Integrand size = 109, antiderivative size = 19 \[ \int \frac {-24 x^3 \log (20)+e^x \left (x^3-2 x^4\right ) \log (20)}{-2641807540224 e^2+990677827584 e^{2+x}-165112971264 e^{2+2 x}+16052649984 e^{2+3 x}-1003290624 e^{2+4 x}+41803776 e^{2+5 x}-1161216 e^{2+6 x}+20736 e^{2+7 x}-216 e^{2+8 x}+e^{2+9 x}} \, dx=\frac {x^4 \log (20)}{4 e^2 \left (-24+e^x\right )^8} \]
Time = 0.79 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-24 x^3 \log (20)+e^x \left (x^3-2 x^4\right ) \log (20)}{-2641807540224 e^2+990677827584 e^{2+x}-165112971264 e^{2+2 x}+16052649984 e^{2+3 x}-1003290624 e^{2+4 x}+41803776 e^{2+5 x}-1161216 e^{2+6 x}+20736 e^{2+7 x}-216 e^{2+8 x}+e^{2+9 x}} \, dx=\frac {x^4 \log (20)}{4 e^2 \left (-24+e^x\right )^8} \]
Integrate[(-24*x^3*Log[20] + E^x*(x^3 - 2*x^4)*Log[20])/(-2641807540224*E^ 2 + 990677827584*E^(2 + x) - 165112971264*E^(2 + 2*x) + 16052649984*E^(2 + 3*x) - 1003290624*E^(2 + 4*x) + 41803776*E^(2 + 5*x) - 1161216*E^(2 + 6*x ) + 20736*E^(2 + 7*x) - 216*E^(2 + 8*x) + E^(2 + 9*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.
| \(\displaystyle \int \frac {e^x \left (x^3-2 x^4\right ) \log (20)-24 x^3 \log (20)}{990677827584 e^{x+2}-165112971264 e^{2 x+2}+16052649984 e^{3 x+2}-1003290624 e^{4 x+2}+41803776 e^{5 x+2}-1161216 e^{6 x+2}+20736 e^{7 x+2}-216 e^{8 x+2}+e^{9 x+2}-2641807540224 e^2} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^3 \left (e^x (2 x-1)+24\right ) \log (20)}{e^2 \left (24-e^x\right )^9}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\log (20) \int \frac {\left (24-e^x (1-2 x)\right ) x^3}{\left (24-e^x\right )^9}dx}{e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\log (20) \int \left (-\frac {48 x^4}{\left (-24+e^x\right )^9}-\frac {(2 x-1) x^3}{\left (-24+e^x\right )^8}\right )dx}{e^2}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {\log (20) \int \left (-\frac {48 x^4}{\left (-24+e^x\right )^9}-\frac {(2 x-1) x^3}{\left (-24+e^x\right )^8}\right )dx}{e^2}\) |
Int[(-24*x^3*Log[20] + E^x*(x^3 - 2*x^4)*Log[20])/(-2641807540224*E^2 + 99 0677827584*E^(2 + x) - 165112971264*E^(2 + 2*x) + 16052649984*E^(2 + 3*x) - 1003290624*E^(2 + 4*x) + 41803776*E^(2 + 5*x) - 1161216*E^(2 + 6*x) + 20 736*E^(2 + 7*x) - 216*E^(2 + 8*x) + E^(2 + 9*x)),x]
3.23.94.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(\frac {x^{4} \left (2 \ln \left (2\right )+\ln \left (5\right )\right ) {\mathrm e}^{-2}}{4 \left (-24+{\mathrm e}^{x}\right )^{8}}\) | \(21\) |
| parallelrisch | \(\frac {\ln \left (20\right ) x^{4} {\mathrm e}^{-2}}{4 \,{\mathrm e}^{8 x}-768 \,{\mathrm e}^{7 x}+64512 \,{\mathrm e}^{6 x}-3096576 \,{\mathrm e}^{5 x}+92897280 \,{\mathrm e}^{4 x}-1783627776 \,{\mathrm e}^{3 x}+21403533312 \,{\mathrm e}^{2 x}-146767085568 \,{\mathrm e}^{x}+440301256704}\) | \(60\) |
int(((-2*x^4+x^3)*ln(20)*exp(x)-24*x^3*ln(20))/(exp(1)^2*exp(x)^9-216*exp( 1)^2*exp(x)^8+20736*exp(1)^2*exp(x)^7-1161216*exp(1)^2*exp(x)^6+41803776*e xp(1)^2*exp(x)^5-1003290624*exp(1)^2*exp(x)^4+16052649984*exp(1)^2*exp(x)^ 3-165112971264*exp(1)^2*exp(x)^2+990677827584*exp(1)^2*exp(x)-264180754022 4*exp(1)^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (15) = 30\).
