Integrand size = 322, antiderivative size = 36 \[ \int \frac {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} \left (-50 x-20 x^3-2 x^5\right )+e^{\frac {5}{5+x^2}} \left (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6\right )+e^x \left (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} \left (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6\right )\right )}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} \left (50-5 x^2-8 x^4-x^6\right )+e^{\frac {5}{5+x^2}} \left (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7\right )+e^x \left (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} \left (25 x^2+10 x^4+x^6\right )\right )} \, dx=\log \left (-2+x^2-\frac {x^2 \left (5-e^x+x\right )}{4-e^{\frac {5}{5+x^2}}}\right ) \] Output:
ln(x^2-x^2/(4-exp(5/(x^2+5)))*(x-exp(x)+5)-2)
Time = 0.17 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.78 \[ \int \frac {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} \left (-50 x-20 x^3-2 x^5\right )+e^{\frac {5}{5+x^2}} \left (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6\right )+e^x \left (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} \left (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6\right )\right )}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} \left (50-5 x^2-8 x^4-x^6\right )+e^{\frac {5}{5+x^2}} \left (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7\right )+e^x \left (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} \left (25 x^2+10 x^4+x^6\right )\right )} \, dx=-\log \left (4-e^{\frac {5}{5+x^2}}\right )+\log \left (8-2 e^{\frac {5}{5+x^2}}+x^2-e^x x^2+e^{\frac {5}{5+x^2}} x^2+x^3\right ) \] Input:
Integrate[(200*x + 300*x^2 + 80*x^3 + 120*x^4 + 8*x^5 + 12*x^6 + E^(10/(5 + x^2))*(-50*x - 20*x^3 - 2*x^5) + E^(5/(5 + x^2))*(150*x - 75*x^2 + 10*x^ 3 - 40*x^4 + 6*x^5 - 3*x^6) + E^x*(-200*x - 100*x^2 - 80*x^3 - 40*x^4 - 8* x^5 - 4*x^6 + E^(5/(5 + x^2))*(50*x + 25*x^2 + 30*x^3 + 10*x^4 + 2*x^5 + x ^6)))/(800 + 420*x^2 + 100*x^3 + 72*x^4 + 40*x^5 + 4*x^6 + 4*x^7 + E^(10/( 5 + x^2))*(50 - 5*x^2 - 8*x^4 - x^6) + E^(5/(5 + x^2))*(-400 - 85*x^2 - 25 *x^3 + 14*x^4 - 10*x^5 + 3*x^6 - x^7) + E^x*(-100*x^2 - 40*x^4 - 4*x^6 + E ^(5/(5 + x^2))*(25*x^2 + 10*x^4 + x^6))),x]
Output:
-Log[4 - E^(5/(5 + x^2))] + Log[8 - 2*E^(5/(5 + x^2)) + x^2 - E^x*x^2 + E^ (5/(5 + 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.
| \(\displaystyle \int \frac {12 x^6+8 x^5+120 x^4+80 x^3+300 x^2+e^{\frac {10}{x^2+5}} \left (-2 x^5-20 x^3-50 x\right )+e^{\frac {5}{x^2+5}} \left (-3 x^6+6 x^5-40 x^4+10 x^3-75 x^2+150 x\right )+e^x \left (-4 x^6-8 x^5-40 x^4-80 x^3-100 x^2+e^{\frac {5}{x^2+5}} \left (x^6+2 x^5+10 x^4+30 x^3+25 x^2+50 x\right )-200 x\right )+200 x}{4 x^7+4 x^6+40 x^5+72 x^4+100 x^3+420 x^2+e^{\frac {10}{x^2+5}} \left (-x^6-8 x^4-5 x^2+50\right )+e^x \left (-4 x^6-40 x^4-100 x^2+e^{\frac {5}{x^2+5}} \left (x^6+10 x^4+25 x^2\right )\right )+e^{\frac {5}{x^2+5}} \left (-x^7+3 x^6-10 x^5+14 x^4-25 x^3-85 x^2-400\right )+800} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-2 e^{\frac {10}{x^2+5}} \left (x^2+5\right )^2-4 e^x (x+2) \left (x^2+5\right )^2+4 (3 x+2) \left (x^2+5\right )^2+e^{\frac {5}{x^2+5}} \left (-3 x^5+6 x^4-40 x^3+10 x^2-75 x+150\right )+e^{\frac {5}{x^2+5}+x} \left (x^5+2 x^4+10 x^3+30 x^2+25 x+50\right )\right )}{\left (4-e^{\frac {5}{x^2+5}}\right ) \left (x^2+5\right )^2 \left (x^3-e^x x^2+x^2+e^{\frac {5}{x^2+5}} \left (x^2-2\right )+8\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x}{x^2-2}+\frac {40 x}{\left (e^{\frac {5}{x^2+5}}-4\right ) \left (x^2+5\right )^2}-\frac {\left (e^x x^7-x^7+8 e^x x^5-14 x^5+6 e^x x^4+10 x^4+5 e^x x^3+55 x^3-60 e^x x^2+140 x^2-50 e^x x+150 x-100 e^x+660\right ) x}{\left (x^2-2\right ) \left (x^2+5\right )^2 \left (x^3-e^x x^2+e^{\frac {5}{x^2+5}} x^2+x^2-2 e^{\frac {5}{x^2+5}}+8\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {2 x}{x^2-2}+\frac {40 x}{\left (e^{\frac {5}{x^2+5}}-4\right ) \left (x^2+5\right )^2}-\frac {\left (e^x x^7-x^7+8 e^x x^5-14 x^5+6 e^x x^4+10 x^4+5 e^x x^3+55 x^3-60 e^x x^2+140 x^2-50 e^x x+150 x-100 e^x+660\right ) x}{\left (x^2-2\right ) \left (x^2+5\right )^2 \left (x^3-e^x x^2+e^{\frac {5}{x^2+5}} x^2+x^2-2 e^{\frac {5}{x^2+5}}+8\right )}\right )dx\) |
Input:
Int[(200*x + 300*x^2 + 80*x^3 + 120*x^4 + 8*x^5 + 12*x^6 + E^(10/(5 + x^2) )*(-50*x - 20*x^3 - 2*x^5) + E^(5/(5 + x^2))*(150*x - 75*x^2 + 10*x^3 - 40 *x^4 + 6*x^5 - 3*x^6) + E^x*(-200*x - 100*x^2 - 80*x^3 - 40*x^4 - 8*x^5 - 4*x^6 + E^(5/(5 + x^2))*(50*x + 25*x^2 + 30*x^3 + 10*x^4 + 2*x^5 + x^6)))/ (800 + 420*x^2 + 100*x^3 + 72*x^4 + 40*x^5 + 4*x^6 + 4*x^7 + E^(10/(5 + x^ 2))*(50 - 5*x^2 - 8*x^4 - x^6) + E^(5/(5 + x^2))*(-400 - 85*x^2 - 25*x^3 + 14*x^4 - 10*x^5 + 3*x^6 - x^7) + E^x*(-100*x^2 - 40*x^4 - 4*x^6 + E^(5/(5 + x^2))*(25*x^2 + 10*x^4 + x^6))),x]
Output:
$Aborted
Time = 67.72 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61
| method | result | size |
| risch | \(\ln \left (x^{2}-2\right )+\ln \left ({\mathrm e}^{\frac {5}{x^{2}+5}}+\frac {x^{3}-{\mathrm e}^{x} x^{2}+x^{2}+8}{x^{2}-2}\right )-\ln \left ({\mathrm e}^{\frac {5}{x^{2}+5}}-4\right )\) | \(58\) |
| parallelrisch | \(-\ln \left ({\mathrm e}^{\frac {5}{x^{2}+5}}-4\right )+\ln \left (x^{3}-{\mathrm e}^{x} x^{2}+x^{2} {\mathrm e}^{\frac {5}{x^{2}+5}}+x^{2}-2 \,{\mathrm e}^{\frac {5}{x^{2}+5}}+8\right )\) | \(59\) |
Input:
int((((x^6+2*x^5+10*x^4+30*x^3+25*x^2+50*x)*exp(5/(x^2+5))-4*x^6-8*x^5-40* x^4-80*x^3-100*x^2-200*x)*exp(x)+(-2*x^5-20*x^3-50*x)*exp(5/(x^2+5))^2+(-3 *x^6+6*x^5-40*x^4+10*x^3-75*x^2+150*x)*exp(5/(x^2+5))+12*x^6+8*x^5+120*x^4 +80*x^3+300*x^2+200*x)/(((x^6+10*x^4+25*x^2)*exp(5/(x^2+5))-4*x^6-40*x^4-1 00*x^2)*exp(x)+(-x^6-8*x^4-5*x^2+50)*exp(5/(x^2+5))^2+(-x^7+3*x^6-10*x^5+1 4*x^4-25*x^3-85*x^2-400)*exp(5/(x^2+5))+4*x^7+4*x^6+40*x^5+72*x^4+100*x^3+ 420*x^2+800),x,method=_RETURNVERBOSE)
Output:
