Integrand size = 105, antiderivative size = 25 \[ \int \frac {e^{\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )} \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{2 x^7} \, dx=e^{e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2} \] Output:
exp((x^2+x-4)^2*exp(1/4*exp(1+x)/x^6)^2)
Time = 5.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )} \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{2 x^7} \, dx=e^{e^{\frac {e^{1+x}}{2 x^6}} \left (-4+x+x^2\right )^2} \] Input:
Integrate[(E^(E^(1 + x)/(2*x^6) + E^(E^(1 + x)/(2*x^6))*(16 - 8*x - 7*x^2 + 2*x^3 + x^4))*(-16*x^7 - 28*x^8 + 12*x^9 + 8*x^10 + E^(1 + x)*(-96 + 64* x + 34*x^2 - 19*x^3 - 4*x^4 + x^5)))/(2*x^7),x]
Output:
E^(E^(E^(1 + x)/(2*x^6))*(-4 + x + x^2)^2)
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.
| \(\displaystyle \int \frac {\left (8 x^{10}+12 x^9-28 x^8-16 x^7+e^{x+1} \left (x^5-4 x^4-19 x^3+34 x^2+64 x-96\right )\right ) \exp \left (\frac {e^{x+1}}{2 x^6}+e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )\right )}{2 x^7} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {\exp \left (e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+\frac {e^{x+1}}{2 x^6}\right ) \left (-8 x^{10}-12 x^9+28 x^8+16 x^7+e^{x+1} \left (-x^5+4 x^4+19 x^3-34 x^2-64 x+96\right )\right )}{x^7}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {\exp \left (e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+\frac {e^{x+1}}{2 x^6}\right ) \left (-8 x^{10}-12 x^9+28 x^8+16 x^7+e^{x+1} \left (-x^5+4 x^4+19 x^3-34 x^2-64 x+96\right )\right )}{x^7}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {1}{2} \int \frac {\exp \left (e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+\frac {e^{x+1}}{2 x^6}\right ) \left (-x^2-x+4\right ) \left (8 x^8+4 x^7+e^{x+1} x^3-5 e^{x+1} x^2-10 e^{x+1} x+24 e^{x+1}\right )}{x^7}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (-\frac {\exp \left (x+e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+1+\frac {e^{x+1}}{2 x^6}\right ) (x-6) \left (x^2+x-4\right )^2}{x^7}-4 \exp \left (e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+\frac {e^{x+1}}{2 x^6}\right ) \left (2 x^3+3 x^2-7 x-4\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-16 \int \exp \left (e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+\frac {e^{x+1}}{2 x^6}\right )dx+64 \int \frac {\exp \left (x+e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+1+\frac {e^{x+1}}{2 x^6}\right )}{x^6}dx-19 \int \frac {\exp \left (x+e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+1+\frac {e^{x+1}}{2 x^6}\right )}{x^4}dx-4 \int \frac {\exp \left (x+e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+1+\frac {e^{x+1}}{2 x^6}\right )}{x^3}dx+\int \frac {\exp \left (x+e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+1+\frac {e^{x+1}}{2 x^6}\right )}{x^2}dx-28 \int \exp \left (e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+\frac {e^{x+1}}{2 x^6}\right ) xdx+12 \int \exp \left (e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+\frac {e^{x+1}}{2 x^6}\right ) x^2dx+8 \int \exp \left (e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+\frac {e^{x+1}}{2 x^6}\right ) x^3dx-96 \int \frac {\exp \left (x+e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+1+\frac {e^{x+1}}{2 x^6}\right )}{x^7}dx+34 \int \frac {\exp \left (x+e^{\frac {e^{x+1}}{2 x^6}} \left (x^4+2 x^3-7 x^2-8 x+16\right )+1+\frac {e^{x+1}}{2 x^6}\right )}{x^5}dx\right )\) |
Input:
Int[(E^(E^(1 + x)/(2*x^6) + E^(E^(1 + x)/(2*x^6))*(16 - 8*x - 7*x^2 + 2*x^ 3 + x^4))*(-16*x^7 - 28*x^8 + 12*x^9 + 8*x^10 + E^(1 + x)*(-96 + 64*x + 34 *x^2 - 19*x^3 - 4*x^4 + x^5)))/(2*x^7),x]
Output:
$Aborted
Time = 55.32 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \({\mathrm e}^{\left (x^{2}+x -4\right )^{2} {\mathrm e}^{\frac {{\mathrm e}^{1+x}}{2 x^{6}}}}\) | \(21\) |
| parallelrisch | \({\mathrm e}^{\left (x^{4}+2 x^{3}-7 x^{2}-8 x +16\right ) {\mathrm e}^{\frac {{\mathrm e}^{1+x}}{2 x^{6}}}}\) | \(33\) |
Input:
int(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(1+x)+8*x^10+12*x^9-28*x^8-1 6*x^7)*exp(1/4*exp(1+x)/x^6)^2*exp((x^4+2*x^3-7*x^2-8*x+16)*exp(1/4*exp(1+ x)/x^6)^2)/x^7,x,method=_RETURNVERBOSE)
Output:
exp((x^2+x-4)^2*exp(1/2*exp(1+x)/x^6))
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (20) = 40\).
Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.28 \[ \int \frac {e^{\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )} \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{2 x^7} \, dx=e^{\left (\frac {2 \, {\left (x^{10} + 2 \, x^{9} - 7 \, x^{8} - 8 \, x^{7} + 16 \, x^{6}\right )} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} + e^{\left (x + 1\right )}}{2 \, x^{6}} - \frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} \] Input:
integrate(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(1+x)+8*x^10+12*x^9-28 *x^8-16*x^7)*exp(1/4*exp(1+x)/x^6)^2*exp((x^4+2*x^3-7*x^2-8*x+16)*exp(1/4* exp(1+x)/x^6)^2)/x^7,x, algorithm="fricas")
Output:
e^(1/2*(2*(x^10 + 2*x^9 - 7*x^8 - 8*x^7 + 16*x^6)*e^(1/2*e^(x + 1)/x^6) + e^(x + 1))/x^6 - 1/2*e^(x + 1)/x^6)
Time = 5.34 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )} \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{2 x^7} \, dx=e^{\left (x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 16\right ) e^{\frac {e^{x + 1}}{2 x^{6}}}} \] Input:
integrate(1/2*((x**5-4*x**4-19*x**3+34*x**2+64*x-96)*exp(1+x)+8*x**10+12*x **9-28*x**8-16*x**7)*exp(1/4*exp(1+x)/x**6)**2*exp((x**4+2*x**3-7*x**2-8*x +16)*exp(1/4*exp(1+x)/x**6)**2)/x**7,x)
Output:
exp((x**4 + 2*x**3 - 7*x**2 - 8*x + 16)*exp(exp(x + 1)/(2*x**6)))
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (20) = 40\).
Time = 0.36 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \frac {e^{\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )} \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{2 x^7} \, dx=e^{\left (x^{4} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} + 2 \, x^{3} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} - 7 \, x^{2} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} - 8 \, x e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} + 16 \, e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )}\right )} \] Input:
integrate(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(1+x)+8*x^10+12*x^9-28 *x^8-16*x^7)*exp(1/4*exp(1+x)/x^6)^2*exp((x^4+2*x^3-7*x^2-8*x+16)*exp(1/4* exp(1+x)/x^6)^2)/x^7,x, algorithm="maxima")
Output:
e^(x^4*e^(1/2*e^(x + 1)/x^6) + 2*x^3*e^(1/2*e^(x + 1)/x^6) - 7*x^2*e^(1/2* e^(x + 1)/x^6) - 8*x*e^(1/2*e^(x + 1)/x^6) + 16*e^(1/2*e^(x + 1)/x^6))
\[ \int \frac {e^{\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )} \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{2 x^7} \, dx=\int { \frac {{\left (8 \, x^{10} + 12 \, x^{9} - 28 \, x^{8} - 16 \, x^{7} + {\left (x^{5} - 4 \, x^{4} - 19 \, x^{3} + 34 \, x^{2} + 64 \, x - 96\right )} e^{\left (x + 1\right )}\right )} e^{\left ({\left (x^{4} + 2 \, x^{3} - 7 \, x^{2} - 8 \, x + 16\right )} e^{\left (\frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )} + \frac {e^{\left (x + 1\right )}}{2 \, x^{6}}\right )}}{2 \, x^{7}} \,d x } \] Input:
integrate(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(1+x)+8*x^10+12*x^9-28 *x^8-16*x^7)*exp(1/4*exp(1+x)/x^6)^2*exp((x^4+2*x^3-7*x^2-8*x+16)*exp(1/4* exp(1+x)/x^6)^2)/x^7,x, algorithm="giac")
Output:
integrate(1/2*(8*x^10 + 12*x^9 - 28*x^8 - 16*x^7 + (x^5 - 4*x^4 - 19*x^3 + 34*x^2 + 64*x - 96)*e^(x + 1))*e^((x^4 + 2*x^3 - 7*x^2 - 8*x + 16)*e^(1/2 *e^(x + 1)/x^6) + 1/2*e^(x + 1)/x^6)/x^7, x)
Time = 1.97 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.00 \[ \int \frac {e^{\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )} \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{2 x^7} \, dx={\mathrm {e}}^{-8\,x\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}}\,{\mathrm {e}}^{x^4\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}}\,{\mathrm {e}}^{2\,x^3\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}}\,{\mathrm {e}}^{-7\,x^2\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}}\,{\mathrm {e}}^{16\,{\mathrm {e}}^{\frac {\mathrm {e}\,{\mathrm {e}}^x}{2\,x^6}}} \] Input:
int((exp(exp(exp(x + 1)/(2*x^6))*(2*x^3 - 7*x^2 - 8*x + x^4 + 16))*exp(exp (x + 1)/(2*x^6))*(exp(x + 1)*(64*x + 34*x^2 - 19*x^3 - 4*x^4 + x^5 - 96) - 16*x^7 - 28*x^8 + 12*x^9 + 8*x^10))/(2*x^7),x)
Output:
exp(-8*x*exp((exp(1)*exp(x))/(2*x^6)))*exp(x^4*exp((exp(1)*exp(x))/(2*x^6) ))*exp(2*x^3*exp((exp(1)*exp(x))/(2*x^6)))*exp(-7*x^2*exp((exp(1)*exp(x))/ (2*x^6)))*exp(16*exp((exp(1)*exp(x))/(2*x^6)))
\[ \int \frac {e^{\frac {e^{1+x}}{2 x^6}+e^{\frac {e^{1+x}}{2 x^6}} \left (16-8 x-7 x^2+2 x^3+x^4\right )} \left (-16 x^7-28 x^8+12 x^9+8 x^{10}+e^{1+x} \left (-96+64 x+34 x^2-19 x^3-4 x^4+x^5\right )\right )}{2 x^7} \, dx=\int \frac {\left (\left (x^{5}-4 x^{4}-19 x^{3}+34 x^{2}+64 x -96\right ) {\mathrm e}^{x +1}+8 x^{10}+12 x^{9}-28 x^{8}-16 x^{7}\right ) \left ({\mathrm e}^{\frac {{\mathrm e}^{x +1}}{4 x^{6}}}\right )^{2} {\mathrm e}^{\left (x^{4}+2 x^{3}-7 x^{2}-8 x +16\right ) \left ({\mathrm e}^{\frac {{\mathrm e}^{x +1}}{4 x^{6}}}\right )^{2}}}{2 x^{7}}d x \] Input:
int(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(1+x)+8*x^10+12*x^9-28*x^8-1 6*x^7)*exp(1/4*exp(1+x)/x^6)^2*exp((x^4+2*x^3-7*x^2-8*x+16)*exp(1/4*exp(1+ x)/x^6)^2)/x^7,x)
Output:
int(1/2*((x^5-4*x^4-19*x^3+34*x^2+64*x-96)*exp(1+x)+8*x^10+12*x^9-28*x^8-1 6*x^7)*exp(1/4*exp(1+x)/x^6)^2*exp((x^4+2*x^3-7*x^2-8*x+16)*exp(1/4*exp(1+ x)/x^6)^2)/x^7,x)