^3))/(x^3 - 15) 
)*(exp(x)*(20*x^3 + 150) - 2025*x^2 + 675*x^3 - 45*x^6))/(x^6 - 30*x^3 + 2 
25),x)
 

Output:

exp(-x^3/(x^3 - 15))*exp(15/(x^3 - 15))*exp((45*x^3*exp(-x))/(x^3 - 15))*e 
xp(-(10*x)/(x^3 - 15))
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {e^{-x+\frac {e^{-x} \left (45 x^3+e^x \left (15-10 x-x^3\right )\right )}{-15+x^3}} \left (-2025 x^2+675 x^3-45 x^6+e^x \left (150+20 x^3\right )\right )}{225-30 x^3+x^6} \, dx=\frac {e^{\frac {45 x^{3}}{e^{x} x^{3}-15 e^{x}}}}{e^{\frac {10 x}{x^{3}-15}} e} \] Input:

int(((20*x^3+150)*exp(x)-45*x^6+675*x^3-2025*x^2)*exp(((-x^3-10*x+15)*exp( 
x)+45*x^3)/(x^3-15)/exp(x))/(x^6-30*x^3+225)/exp(x),x)
 

Output:

e**((45*x**3)/(e**x*x**3 - 15*e**x))/(e**((10*x)/(x**3 - 15))*e)