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 4.00 \[ \int \frac {-24 x^3 \log (20)+e^x \left (x^3-2 x^4\right ) \log (20)}{-2641807540224 e^2+990677827584 e^{2+x}-165112971264 e^{2+2 x}+16052649984 e^{2+3 x}-1003290624 e^{2+4 x}+41803776 e^{2+5 x}-1161216 e^{2+6 x}+20736 e^{2+7 x}-216 e^{2+8 x}+e^{2+9 x}} \, dx=\frac {x^{4} e^{14} \log \left (20\right )}{4 \, {\left (110075314176 \, e^{16} + e^{\left (8 \, x + 16\right )} - 192 \, e^{\left (7 \, x + 16\right )} + 16128 \, e^{\left (6 \, x + 16\right )} - 774144 \, e^{\left (5 \, x + 16\right )} + 23224320 \, e^{\left (4 \, x + 16\right )} - 445906944 \, e^{\left (3 \, x + 16\right )} + 5350883328 \, e^{\left (2 \, x + 16\right )} - 36691771392 \, e^{\left (x + 16\right )}\right )}} \]
integrate(((-2*x^4+x^3)*log(20)*exp(x)-24*x^3*log(20))/(exp(1)^2*exp(x)^9- 216*exp(1)^2*exp(x)^8+20736*exp(1)^2*exp(x)^7-1161216*exp(1)^2*exp(x)^6+41 803776*exp(1)^2*exp(x)^5-1003290624*exp(1)^2*exp(x)^4+16052649984*exp(1)^2 *exp(x)^3-165112971264*exp(1)^2*exp(x)^2+990677827584*exp(1)^2*exp(x)-2641 807540224*exp(1)^2),x, algorithm=\
1/4*x^4*e^14*log(20)/(110075314176*e^16 + e^(8*x + 16) - 192*e^(7*x + 16) + 16128*e^(6*x + 16) - 774144*e^(5*x + 16) + 23224320*e^(4*x + 16) - 44590 6944*e^(3*x + 16) + 5350883328*e^(2*x + 16) - 36691771392*e^(x + 16))
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (17) = 34\).
Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.74 \[ \int \frac {-24 x^3 \log (20)+e^x \left (x^3-2 x^4\right ) \log (20)}{-2641807540224 e^2+990677827584 e^{2+x}-165112971264 e^{2+2 x}+16052649984 e^{2+3 x}-1003290624 e^{2+4 x}+41803776 e^{2+5 x}-1161216 e^{2+6 x}+20736 e^{2+7 x}-216 e^{2+8 x}+e^{2+9 x}} \, dx=\frac {x^{4} \log {\left (20 \right )}}{4 e^{2} e^{8 x} - 768 e^{2} e^{7 x} + 64512 e^{2} e^{6 x} - 3096576 e^{2} e^{5 x} + 92897280 e^{2} e^{4 x} - 1783627776 e^{2} e^{3 x} + 21403533312 e^{2} e^{2 x} - 146767085568 e^{2} e^{x} + 440301256704 e^{2}} \]
integrate(((-2*x**4+x**3)*ln(20)*exp(x)-24*x**3*ln(20))/(exp(1)**2*exp(x)* *9-216*exp(1)**2*exp(x)**8+20736*exp(1)**2*exp(x)**7-1161216*exp(1)**2*exp (x)**6+41803776*exp(1)**2*exp(x)**5-1003290624*exp(1)**2*exp(x)**4+1605264 9984*exp(1)**2*exp(x)**3-165112971264*exp(1)**2*exp(x)**2+990677827584*exp (1)**2*exp(x)-2641807540224*exp(1)**2),x)
x**4*log(20)/(4*exp(2)*exp(8*x) - 768*exp(2)*exp(7*x) + 64512*exp(2)*exp(6 *x) - 3096576*exp(2)*exp(5*x) + 92897280*exp(2)*exp(4*x) - 1783627776*exp( 2)*exp(3*x) + 21403533312*exp(2)*exp(2*x) - 146767085568*exp(2)*exp(x) + 4 40301256704*exp(2))
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (15) = 30\).
