\(\int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} (e^x+10 x)}{e^{-e^x+x}-x+2 x^3-x^4} \, dx\) [385]

Optimal result

Integrand size = 68, antiderivative size = 30 \[ \int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} \left (e^x+10 x\right )}{e^{-e^x+x}-x+2 x^3-x^4} \, dx=x+5 x^2-\log \left (-e^{-e^x+x}+x+(-2+x) x^3\right ) \] Output:

x-ln(x-exp(x-exp(x))+(-2+x)*x^3)+5*x^2
 

Mathematica [A] (verified)

Time = 5.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} \left (e^x+10 x\right )}{e^{-e^x+x}-x+2 x^3-x^4} \, dx=e^x+x+5 x^2-\log \left (e^x-e^{e^x} x \left (1-2 x^2+x^3\right )\right ) \] Input:

Integrate[(1 - x - 16*x^2 + 6*x^3 + 19*x^4 - 10*x^5 + E^(-E^x + x)*(E^x + 
10*x))/(E^(-E^x + x) - x + 2*x^3 - x^4),x]
 

Output:

E^x + x + 5*x^2 - Log[E^x - E^E^x*x*(1 - 2*x^2 + 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.

\(\displaystyle \int \frac {-10 x^5+19 x^4+6 x^3-16 x^2-x+e^{x-e^x} \left (10 x+e^x\right )+1}{-x^4+2 x^3-x+e^{x-e^x}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (x \left (e^{e^x} x^3-2 e^{e^x} x^2+e^{e^x}+10\right )-\frac {e^{e^x} \left (e^{e^x} x^8-4 e^{e^x} x^7+4 e^{e^x} x^6+2 e^{e^x} x^5-4 e^{e^x} x^4-x^4+6 x^3+e^{e^x} x^2-6 x^2-x+1\right )}{e^{e^x} x^4-2 e^{e^x} x^3+e^{e^x} x-e^x}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-e^{e^x} \left (10 x^5-19 x^4-6 x^3+16 x^2+x-1\right )+10 e^x x+e^{2 x}}{e^x-e^{e^x} x \left (x^3-2 x^2+1\right )}dx\)

Input:

Int[(1 - x - 16*x^2 + 6*x^3 + 19*x^4 - 10*x^5 + E^(-E^x + x)*(E^x + 10*x)) 
/(E^(-E^x + x) - x + 2*x^3 - x^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00

method	result	size
norman	\(5 x^{2}+x -\ln \left (x^{4}-2 x^{3}+x -{\mathrm e}^{x -{\mathrm e}^{x}}\right )\)	\(30\)
parallelrisch	\(5 x^{2}+x -\ln \left (x^{4}-2 x^{3}+x -{\mathrm e}^{x -{\mathrm e}^{x}}\right )\)	\(30\)
risch	\(5 x^{2}+x -\ln \left ({\mathrm e}^{x -{\mathrm e}^{x}}-x^{4}+2 x^{3}-x \right )\)	\(32\)

Input:

int(((exp(x)+10*x)*exp(x-exp(x))-10*x^5+19*x^4+6*x^3-16*x^2-x+1)/(exp(x-ex 
p(x))-x^4+2*x^3-x),x,method=_RETURNVERBOSE)
 

Output:

5*x^2+x-ln(x^4-2*x^3+x-exp(x-exp(x)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} \left (e^x+10 x\right )}{e^{-e^x+x}-x+2 x^3-x^4} \, dx=5 \, x^{2} + x - \log \left (-x^{4} + 2 \, x^{3} - x + e^{\left (x - e^{x}\right )}\right ) \] Input:

integrate(((exp(x)+10*x)*exp(x-exp(x))-10*x^5+19*x^4+6*x^3-16*x^2-x+1)/(ex 
p(x-exp(x))-x^4+2*x^3-x),x, algorithm="fricas")
 

Output:

5*x^2 + x - log(-x^4 + 2*x^3 - x + e^(x - e^x))
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} \left (e^x+10 x\right )}{e^{-e^x+x}-x+2 x^3-x^4} \, dx=5 x^{2} + x - \log {\left (- x^{4} + 2 x^{3} - x + e^{x - e^{x}} \right )} \] Input:

integrate(((exp(x)+10*x)*exp(x-exp(x))-10*x**5+19*x**4+6*x**3-16*x**2-x+1) 
/(exp(x-exp(x))-x**4+2*x**3-x),x)
 

Output:

5*x**2 + x - log(-x**4 + 2*x**3 - x + exp(x - exp(x)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (28) = 56\).

Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.17 \[ \int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} \left (e^x+10 x\right )}{e^{-e^x+x}-x+2 x^3-x^4} \, dx=5 \, x^{2} + x + e^{x} - \log \left (x^{2} - x - 1\right ) - \log \left (x - 1\right ) - \log \left (x\right ) - \log \left (\frac {{\left (x^{4} - 2 \, x^{3} + x\right )} e^{\left (e^{x}\right )} - e^{x}}{x^{4} - 2 \, x^{3} + x}\right ) \] Input:

integrate(((exp(x)+10*x)*exp(x-exp(x))-10*x^5+19*x^4+6*x^3-16*x^2-x+1)/(ex 
p(x-exp(x))-x^4+2*x^3-x),x, algorithm="maxima")
 

Output:

5*x^2 + x + e^x - log(x^2 - x - 1) - log(x - 1) - log(x) - log(((x^4 - 2*x 
^3 + x)*e^(e^x) - e^x)/(x^4 - 2*x^3 + x))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} \left (e^x+10 x\right )}{e^{-e^x+x}-x+2 x^3-x^4} \, dx=5 \, x^{2} + x - \log \left (-x^{4} + 2 \, x^{3} - x + e^{\left (x - e^{x}\right )}\right ) \] Input:

integrate(((exp(x)+10*x)*exp(x-exp(x))-10*x^5+19*x^4+6*x^3-16*x^2-x+1)/(ex 
p(x-exp(x))-x^4+2*x^3-x),x, algorithm="giac")
 

Output:

5*x^2 + x - log(-x^4 + 2*x^3 - x + e^(x - e^x))
 

Mupad [B] (verification not implemented)

Time = 2.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} \left (e^x+10 x\right )}{e^{-e^x+x}-x+2 x^3-x^4} \, dx=x-\ln \left (x-{\mathrm {e}}^{x-{\mathrm {e}}^x}-2\,x^3+x^4\right )+5\,x^2 \] Input:

int(-(6*x^3 - 16*x^2 - x + 19*x^4 - 10*x^5 + exp(x - exp(x))*(10*x + exp(x 
)) + 1)/(x - exp(x - exp(x)) - 2*x^3 + x^4),x)
 

Output:

x - log(x - exp(x - exp(x)) - 2*x^3 + x^4) + 5*x^2
 

Reduce [F]

\[ \int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} \left (e^x+10 x\right )}{e^{-e^x+x}-x+2 x^3-x^4} \, dx=-\left (\int \frac {e^{e^{x}}}{e^{e^{x}} x^{4}-2 e^{e^{x}} x^{3}+e^{e^{x}} x -e^{x}}d x \right )-\left (\int \frac {e^{2 x}}{e^{e^{x}} x^{4}-2 e^{e^{x}} x^{3}+e^{e^{x}} x -e^{x}}d x \right )+10 \left (\int \frac {e^{e^{x}} x^{5}}{e^{e^{x}} x^{4}-2 e^{e^{x}} x^{3}+e^{e^{x}} x -e^{x}}d x \right )-19 \left (\int \frac {e^{e^{x}} x^{4}}{e^{e^{x}} x^{4}-2 e^{e^{x}} x^{3}+e^{e^{x}} x -e^{x}}d x \right )-6 \left (\int \frac {e^{e^{x}} x^{3}}{e^{e^{x}} x^{4}-2 e^{e^{x}} x^{3}+e^{e^{x}} x -e^{x}}d x \right )+16 \left (\int \frac {e^{e^{x}} x^{2}}{e^{e^{x}} x^{4}-2 e^{e^{x}} x^{3}+e^{e^{x}} x -e^{x}}d x \right )+\int \frac {e^{e^{x}} x}{e^{e^{x}} x^{4}-2 e^{e^{x}} x^{3}+e^{e^{x}} x -e^{x}}d x -10 \left (\int \frac {e^{x} x}{e^{e^{x}} x^{4}-2 e^{e^{x}} x^{3}+e^{e^{x}} x -e^{x}}d x \right ) \] Input:

int(((exp(x)+10*x)*exp(x-exp(x))-10*x^5+19*x^4+6*x^3-16*x^2-x+1)/(exp(x-ex 
p(x))-x^4+2*x^3-x),x)
 

Output:

 - int(e**(e**x)/(e**(e**x)*x**4 - 2*e**(e**x)*x**3 + e**(e**x)*x - e**x), 
x) - int(e**(2*x)/(e**(e**x)*x**4 - 2*e**(e**x)*x**3 + e**(e**x)*x - e**x) 
,x) + 10*int((e**(e**x)*x**5)/(e**(e**x)*x**4 - 2*e**(e**x)*x**3 + e**(e** 
x)*x - e**x),x) - 19*int((e**(e**x)*x**4)/(e**(e**x)*x**4 - 2*e**(e**x)*x* 
*3 + e**(e**x)*x - e**x),x) - 6*int((e**(e**x)*x**3)/(e**(e**x)*x**4 - 2*e 
**(e**x)*x**3 + e**(e**x)*x - e**x),x) + 16*int((e**(e**x)*x**2)/(e**(e**x 
)*x**4 - 2*e**(e**x)*x**3 + e**(e**x)*x - e**x),x) + int((e**(e**x)*x)/(e* 
*(e**x)*x**4 - 2*e**(e**x)*x**3 + e**(e**x)*x - e**x),x) - 10*int((e**x*x) 
/(e**(e**x)*x**4 - 2*e**(e**x)*x**3 + e**(e**x)*x - e**x),x)