Integrand size = 112, antiderivative size = 28 \[ \int \frac {x^2-2 x^3-\log (2)+e^{2+x} \left (9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)\right )}{9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)} \, dx=e^{2+x}+\frac {x}{3 x-x \left (x-x^2\right )-\log (2)} \] Output:
exp(2+x)+x/(3*x-ln(2)-x*(-x^2+x))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 66.52 (sec) , antiderivative size = 3869, normalized size of antiderivative = 138.18 \[ \int \frac {x^2-2 x^3-\log (2)+e^{2+x} \left (9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)\right )}{9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)} \, dx=\text {Result too large to show} \] Input:
Integrate[(x^2 - 2*x^3 - Log[2] + E^(2 + x)*(9*x^2 - 6*x^3 + 7*x^4 - 2*x^5 + x^6 + (-6*x + 2*x^2 - 2*x^3)*Log[2] + Log[2]^2))/(9*x^2 - 6*x^3 + 7*x^4 - 2*x^5 + x^6 + (-6*x + 2*x^2 - 2*x^3)*Log[2] + Log[2]^2),x]
Output:
E^(2 + x) + x/(3*x - x^2 + x^3 - Log[2]) - 2*E^2*Log[2]*(((E^Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 1]*ExpIntegralEi[x - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 1]] - E^x/(x - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 1]))*Root[- Log[2] + 3*#1 - #1^2 + #1^3 & , 1]^2)/((Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 1] - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2])^2*(Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 1] - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 3])^2) + ((E^ Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2]*ExpIntegralEi[x - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2]] - E^x/(x - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2]))*Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2]^2)/((Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 1] - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2])^2*(Root[-L og[2] + 3*#1 - #1^2 + #1^3 & , 2] - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 3])^2) + ((E^Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 3]*ExpIntegralEi[x - Ro ot[-Log[2] + 3*#1 - #1^2 + #1^3 & , 3]] - E^x/(x - Root[-Log[2] + 3*#1 - # 1^2 + #1^3 & , 3]))*Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 3]^2)/((Root[-Lo g[2] + 3*#1 - #1^2 + #1^3 & , 1] - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 3 ])^2*(-Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2] + Root[-Log[2] + 3*#1 - #1 ^2 + #1^3 & , 3])^2) - (2*E^Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2]*ExpIn tegralEi[x - Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2]]*Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2]*(-Root[-Log[2] + 3*#1 - #1^2 + #1^3 & , 2]^2 + Root[ -Log[2] + 3*#1 - #1^2 + #1^3 & , 1]*Root[-Log[2] + 3*#1 - #1^2 + #1^3 &...
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.
| \(\displaystyle \int \frac {-2 x^3+x^2+e^{x+2} \left (x^6-2 x^5+7 x^4-6 x^3+9 x^2+\left (-2 x^3+2 x^2-6 x\right ) \log (2)+\log ^2(2)\right )-\log (2)}{x^6-2 x^5+7 x^4-6 x^3+9 x^2+\left (-2 x^3+2 x^2-6 x\right ) \log (2)+\log ^2(2)} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {-2 x^3+x^2+e^{x+2} \left (x^6-2 x^5+7 x^4-6 x^3+9 x^2+\left (-2 x^3+2 x^2-6 x\right ) \log (2)+\log ^2(2)\right )-\log (2)}{\left (x^3-x^2+3 x-\log (2)\right )^2}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-2 x^3+x^2+e^{x+2} \left (-x^3+x^2-3 x+\log (2)\right )^2-\log (2)}{\left (x^3-x^2+3 x-\log (2)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^3}{\left (x^3-x^2+3 x-\log (2)\right )^2}+\frac {x^2}{\left (x^3-x^2+3 x-\log (2)\right )^2}-\frac {\log (2)}{\left (-x^3+x^2-3 x+\log (2)\right )^2}+e^{x+2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {2 x^3}{\left (x^3-x^2+3 x-\log (2)\right )^2}+\frac {x^2}{\left (x^3-x^2+3 x-\log (2)\right )^2}-\frac {\log (2)}{\left (-x^3+x^2-3 x+\log (2)\right )^2}+e^{x+2}\right )dx\) |