Time = 0.35 (sec) , antiderivative size = 79, normalized size of antiderivative = 4.16 \[ \int \frac {-24 x^3 \log (20)+e^x \left (x^3-2 x^4\right ) \log (20)}{-2641807540224 e^2+990677827584 e^{2+x}-165112971264 e^{2+2 x}+16052649984 e^{2+3 x}-1003290624 e^{2+4 x}+41803776 e^{2+5 x}-1161216 e^{2+6 x}+20736 e^{2+7 x}-216 e^{2+8 x}+e^{2+9 x}} \, dx=\frac {x^{4} {\left (\log \left (5\right ) + 2 \, \log \left (2\right )\right )}}{4 \, {\left (110075314176 \, e^{2} + e^{\left (8 \, x + 2\right )} - 192 \, e^{\left (7 \, x + 2\right )} + 16128 \, e^{\left (6 \, x + 2\right )} - 774144 \, e^{\left (5 \, x + 2\right )} + 23224320 \, e^{\left (4 \, x + 2\right )} - 445906944 \, e^{\left (3 \, x + 2\right )} + 5350883328 \, e^{\left (2 \, x + 2\right )} - 36691771392 \, e^{\left (x + 2\right )}\right )}} \]
integrate(((-2*x^4+x^3)*log(20)*exp(x)-24*x^3*log(20))/(exp(1)^2*exp(x)^9- 216*exp(1)^2*exp(x)^8+20736*exp(1)^2*exp(x)^7-1161216*exp(1)^2*exp(x)^6+41 803776*exp(1)^2*exp(x)^5-1003290624*exp(1)^2*exp(x)^4+16052649984*exp(1)^2 *exp(x)^3-165112971264*exp(1)^2*exp(x)^2+990677827584*exp(1)^2*exp(x)-2641 807540224*exp(1)^2),x, algorithm=\
1/4*x^4*(log(5) + 2*log(2))/(110075314176*e^2 + e^(8*x + 2) - 192*e^(7*x + 2) + 16128*e^(6*x + 2) - 774144*e^(5*x + 2) + 23224320*e^(4*x + 2) - 4459 06944*e^(3*x + 2) + 5350883328*e^(2*x + 2) - 36691771392*e^(x + 2))
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (15) = 30\).
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.89 \[ \int \frac {-24 x^3 \log (20)+e^x \left (x^3-2 x^4\right ) \log (20)}{-2641807540224 e^2+990677827584 e^{2+x}-165112971264 e^{2+2 x}+16052649984 e^{2+3 x}-1003290624 e^{2+4 x}+41803776 e^{2+5 x}-1161216 e^{2+6 x}+20736 e^{2+7 x}-216 e^{2+8 x}+e^{2+9 x}} \, dx=\frac {x^{4} \log \left (20\right )}{4 \, {\left (110075314176 \, e^{2} + e^{\left (8 \, x + 2\right )} - 192 \, e^{\left (7 \, x + 2\right )} + 16128 \, e^{\left (6 \, x + 2\right )} - 774144 \, e^{\left (5 \, x + 2\right )} + 23224320 \, e^{\left (4 \, x + 2\right )} - 445906944 \, e^{\left (3 \, x + 2\right )} + 5350883328 \, e^{\left (2 \, x + 2\right )} - 36691771392 \, e^{\left (x + 2\right )}\right )}} \]
integrate(((-2*x^4+x^3)*log(20)*exp(x)-24*x^3*log(20))/(exp(1)^2*exp(x)^9- 216*exp(1)^2*exp(x)^8+20736*exp(1)^2*exp(x)^7-1161216*exp(1)^2*exp(x)^6+41 803776*exp(1)^2*exp(x)^5-1003290624*exp(1)^2*exp(x)^4+16052649984*exp(1)^2 *exp(x)^3-165112971264*exp(1)^2*exp(x)^2+990677827584*exp(1)^2*exp(x)-2641 807540224*exp(1)^2),x, algorithm=\
1/4*x^4*log(20)/(110075314176*e^2 + e^(8*x + 2) - 192*e^(7*x + 2) + 16128* e^(6*x + 2) - 774144*e^(5*x + 2) + 23224320*e^(4*x + 2) - 445906944*e^(3*x + 2) + 5350883328*e^(2*x + 2) - 36691771392*e^(x + 2))
Time = 14.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.05 \[ \int \frac {-24 x^3 \log (20)+e^x \left (x^3-2 x^4\right ) \log (20)}{-2641807540224 e^2+990677827584 e^{2+x}-165112971264 e^{2+2 x}+16052649984 e^{2+3 x}-1003290624 e^{2+4 x}+41803776 e^{2+5 x}-1161216 e^{2+6 x}+20736 e^{2+7 x}-216 e^{2+8 x}+e^{2+9 x}} \, dx=\frac {x^4\,{\mathrm {e}}^{-2}\,\ln \left (20\right )}{4\,\left (5350883328\,{\mathrm {e}}^{2\,x}-445906944\,{\mathrm {e}}^{3\,x}+23224320\,{\mathrm {e}}^{4\,x}-774144\,{\mathrm {e}}^{5\,x}+16128\,{\mathrm {e}}^{6\,x}-192\,{\mathrm {e}}^{7\,x}+{\mathrm {e}}^{8\,x}-36691771392\,{\mathrm {e}}^x+110075314176\right )} \]
int((24*x^3*log(20) - exp(x)*log(20)*(x^3 - 2*x^4))/(2641807540224*exp(2) + 165112971264*exp(2*x)*exp(2) - 16052649984*exp(3*x)*exp(2) + 1003290624* exp(4*x)*exp(2) - 41803776*exp(5*x)*exp(2) + 1161216*exp(6*x)*exp(2) - 207 36*exp(7*x)*exp(2) + 216*exp(8*x)*exp(2) - exp(9*x)*exp(2) - 990677827584* exp(2)*exp(x)),x)