Integrand size = 259, antiderivative size = 35 \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=1+\log \left (1-x-e^4 x+\frac {3}{\frac {x}{1-e^x x^2}+\log (3)}\right ) \] Output:
ln(3/(x/(1-exp(x)*x^2)+ln(3))-x-x*exp(4)+1)+1
\[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx \] Input:
Integrate[(3 + x^2 + E^4*x^2 + (2*x + 2*E^4*x)*Log[3] + (1 + E^4)*Log[3]^2 + E^(2*x)*(x^4 + E^4*x^4)*Log[3]^2 + E^x*(3*x^2 + 3*x^3 + (-2*x^3 - 2*E^4 *x^3)*Log[3] + (-2*x^2 - 2*E^4*x^2)*Log[3]^2))/(-3*x - x^2 + x^3 + E^4*x^3 + (-3 - 2*x + 2*x^2 + 2*E^4*x^2)*Log[3] + (-1 + x + E^4*x)*Log[3]^2 + E^x *(3*x^3 + (6*x^2 + 2*x^3 - 2*x^4 - 2*E^4*x^4)*Log[3] + (2*x^2 - 2*x^3 - 2* E^4*x^3)*Log[3]^2) + E^(2*x)*(-3*x^4*Log[3] + (-x^4 + x^5 + E^4*x^5)*Log[3 ]^2)),x]
Output:
Integrate[(3 + x^2 + E^4*x^2 + (2*x + 2*E^4*x)*Log[3] + (1 + E^4)*Log[3]^2 + E^(2*x)*(x^4 + E^4*x^4)*Log[3]^2 + E^x*(3*x^2 + 3*x^3 + (-2*x^3 - 2*E^4 *x^3)*Log[3] + (-2*x^2 - 2*E^4*x^2)*Log[3]^2))/(-3*x - x^2 + x^3 + E^4*x^3 + (-3 - 2*x + 2*x^2 + 2*E^4*x^2)*Log[3] + (-1 + x + E^4*x)*Log[3]^2 + E^x *(3*x^3 + (6*x^2 + 2*x^3 - 2*x^4 - 2*E^4*x^4)*Log[3] + (2*x^2 - 2*x^3 - 2* E^4*x^3)*Log[3]^2) + E^(2*x)*(-3*x^4*Log[3] + (-x^4 + x^5 + E^4*x^5)*Log[3 ]^2)), 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.
| \(\displaystyle \int \frac {e^{2 x} \left (e^4 x^4+x^4\right ) \log ^2(3)+e^4 x^2+x^2+e^x \left (3 x^3+\left (-2 e^4 x^3-2 x^3\right ) \log (3)+3 x^2+\left (-2 e^4 x^2-2 x^2\right ) \log ^2(3)\right )+\left (2 e^4 x+2 x\right ) \log (3)+3+\left (1+e^4\right ) \log ^2(3)}{e^4 x^3+x^3-x^2+\left (2 e^4 x^2+2 x^2-2 x-3\right ) \log (3)+e^{2 x} \left (\left (e^4 x^5+x^5-x^4\right ) \log ^2(3)-3 x^4 \log (3)\right )+e^x \left (3 x^3+\left (-2 e^4 x^3-2 x^3+2 x^2\right ) \log ^2(3)+\left (-2 e^4 x^4-2 x^4+2 x^3+6 x^2\right ) \log (3)\right )-3 x+\left (e^4 x+x-1\right ) \log ^2(3)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^{2 x} \left (e^4 x^4+x^4\right ) \log ^2(3)+\left (1+e^4\right ) x^2+e^x \left (3 x^3+\left (-2 e^4 x^3-2 x^3\right ) \log (3)+3 x^2+\left (-2 e^4 x^2-2 x^2\right ) \log ^2(3)\right )+\left (2 e^4 x+2 x\right ) \log (3)+3+\left (1+e^4\right ) \log ^2(3)}{e^4 x^3+x^3-x^2+\left (2 e^4 x^2+2 x^2-2 x-3\right ) \log (3)+e^{2 x} \left (\left (e^4 x^5+x^5-x^4\right ) \log ^2(3)-3 x^4 \log (3)\right )+e^x \left (3 x^3+\left (-2 e^4 x^3-2 x^3+2 x^2\right ) \log ^2(3)+\left (-2 e^4 x^4-2 x^4+2 x^3+6 x^2\right ) \log (3)\right )-3 x+\left (e^4 x+x-1\right ) \log ^2(3)}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^{2 x} \left (e^4 x^4+x^4\right ) \log ^2(3)+\left (1+e^4\right ) x^2+e^x \left (3 x^3+\left (-2 e^4 x^3-2 x^3\right ) \log (3)+3 x^2+\left (-2 e^4 x^2-2 x^2\right ) \log ^2(3)\right )+\left (2 e^4 x+2 x\right ) \log (3)+3+\left (1+e^4\right ) \log ^2(3)}{\left (1+e^4\right ) x^3-x^2+\left (2 e^4 x^2+2 x^2-2 x-3\right ) \log (3)+e^{2 x} \left (\left (e^4 x^5+x^5-x^4\right ) \log ^2(3)-3 x^4 \log (3)\right )+e^x \left (3 x^3+\left (-2 e^4 x^3-2 x^3+2 x^2\right ) \log ^2(3)+\left (-2 e^4 x^4-2 x^4+2 x^3+6 x^2\right ) \log (3)\right )-3 x+\left (e^4 x+x-1\right ) \log ^2(3)}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 x} \left (e^4 x^4+x^4\right ) \log ^2(3)+\left (1+e^4\right ) x^2+e^x \left (3 x^3+\left (-2 e^4 x^3-2 x^3\right ) \log (3)+3 x^2+\left (-2 e^4 x^2-2 x^2\right ) \log ^2(3)\right )+\left (2 e^4 x+2 x\right ) \log (3)+3 \left (1+\frac {1}{3} \left (1+e^4\right ) \log ^2(3)\right )}{\left (-e^x x^2 \log (3)+x+\log (3)\right ) \left (\left (1+e^4\right ) \left (-e^x\right ) x^3 \log (3)+\left (1+e^4\right ) x^2+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-x \left (1-\left (1+e^4\right ) \log (3)\right )-3 \left (1+\frac {\log (3)}{3}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2+x (1+\log (3))+\log (9)}{x \left (-e^x x^2 \log (3)+x+\log (3)\right )}+\frac {\left (1+e^4\right )^2 x^4 \log (3)-\left (1+e^4\right ) x^3 \left (3-\log ^2(3)+\log (3)-e^4 \log (3) (1+\log (3))\right )+x^2 \left (3+2 e^8 \log ^2(3)-7 \log (3)-2 e^4 (4-\log (3)) \log (3)\right )+x (3+\log (3)) \left (4-\log (27)-e^4 \log (81)\right )+2 (3+\log (3))^2}{x \left (-\left (\left (1+e^4\right ) x \log (3)\right )+3+\log (3)\right ) \left (\left (1+e^4\right ) \left (-e^x\right ) x^3 \log (3)+\left (1+e^4\right ) x^2+3 e^x x^2 \left (1+\frac {\log (3)}{3}\right )-x \left (1-\left (1+e^4\right ) \log (3)\right )-3 \left (1+\frac {\log (3)}{3}\right )\right )}+\frac {\left (-1-e^4\right ) \log (3)}{-\left (\left (1+e^4\right ) x \log (3)\right )+3+\log (3)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -(1+\log (3)) \int \frac {1}{e^x \log (3) x^2-x-\log (3)}dx-\log (9) \int \frac {1}{x \left (e^x \log (3) x^2-x-\log (3)\right )}dx-\int \frac {x}{e^x \log (3) x^2-x-\log (3)}dx-\frac {\left (3+\log ^2(3)+2 e^4 \log ^2(3)-4 \log (3)\right ) \int \frac {1}{-e^x \left (1+e^4\right ) \log (3) x^3+3 e^x \left (1+\frac {\log (3)}{3}\right ) x^2+\left (1+e^4\right ) x^2-\left (1-\left (1+e^4\right ) \log (3)\right ) x-3 \left (1+\frac {\log (3)}{3}\right )}dx}{\log (3)}+\frac {(3+\log (3)) \left (3+3 \log ^2(3)-\log (3) \log (27)\right ) \int \frac {1}{\left (-\left (\left (1+e^4\right ) \log (3) x\right )+\log (3)+3\right ) \left (-e^x \left (1+e^4\right ) \log (3) x^3+3 e^x \left (1+\frac {\log (3)}{3}\right ) x^2+\left (1+e^4\right ) x^2-\left (1-\left (1+e^4\right ) \log (3)\right ) x-3 \left (1+\frac {\log (3)}{3}\right )\right )}dx}{\log (3)}+2 (3+\log (3)) \int \frac {1}{x \left (-e^x \left (1+e^4\right ) \log (3) x^3+3 e^x \left (1+\frac {\log (3)}{3}\right ) x^2+\left (1+e^4\right ) x^2-\left (1-\left (1+e^4\right ) \log (3)\right ) x-3 \left (1+\frac {\log (3)}{3}\right )\right )}dx-\left (\log (3)+e^4 (1+\log (3))\right ) \int \frac {x}{-e^x \left (1+e^4\right ) \log (3) x^3+3 e^x \left (1+\frac {\log (3)}{3}\right ) x^2+\left (1+e^4\right ) x^2-\left (1-\left (1+e^4\right ) \log (3)\right ) x-3 \left (1+\frac {\log (3)}{3}\right )}dx-\left (1+e^4\right ) \int \frac {x^2}{-e^x \left (1+e^4\right ) \log (3) x^3+3 e^x \left (1+\frac {\log (3)}{3}\right ) x^2+\left (1+e^4\right ) x^2-\left (1-\left (1+e^4\right ) \log (3)\right ) x-3 \left (1+\frac {\log (3)}{3}\right )}dx+\log \left (-\left (\left (1+e^4\right ) x \log (3)\right )+3+\log (3)\right )\) |
Input:
Int[(3 + x^2 + E^4*x^2 + (2*x + 2*E^4*x)*Log[3] + (1 + E^4)*Log[3]^2 + E^( 2*x)*(x^4 + E^4*x^4)*Log[3]^2 + E^x*(3*x^2 + 3*x^3 + (-2*x^3 - 2*E^4*x^3)* Log[3] + (-2*x^2 - 2*E^4*x^2)*Log[3]^2))/(-3*x - x^2 + x^3 + E^4*x^3 + (-3 - 2*x + 2*x^2 + 2*E^4*x^2)*Log[3] + (-1 + x + E^4*x)*Log[3]^2 + E^x*(3*x^ 3 + (6*x^2 + 2*x^3 - 2*x^4 - 2*E^4*x^4)*Log[3] + (2*x^2 - 2*x^3 - 2*E^4*x^ 3)*Log[3]^2) + E^(2*x)*(-3*x^4*Log[3] + (-x^4 + x^5 + E^4*x^5)*Log[3]^2)), x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs. \(2(33)=66\).
Time = 1.75 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.43
| method | result | size |
| norman | \(-\ln \left ({\mathrm e}^{x} \ln \left (3\right ) x^{2}-\ln \left (3\right )-x \right )+\ln \left ({\mathrm e}^{x} {\mathrm e}^{4} \ln \left (3\right ) x^{3}+{\mathrm e}^{x} \ln \left (3\right ) x^{3}-{\mathrm e}^{x} \ln \left (3\right ) x^{2}-{\mathrm e}^{4} \ln \left (3\right ) x -x^{2} {\mathrm e}^{4}-3 \,{\mathrm e}^{x} x^{2}-x \ln \left (3\right )-x^{2}+\ln \left (3\right )+x +3\right )\) | \(85\) |
| risch | \(\ln \left (\left (\ln \left (3\right ) {\mathrm e}^{4}+\ln \left (3\right )\right ) x -\ln \left (3\right )-3\right )+\ln \left ({\mathrm e}^{x}-\frac {{\mathrm e}^{4} \ln \left (3\right ) x +x^{2} {\mathrm e}^{4}+x \ln \left (3\right )+x^{2}-\ln \left (3\right )-x -3}{x^{2} \left ({\mathrm e}^{4} \ln \left (3\right ) x +x \ln \left (3\right )-\ln \left (3\right )-3\right )}\right )-\ln \left ({\mathrm e}^{x}-\frac {\ln \left (3\right )+x}{\ln \left (3\right ) x^{2}}\right )\) | \(93\) |
| parallelrisch | \(-\ln \left (\frac {{\mathrm e}^{x} \ln \left (3\right ) x^{2}-\ln \left (3\right )-x}{\ln \left (3\right )}\right )+\ln \left (\frac {{\mathrm e}^{x} {\mathrm e}^{4} \ln \left (3\right ) x^{3}+{\mathrm e}^{x} \ln \left (3\right ) x^{3}-{\mathrm e}^{x} \ln \left (3\right ) x^{2}-{\mathrm e}^{4} \ln \left (3\right ) x -x^{2} {\mathrm e}^{4}-3 \,{\mathrm e}^{x} x^{2}-x \ln \left (3\right )-x^{2}+\ln \left (3\right )+x +3}{\left ({\mathrm e}^{4}+1\right ) \ln \left (3\right )}\right )\) | \(101\) |