ln(x^2-2)+ln(exp(5/(x^2+5))+(x^3-exp(x)*x^2+x^2+8)/(x^2-2))-ln(exp(5/(x^2+ 5))-4)
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.58 \[ \int \frac {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} \left (-50 x-20 x^3-2 x^5\right )+e^{\frac {5}{5+x^2}} \left (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6\right )+e^x \left (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} \left (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6\right )\right )}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} \left (50-5 x^2-8 x^4-x^6\right )+e^{\frac {5}{5+x^2}} \left (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7\right )+e^x \left (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} \left (25 x^2+10 x^4+x^6\right )\right )} \, dx=2 \, \log \left (x\right ) + \log \left (-\frac {x^{3} - x^{2} e^{x} + x^{2} + {\left (x^{2} - 2\right )} e^{\left (\frac {5}{x^{2} + 5}\right )} + 8}{x^{2}}\right ) - \log \left (e^{\left (\frac {5}{x^{2} + 5}\right )} - 4\right ) \] Input:
integrate((((x^6+2*x^5+10*x^4+30*x^3+25*x^2+50*x)*exp(5/(x^2+5))-4*x^6-8*x ^5-40*x^4-80*x^3-100*x^2-200*x)*exp(x)+(-2*x^5-20*x^3-50*x)*exp(5/(x^2+5)) ^2+(-3*x^6+6*x^5-40*x^4+10*x^3-75*x^2+150*x)*exp(5/(x^2+5))+12*x^6+8*x^5+1 20*x^4+80*x^3+300*x^2+200*x)/(((x^6+10*x^4+25*x^2)*exp(5/(x^2+5))-4*x^6-40 *x^4-100*x^2)*exp(x)+(-x^6-8*x^4-5*x^2+50)*exp(5/(x^2+5))^2+(-x^7+3*x^6-10 *x^5+14*x^4-25*x^3-85*x^2-400)*exp(5/(x^2+5))+4*x^7+4*x^6+40*x^5+72*x^4+10 0*x^3+420*x^2+800),x, algorithm="fricas")
Output:
2*log(x) + log(-(x^3 - x^2*e^x + x^2 + (x^2 - 2)*e^(5/(x^2 + 5)) + 8)/x^2) - log(e^(5/(x^2 + 5)) - 4)
Exception generated. \[ \int \frac {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} \left (-50 x-20 x^3-2 x^5\right )+e^{\frac {5}{5+x^2}} \left (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6\right )+e^x \left (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} \left (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6\right )\right )}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} \left (50-5 x^2-8 x^4-x^6\right )+e^{\frac {5}{5+x^2}} \left (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7\right )+e^x \left (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} \left (25 x^2+10 x^4+x^6\right )\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((((x**6+2*x**5+10*x**4+30*x**3+25*x**2+50*x)*exp(5/(x**2+5))-4*x **6-8*x**5-40*x**4-80*x**3-100*x**2-200*x)*exp(x)+(-2*x**5-20*x**3-50*x)*e xp(5/(x**2+5))**2+(-3*x**6+6*x**5-40*x**4+10*x**3-75*x**2+150*x)*exp(5/(x* *2+5))+12*x**6+8*x**5+120*x**4+80*x**3+300*x**2+200*x)/(((x**6+10*x**4+25* x**2)*exp(5/(x**2+5))-4*x**6-40*x**4-100*x**2)*exp(x)+(-x**6-8*x**4-5*x**2 +50)*exp(5/(x**2+5))**2+(-x**7+3*x**6-10*x**5+14*x**4-25*x**3-85*x**2-400) *exp(5/(x**2+5))+4*x**7+4*x**6+40*x**5+72*x**4+100*x**3+420*x**2+800),x)
Output:
Exception raised: PolynomialError >> 1/(x**8 + 6*x**6 - 11*x**4 - 60*x**2 + 100) contains an element of the set of generators.