Input:
Int[(x^2 - 2*x^3 - Log[2] + E^(2 + x)*(9*x^2 - 6*x^3 + 7*x^4 - 2*x^5 + x^6 + (-6*x + 2*x^2 - 2*x^3)*Log[2] + Log[2]^2))/(9*x^2 - 6*x^3 + 7*x^4 - 2*x ^5 + x^6 + (-6*x + 2*x^2 - 2*x^3)*Log[2] + Log[2]^2),x]
Output:
$Aborted
Time = 1.64 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
| method | result | size |
| risch | \(-\frac {x}{-x^{3}+x^{2}+\ln \left (2\right )-3 x}+{\mathrm e}^{2+x}\) | \(25\) |
| parts | \(-\frac {x}{-x^{3}+x^{2}+\ln \left (2\right )-3 x}+{\mathrm e}^{2+x}\) | \(25\) |
| norman | \(\frac {x^{2} {\mathrm e}^{2+x}-x +\ln \left (2\right ) {\mathrm e}^{2+x}-3 x \,{\mathrm e}^{2+x}-{\mathrm e}^{2+x} x^{3}}{-x^{3}+x^{2}+\ln \left (2\right )-3 x}\) | \(53\) |
| parallelrisch | \(\frac {x^{2} {\mathrm e}^{2+x}-x +\ln \left (2\right ) {\mathrm e}^{2+x}-3 x \,{\mathrm e}^{2+x}-{\mathrm e}^{2+x} x^{3}}{-x^{3}+x^{2}+\ln \left (2\right )-3 x}\) | \(53\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3027\) |
| default | \(\text {Expression too large to display}\) | \(3027\) |
Input:
int(((ln(2)^2+(-2*x^3+2*x^2-6*x)*ln(2)+x^6-2*x^5+7*x^4-6*x^3+9*x^2)*exp(2+ x)-ln(2)-2*x^3+x^2)/(ln(2)^2+(-2*x^3+2*x^2-6*x)*ln(2)+x^6-2*x^5+7*x^4-6*x^ 3+9*x^2),x,method=_RETURNVERBOSE)
Output:
-x/(-x^3+x^2+ln(2)-3*x)+exp(2+x)
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {x^2-2 x^3-\log (2)+e^{2+x} \left (9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)\right )}{9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)} \, dx=\frac {{\left (x^{3} - x^{2} + 3 \, x - \log \left (2\right )\right )} e^{\left (x + 2\right )} + x}{x^{3} - x^{2} + 3 \, x - \log \left (2\right )} \] Input:
integrate(((log(2)^2+(-2*x^3+2*x^2-6*x)*log(2)+x^6-2*x^5+7*x^4-6*x^3+9*x^2 )*exp(2+x)-log(2)-2*x^3+x^2)/(log(2)^2+(-2*x^3+2*x^2-6*x)*log(2)+x^6-2*x^5 +7*x^4-6*x^3+9*x^2),x, algorithm="fricas")
Output:
((x^3 - x^2 + 3*x - log(2))*e^(x + 2) + x)/(x^3 - x^2 + 3*x - log(2))
Time = 0.48 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {x^2-2 x^3-\log (2)+e^{2+x} \left (9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)\right )}{9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)} \, dx=\frac {x}{x^{3} - x^{2} + 3 x - \log {\left (2 \right )}} + e^{x + 2} \] Input:
integrate(((ln(2)**2+(-2*x**3+2*x**2-6*x)*ln(2)+x**6-2*x**5+7*x**4-6*x**3+ 9*x**2)*exp(2+x)-ln(2)-2*x**3+x**2)/(ln(2)**2+(-2*x**3+2*x**2-6*x)*ln(2)+x **6-2*x**5+7*x**4-6*x**3+9*x**2),x)
Output:
x/(x**3 - x**2 + 3*x - log(2)) + exp(x + 2)
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {x^2-2 x^3-\log (2)+e^{2+x} \left (9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)\right )}{9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)} \, dx=\frac {{\left (x^{3} e^{2} - x^{2} e^{2} + 3 \, x e^{2} - e^{2} \log \left (2\right )\right )} e^{x} + x}{x^{3} - x^{2} + 3 \, x - \log \left (2\right )} \] Input:
integrate(((log(2)^2+(-2*x^3+2*x^2-6*x)*log(2)+x^6-2*x^5+7*x^4-6*x^3+9*x^2 )*exp(2+x)-log(2)-2*x^3+x^2)/(log(2)^2+(-2*x^3+2*x^2-6*x)*log(2)+x^6-2*x^5 +7*x^4-6*x^3+9*x^2),x, algorithm="maxima")
Output:
((x^3*e^2 - x^2*e^2 + 3*x*e^2 - e^2*log(2))*e^x + x)/(x^3 - x^2 + 3*x - lo g(2))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {x^2-2 x^3-\log (2)+e^{2+x} \left (9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)\right )}{9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)} \, dx=\frac {x^{3} e^{\left (x + 2\right )} - x^{2} e^{\left (x + 2\right )} + 3 \, x e^{\left (x + 2\right )} - e^{\left (x + 2\right )} \log \left (2\right ) + x}{x^{3} - x^{2} + 3 \, x - \log \left (2\right )} \] Input:
integrate(((log(2)^2+(-2*x^3+2*x^2-6*x)*log(2)+x^6-2*x^5+7*x^4-6*x^3+9*x^2 )*exp(2+x)-log(2)-2*x^3+x^2)/(log(2)^2+(-2*x^3+2*x^2-6*x)*log(2)+x^6-2*x^5 +7*x^4-6*x^3+9*x^2),x, algorithm="giac")
Output:
(x^3*e^(x + 2) - x^2*e^(x + 2) + 3*x*e^(x + 2) - e^(x + 2)*log(2) + x)/(x^ 3 - x^2 + 3*x - log(2))
Time = 3.48 (sec) , antiderivative size = 368, normalized size of antiderivative = 13.14 \[ \int \frac {x^2-2 x^3-\log (2)+e^{2+x} \left (9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)\right )}{9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)} \, dx={\mathrm {e}}^{x+2}+\left (\sum _{k=1}^6\ln \left (-1089\,\ln \left (2\right )+\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\,\ln \left (2\right )\,6534-\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\,x\,19602+1452\,x\,\ln \left (2\right )-\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\,{\ln \left (2\right )}^2\,3894+\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\,{\ln \left (2\right )}^3\,2082-\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\,{\ln \left (2\right )}^4\,162-1832\,x\,{\ln \left (2\right )}^2+108\,x\,{\ln \left (2\right )}^3-250\,{\ln \left (2\right )}^2+567\,{\ln \left (2\right )}^3+\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\,x\,\ln \left (2\right )\,15444-\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\,x\,{\ln \left (2\right )}^2\,9928+\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\,x\,{\ln \left (2\right )}^3\,2412-\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\,x\,{\ln \left (2\right )}^4\,486\right )\,\mathrm {root}\left (9900\,\ln \left (2\right )-7846\,{\ln \left (2\right )}^2+2700\,{\ln \left (2\right )}^3-729\,{\ln \left (2\right )}^4-9801,z,k\right )\right ) \] Input:
int(-(log(2) - x^2 + 2*x^3 - exp(x + 2)*(log(2)^2 - log(2)*(6*x - 2*x^2 + 2*x^3) + 9*x^2 - 6*x^3 + 7*x^4 - 2*x^5 + x^6))/(log(2)^2 - log(2)*(6*x - 2 *x^2 + 2*x^3) + 9*x^2 - 6*x^3 + 7*x^4 - 2*x^5 + x^6),x)
Output:
exp(x + 2) + symsum(log(6534*root(9900*log(2) - 7846*log(2)^2 + 2700*log(2 )^3 - 729*log(2)^4 - 9801, z, k)*log(2) - 1089*log(2) - 19602*root(9900*lo g(2) - 7846*log(2)^2 + 2700*log(2)^3 - 729*log(2)^4 - 9801, z, k)*x + 1452 *x*log(2) - 3894*root(9900*log(2) - 7846*log(2)^2 + 2700*log(2)^3 - 729*lo g(2)^4 - 9801, z, k)*log(2)^2 + 2082*root(9900*log(2) - 7846*log(2)^2 + 27 00*log(2)^3 - 729*log(2)^4 - 9801, z, k)*log(2)^3 - 162*root(9900*log(2) - 7846*log(2)^2 + 2700*log(2)^3 - 729*log(2)^4 - 9801, z, k)*log(2)^4 - 183 2*x*log(2)^2 + 108*x*log(2)^3 - 250*log(2)^2 + 567*log(2)^3 + 15444*root(9 900*log(2) - 7846*log(2)^2 + 2700*log(2)^3 - 729*log(2)^4 - 9801, z, k)*x* log(2) - 9928*root(9900*log(2) - 7846*log(2)^2 + 2700*log(2)^3 - 729*log(2 )^4 - 9801, z, k)*x*log(2)^2 + 2412*root(9900*log(2) - 7846*log(2)^2 + 270 0*log(2)^3 - 729*log(2)^4 - 9801, z, k)*x*log(2)^3 - 486*root(9900*log(2) - 7846*log(2)^2 + 2700*log(2)^3 - 729*log(2)^4 - 9801, z, k)*x*log(2)^4)*r oot(9900*log(2) - 7846*log(2)^2 + 2700*log(2)^3 - 729*log(2)^4 - 9801, z, k), k, 1, 6)
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {x^2-2 x^3-\log (2)+e^{2+x} \left (9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)\right )}{9 x^2-6 x^3+7 x^4-2 x^5+x^6+\left (-6 x+2 x^2-2 x^3\right ) \log (2)+\log ^2(2)} \, dx=\frac {3 e^{x} \mathrm {log}\left (2\right ) e^{2}-3 e^{x} e^{2} x^{3}+3 e^{x} e^{2} x^{2}-9 e^{x} e^{2} x -\mathrm {log}\left (2\right )+x^{3}-x^{2}}{3 \,\mathrm {log}\left (2\right )-3 x^{3}+3 x^{2}-9 x} \] Input:
int(((log(2)^2+(-2*x^3+2*x^2-6*x)*log(2)+x^6-2*x^5+7*x^4-6*x^3+9*x^2)*exp( 2+x)-log(2)-2*x^3+x^2)/(log(2)^2+(-2*x^3+2*x^2-6*x)*log(2)+x^6-2*x^5+7*x^4 -6*x^3+9*x^2),x)
Output:
(3*e**x*log(2)*e**2 - 3*e**x*e**2*x**3 + 3*e**x*e**2*x**2 - 9*e**x*e**2*x - log(2) + x**3 - x**2)/(3*(log(2) - x**3 + x**2 - 3*x))