Time = 0.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.72 \[ \int \frac {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} \left (-50 x-20 x^3-2 x^5\right )+e^{\frac {5}{5+x^2}} \left (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6\right )+e^x \left (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} \left (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6\right )\right )}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} \left (50-5 x^2-8 x^4-x^6\right )+e^{\frac {5}{5+x^2}} \left (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7\right )+e^x \left (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} \left (25 x^2+10 x^4+x^6\right )\right )} \, dx=\log \left (x^{2} - 2\right ) + \log \left (\frac {x^{3} - x^{2} e^{x} + x^{2} + {\left (x^{2} - 2\right )} e^{\left (\frac {5}{x^{2} + 5}\right )} + 8}{x^{2} - 2}\right ) - \log \left (e^{\left (\frac {5}{x^{2} + 5}\right )} - 4\right ) \] Input:
integrate((((x^6+2*x^5+10*x^4+30*x^3+25*x^2+50*x)*exp(5/(x^2+5))-4*x^6-8*x ^5-40*x^4-80*x^3-100*x^2-200*x)*exp(x)+(-2*x^5-20*x^3-50*x)*exp(5/(x^2+5)) ^2+(-3*x^6+6*x^5-40*x^4+10*x^3-75*x^2+150*x)*exp(5/(x^2+5))+12*x^6+8*x^5+1 20*x^4+80*x^3+300*x^2+200*x)/(((x^6+10*x^4+25*x^2)*exp(5/(x^2+5))-4*x^6-40 *x^4-100*x^2)*exp(x)+(-x^6-8*x^4-5*x^2+50)*exp(5/(x^2+5))^2+(-x^7+3*x^6-10 *x^5+14*x^4-25*x^3-85*x^2-400)*exp(5/(x^2+5))+4*x^7+4*x^6+40*x^5+72*x^4+10 0*x^3+420*x^2+800),x, algorithm="maxima")
Output:
log(x^2 - 2) + log((x^3 - x^2*e^x + x^2 + (x^2 - 2)*e^(5/(x^2 + 5)) + 8)/( x^2 - 2)) - log(e^(5/(x^2 + 5)) - 4)
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (31) = 62\).
Time = 0.84 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.08 \[ \int \frac {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} \left (-50 x-20 x^3-2 x^5\right )+e^{\frac {5}{5+x^2}} \left (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6\right )+e^x \left (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} \left (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6\right )\right )}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} \left (50-5 x^2-8 x^4-x^6\right )+e^{\frac {5}{5+x^2}} \left (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7\right )+e^x \left (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} \left (25 x^2+10 x^4+x^6\right )\right )} \, dx=\log \left (x^{3} e^{x} - x^{2} e^{\left (2 \, x\right )} + x^{2} e^{x} + x^{2} e^{\left (\frac {x^{3} - x^{2} + 5 \, x}{x^{2} + 5} + 1\right )} + 8 \, e^{x} - 2 \, e^{\left (\frac {x^{3} - x^{2} + 5 \, x}{x^{2} + 5} + 1\right )}\right ) - \log \left (-4 \, e^{x} + e^{\left (\frac {x^{3} - x^{2} + 5 \, x}{x^{2} + 5} + 1\right )}\right ) \] Input:
integrate((((x^6+2*x^5+10*x^4+30*x^3+25*x^2+50*x)*exp(5/(x^2+5))-4*x^6-8*x ^5-40*x^4-80*x^3-100*x^2-200*x)*exp(x)+(-2*x^5-20*x^3-50*x)*exp(5/(x^2+5)) ^2+(-3*x^6+6*x^5-40*x^4+10*x^3-75*x^2+150*x)*exp(5/(x^2+5))+12*x^6+8*x^5+1 20*x^4+80*x^3+300*x^2+200*x)/(((x^6+10*x^4+25*x^2)*exp(5/(x^2+5))-4*x^6-40 *x^4-100*x^2)*exp(x)+(-x^6-8*x^4-5*x^2+50)*exp(5/(x^2+5))^2+(-x^7+3*x^6-10 *x^5+14*x^4-25*x^3-85*x^2-400)*exp(5/(x^2+5))+4*x^7+4*x^6+40*x^5+72*x^4+10 0*x^3+420*x^2+800),x, algorithm="giac")
Output:
log(x^3*e^x - x^2*e^(2*x) + x^2*e^x + x^2*e^((x^3 - x^2 + 5*x)/(x^2 + 5) + 1) + 8*e^x - 2*e^((x^3 - x^2 + 5*x)/(x^2 + 5) + 1)) - log(-4*e^x + e^((x^ 3 - x^2 + 5*x)/(x^2 + 5) + 1))
Time = 5.02 (sec) , antiderivative size = 528, normalized size of antiderivative = 14.67 \[ \int \frac {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} \left (-50 x-20 x^3-2 x^5\right )+e^{\frac {5}{5+x^2}} \left (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6\right )+e^x \left (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} \left (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6\right )\right )}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} \left (50-5 x^2-8 x^4-x^6\right )+e^{\frac {5}{5+x^2}} \left (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7\right )+e^x \left (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} \left (25 x^2+10 x^4+x^6\right )\right )} \, dx=\ln \left (x^7+8\,x^5+6\,x^4+5\,x^3-60\,x^2-50\,x-100\right )-\ln \left (\frac {5280\,x-400\,x^2\,{\mathrm {e}}^x-480\,x^3\,{\mathrm {e}}^x+40\,x^4\,{\mathrm {e}}^x+48\,x^5\,{\mathrm {e}}^x+64\,x^6\,{\mathrm {e}}^x+8\,x^8\,{\mathrm {e}}^x-1320\,x\,{\mathrm {e}}^{\frac {5}{x^2+5}}-300\,x^2\,{\mathrm {e}}^{\frac {5}{x^2+5}}-280\,x^3\,{\mathrm {e}}^{\frac {5}{x^2+5}}-110\,x^4\,{\mathrm {e}}^{\frac {5}{x^2+5}}-20\,x^5\,{\mathrm {e}}^{\frac {5}{x^2+5}}+28\,x^6\,{\mathrm {e}}^{\frac {5}{x^2+5}}+2\,x^8\,{\mathrm {e}}^{\frac {5}{x^2+5}}-800\,x\,{\mathrm {e}}^x+1200\,x^2+1120\,x^3+440\,x^4+80\,x^5-112\,x^6-8\,x^8+200\,x\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x+100\,x^2\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x+120\,x^3\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x-10\,x^4\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x-12\,x^5\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x-16\,x^6\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x-2\,x^8\,{\mathrm {e}}^{\frac {5}{x^2+5}}\,{\mathrm {e}}^x}{x^8+6\,x^6-11\,x^4-60\,x^2+100}\right )-\ln \left (x^2-2\right )+\ln \left (\frac {150\,x-100\,{\mathrm {e}}^x-60\,x^2\,{\mathrm {e}}^x+5\,x^3\,{\mathrm {e}}^x+6\,x^4\,{\mathrm {e}}^x+8\,x^5\,{\mathrm {e}}^x+x^7\,{\mathrm {e}}^x-50\,x\,{\mathrm {e}}^x+140\,x^2+55\,x^3+10\,x^4-14\,x^5-x^7+660}{x^7+8\,x^5+6\,x^4+5\,x^3-60\,x^2-50\,x-100}\right )+\ln \left (\frac {x\,\left (x^2\,{\mathrm {e}}^{\frac {5}{x^2+5}}-x^2\,{\mathrm {e}}^x-2\,{\mathrm {e}}^{\frac {5}{x^2+5}}+x^2+x^3+8\right )}{\left (x^2-2\right )\,{\left (x^2+5\right )}^2}\right ) \] Input:
int((200*x + exp(5/(x^2 + 5))*(150*x - 75*x^2 + 10*x^3 - 40*x^4 + 6*x^5 - 3*x^6) - exp(x)*(200*x - exp(5/(x^2 + 5))*(50*x + 25*x^2 + 30*x^3 + 10*x^4 + 2*x^5 + x^6) + 100*x^2 + 80*x^3 + 40*x^4 + 8*x^5 + 4*x^6) - exp(10/(x^2 + 5))*(50*x + 20*x^3 + 2*x^5) + 300*x^2 + 80*x^3 + 120*x^4 + 8*x^5 + 12*x ^6)/(420*x^2 - exp(10/(x^2 + 5))*(5*x^2 + 8*x^4 + x^6 - 50) - exp(x)*(100* x^2 - exp(5/(x^2 + 5))*(25*x^2 + 10*x^4 + x^6) + 40*x^4 + 4*x^6) - exp(5/( x^2 + 5))*(85*x^2 + 25*x^3 - 14*x^4 + 10*x^5 - 3*x^6 + x^7 + 400) + 100*x^ 3 + 72*x^4 + 40*x^5 + 4*x^6 + 4*x^7 + 800),x)
Output:
log(5*x^3 - 60*x^2 - 50*x + 6*x^4 + 8*x^5 + x^7 - 100) - log((5280*x - 400 *x^2*exp(x) - 480*x^3*exp(x) + 40*x^4*exp(x) + 48*x^5*exp(x) + 64*x^6*exp( x) + 8*x^8*exp(x) - 1320*x*exp(5/(x^2 + 5)) - 300*x^2*exp(5/(x^2 + 5)) - 2 80*x^3*exp(5/(x^2 + 5)) - 110*x^4*exp(5/(x^2 + 5)) - 20*x^5*exp(5/(x^2 + 5 )) + 28*x^6*exp(5/(x^2 + 5)) + 2*x^8*exp(5/(x^2 + 5)) - 800*x*exp(x) + 120 0*x^2 + 1120*x^3 + 440*x^4 + 80*x^5 - 112*x^6 - 8*x^8 + 200*x*exp(5/(x^2 + 5))*exp(x) + 100*x^2*exp(5/(x^2 + 5))*exp(x) + 120*x^3*exp(5/(x^2 + 5))*e xp(x) - 10*x^4*exp(5/(x^2 + 5))*exp(x) - 12*x^5*exp(5/(x^2 + 5))*exp(x) - 16*x^6*exp(5/(x^2 + 5))*exp(x) - 2*x^8*exp(5/(x^2 + 5))*exp(x))/(6*x^6 - 1 1*x^4 - 60*x^2 + x^8 + 100)) - log(x^2 - 2) + log((150*x - 100*exp(x) - 60 *x^2*exp(x) + 5*x^3*exp(x) + 6*x^4*exp(x) + 8*x^5*exp(x) + x^7*exp(x) - 50 *x*exp(x) + 140*x^2 + 55*x^3 + 10*x^4 - 14*x^5 - x^7 + 660)/(5*x^3 - 60*x^ 2 - 50*x + 6*x^4 + 8*x^5 + x^7 - 100)) + log((x*(x^2*exp(5/(x^2 + 5)) - x^ 2*exp(x) - 2*exp(5/(x^2 + 5)) + x^2 + x^3 + 8))/((x^2 - 2)*(x^2 + 5)^2))
\[ \int \frac {200 x+300 x^2+80 x^3+120 x^4+8 x^5+12 x^6+e^{\frac {10}{5+x^2}} \left (-50 x-20 x^3-2 x^5\right )+e^{\frac {5}{5+x^2}} \left (150 x-75 x^2+10 x^3-40 x^4+6 x^5-3 x^6\right )+e^x \left (-200 x-100 x^2-80 x^3-40 x^4-8 x^5-4 x^6+e^{\frac {5}{5+x^2}} \left (50 x+25 x^2+30 x^3+10 x^4+2 x^5+x^6\right )\right )}{800+420 x^2+100 x^3+72 x^4+40 x^5+4 x^6+4 x^7+e^{\frac {10}{5+x^2}} \left (50-5 x^2-8 x^4-x^6\right )+e^{\frac {5}{5+x^2}} \left (-400-85 x^2-25 x^3+14 x^4-10 x^5+3 x^6-x^7\right )+e^x \left (-100 x^2-40 x^4-4 x^6+e^{\frac {5}{5+x^2}} \left (25 x^2+10 x^4+x^6\right )\right )} \, dx=\text {too large to display} \] Input:
int((((x^6+2*x^5+10*x^4+30*x^3+25*x^2+50*x)*exp(5/(x^2+5))-4*x^6-8*x^5-40* x^4-80*x^3-100*x^2-200*x)*exp(x)+(-2*x^5-20*x^3-50*x)*exp(5/(x^2+5))^2+(-3 *x^6+6*x^5-40*x^4+10*x^3-75*x^2+150*x)*exp(5/(x^2+5))+12*x^6+8*x^5+120*x^4 +80*x^3+300*x^2+200*x)/(((x^6+10*x^4+25*x^2)*exp(5/(x^2+5))-4*x^6-40*x^4-1 00*x^2)*exp(x)+(-x^6-8*x^4-5*x^2+50)*exp(5/(x^2+5))^2+(-x^7+3*x^6-10*x^5+1 4*x^4-25*x^3-85*x^2-400)*exp(5/(x^2+5))+4*x^7+4*x^6+40*x^5+72*x^4+100*x^3+ 420*x^2+800),x)
Output:
- 12*int(x**6/(e**(10/(x**2 + 5))*x**6 + 8*e**(10/(x**2 + 5))*x**4 + 5*e* *(10/(x**2 + 5))*x**2 - 50*e**(10/(x**2 + 5)) - e**((x**3 + 5*x + 5)/(x**2 + 5))*x**6 - 10*e**((x**3 + 5*x + 5)/(x**2 + 5))*x**4 - 25*e**((x**3 + 5* x + 5)/(x**2 + 5))*x**2 + e**(5/(x**2 + 5))*x**7 - 3*e**(5/(x**2 + 5))*x** 6 + 10*e**(5/(x**2 + 5))*x**5 - 14*e**(5/(x**2 + 5))*x**4 + 25*e**(5/(x**2 + 5))*x**3 + 85*e**(5/(x**2 + 5))*x**2 + 400*e**(5/(x**2 + 5)) + 4*e**x*x **6 + 40*e**x*x**4 + 100*e**x*x**2 - 4*x**7 - 4*x**6 - 40*x**5 - 72*x**4 - 100*x**3 - 420*x**2 - 800),x) - 8*int(x**5/(e**(10/(x**2 + 5))*x**6 + 8*e **(10/(x**2 + 5))*x**4 + 5*e**(10/(x**2 + 5))*x**2 - 50*e**(10/(x**2 + 5)) - e**((x**3 + 5*x + 5)/(x**2 + 5))*x**6 - 10*e**((x**3 + 5*x + 5)/(x**2 + 5))*x**4 - 25*e**((x**3 + 5*x + 5)/(x**2 + 5))*x**2 + e**(5/(x**2 + 5))*x **7 - 3*e**(5/(x**2 + 5))*x**6 + 10*e**(5/(x**2 + 5))*x**5 - 14*e**(5/(x** 2 + 5))*x**4 + 25*e**(5/(x**2 + 5))*x**3 + 85*e**(5/(x**2 + 5))*x**2 + 400 *e**(5/(x**2 + 5)) + 4*e**x*x**6 + 40*e**x*x**4 + 100*e**x*x**2 - 4*x**7 - 4*x**6 - 40*x**5 - 72*x**4 - 100*x**3 - 420*x**2 - 800),x) - 120*int(x**4 /(e**(10/(x**2 + 5))*x**6 + 8*e**(10/(x**2 + 5))*x**4 + 5*e**(10/(x**2 + 5 ))*x**2 - 50*e**(10/(x**2 + 5)) - e**((x**3 + 5*x + 5)/(x**2 + 5))*x**6 - 10*e**((x**3 + 5*x + 5)/(x**2 + 5))*x**4 - 25*e**((x**3 + 5*x + 5)/(x**2 + 5))*x**2 + e**(5/(x**2 + 5))*x**7 - 3*e**(5/(x**2 + 5))*x**6 + 10*e**(5/( x**2 + 5))*x**5 - 14*e**(5/(x**2 + 5))*x**4 + 25*e**(5/(x**2 + 5))*x**3...