Input:
int(((x^4*exp(4)+x^4)*ln(3)^2*exp(x)^2+((-2*x^2*exp(4)-2*x^2)*ln(3)^2+(-2* x^3*exp(4)-2*x^3)*ln(3)+3*x^3+3*x^2)*exp(x)+(exp(4)+1)*ln(3)^2+(2*x*exp(4) +2*x)*ln(3)+x^2*exp(4)+x^2+3)/(((x^5*exp(4)+x^5-x^4)*ln(3)^2-3*x^4*ln(3))* exp(x)^2+((-2*x^3*exp(4)-2*x^3+2*x^2)*ln(3)^2+(-2*x^4*exp(4)-2*x^4+2*x^3+6 *x^2)*ln(3)+3*x^3)*exp(x)+(x*exp(4)+x-1)*ln(3)^2+(2*x^2*exp(4)+2*x^2-2*x-3 )*ln(3)+x^3*exp(4)+x^3-x^2-3*x),x,method=_RETURNVERBOSE)
Output:
-ln(exp(x)*ln(3)*x^2-ln(3)-x)+ln(exp(x)*exp(4)*ln(3)*x^3+exp(x)*ln(3)*x^3- exp(x)*ln(3)*x^2-exp(4)*ln(3)*x-x^2*exp(4)-3*exp(x)*x^2-x*ln(3)-x^2+ln(3)+ x+3)
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (34) = 68\).
Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.37 \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\log \left ({\left (x e^{4} + x - 1\right )} \log \left (3\right ) - 3\right ) + \log \left (\frac {x^{2} e^{4} + x^{2} + {\left (3 \, x^{2} - {\left (x^{3} e^{4} + x^{3} - x^{2}\right )} \log \left (3\right )\right )} e^{x} + {\left (x e^{4} + x - 1\right )} \log \left (3\right ) - x - 3}{3 \, x^{2} - {\left (x^{3} e^{4} + x^{3} - x^{2}\right )} \log \left (3\right )}\right ) - \log \left (\frac {x^{2} e^{x} \log \left (3\right ) - x - \log \left (3\right )}{x^{2}}\right ) \] Input:
integrate(((x^4*exp(4)+x^4)*log(3)^2*exp(x)^2+((-2*x^2*exp(4)-2*x^2)*log(3 )^2+(-2*x^3*exp(4)-2*x^3)*log(3)+3*x^3+3*x^2)*exp(x)+(exp(4)+1)*log(3)^2+( 2*x*exp(4)+2*x)*log(3)+x^2*exp(4)+x^2+3)/(((x^5*exp(4)+x^5-x^4)*log(3)^2-3 *x^4*log(3))*exp(x)^2+((-2*x^3*exp(4)-2*x^3+2*x^2)*log(3)^2+(-2*x^4*exp(4) -2*x^4+2*x^3+6*x^2)*log(3)+3*x^3)*exp(x)+(x*exp(4)+x-1)*log(3)^2+(2*x^2*ex p(4)+2*x^2-2*x-3)*log(3)+x^3*exp(4)+x^3-x^2-3*x),x, algorithm="fricas")
Output:
log((x*e^4 + x - 1)*log(3) - 3) + log((x^2*e^4 + x^2 + (3*x^2 - (x^3*e^4 + x^3 - x^2)*log(3))*e^x + (x*e^4 + x - 1)*log(3) - x - 3)/(3*x^2 - (x^3*e^ 4 + x^3 - x^2)*log(3))) - log((x^2*e^x*log(3) - x - log(3))/x^2)
Exception generated. \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate(((x**4*exp(4)+x**4)*ln(3)**2*exp(x)**2+((-2*x**2*exp(4)-2*x**2)* ln(3)**2+(-2*x**3*exp(4)-2*x**3)*ln(3)+3*x**3+3*x**2)*exp(x)+(exp(4)+1)*ln (3)**2+(2*x*exp(4)+2*x)*ln(3)+x**2*exp(4)+x**2+3)/(((x**5*exp(4)+x**5-x**4 )*ln(3)**2-3*x**4*ln(3))*exp(x)**2+((-2*x**3*exp(4)-2*x**3+2*x**2)*ln(3)** 2+(-2*x**4*exp(4)-2*x**4+2*x**3+6*x**2)*ln(3)+3*x**3)*exp(x)+(x*exp(4)+x-1 )*ln(3)**2+(2*x**2*exp(4)+2*x**2-2*x-3)*ln(3)+x**3*exp(4)+x**3-x**2-3*x),x )
Output:
Exception raised: PolynomialError >> 1/(x**5*log(3)**3 + 2*x**5*exp(4)*log (3)**3 + x**5*exp(8)*log(3)**3 - 6*x**4*exp(4)*log(3)**2 - 2*x**4*exp(4)*l og(3)**3 - 6*x**4*log(3)**2 - 2*x**4*log(3)**3 + x**3*log(3)**3 + 6*x**3*l og(3)**2 + 9*x*
Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (34) = 68\).
Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.51 \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\log \left ({\left (e^{4} \log \left (3\right ) + \log \left (3\right )\right )} x - \log \left (3\right ) - 3\right ) + \log \left (-\frac {x^{2} {\left (e^{4} + 1\right )} + {\left (e^{4} \log \left (3\right ) + \log \left (3\right ) - 1\right )} x - {\left ({\left (e^{4} \log \left (3\right ) + \log \left (3\right )\right )} x^{3} - x^{2} {\left (\log \left (3\right ) + 3\right )}\right )} e^{x} - \log \left (3\right ) - 3}{{\left (e^{4} \log \left (3\right ) + \log \left (3\right )\right )} x^{3} - x^{2} {\left (\log \left (3\right ) + 3\right )}}\right ) - \log \left (\frac {x^{2} e^{x} \log \left (3\right ) - x - \log \left (3\right )}{x^{2} \log \left (3\right )}\right ) \] Input:
integrate(((x^4*exp(4)+x^4)*log(3)^2*exp(x)^2+((-2*x^2*exp(4)-2*x^2)*log(3 )^2+(-2*x^3*exp(4)-2*x^3)*log(3)+3*x^3+3*x^2)*exp(x)+(exp(4)+1)*log(3)^2+( 2*x*exp(4)+2*x)*log(3)+x^2*exp(4)+x^2+3)/(((x^5*exp(4)+x^5-x^4)*log(3)^2-3 *x^4*log(3))*exp(x)^2+((-2*x^3*exp(4)-2*x^3+2*x^2)*log(3)^2+(-2*x^4*exp(4) -2*x^4+2*x^3+6*x^2)*log(3)+3*x^3)*exp(x)+(x*exp(4)+x-1)*log(3)^2+(2*x^2*ex p(4)+2*x^2-2*x-3)*log(3)+x^3*exp(4)+x^3-x^2-3*x),x, algorithm="maxima")
Output:
log((e^4*log(3) + log(3))*x - log(3) - 3) + log(-(x^2*(e^4 + 1) + (e^4*log (3) + log(3) - 1)*x - ((e^4*log(3) + log(3))*x^3 - x^2*(log(3) + 3))*e^x - log(3) - 3)/((e^4*log(3) + log(3))*x^3 - x^2*(log(3) + 3))) - log((x^2*e^ x*log(3) - x - log(3))/(x^2*log(3)))
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (34) = 68\).
Time = 1.71 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.40 \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\log \left (x^{3} e^{\left (x + 4\right )} \log \left (3\right ) + x^{3} e^{x} \log \left (3\right ) - x^{2} e^{x} \log \left (3\right ) - x^{2} e^{4} - 3 \, x^{2} e^{x} - x e^{4} \log \left (3\right ) - x^{2} - x \log \left (3\right ) + x + \log \left (3\right ) + 3\right ) - \log \left (x^{2} e^{x} \log \left (3\right ) - x - \log \left (3\right )\right ) \] Input:
integrate(((x^4*exp(4)+x^4)*log(3)^2*exp(x)^2+((-2*x^2*exp(4)-2*x^2)*log(3 )^2+(-2*x^3*exp(4)-2*x^3)*log(3)+3*x^3+3*x^2)*exp(x)+(exp(4)+1)*log(3)^2+( 2*x*exp(4)+2*x)*log(3)+x^2*exp(4)+x^2+3)/(((x^5*exp(4)+x^5-x^4)*log(3)^2-3 *x^4*log(3))*exp(x)^2+((-2*x^3*exp(4)-2*x^3+2*x^2)*log(3)^2+(-2*x^4*exp(4) -2*x^4+2*x^3+6*x^2)*log(3)+3*x^3)*exp(x)+(x*exp(4)+x-1)*log(3)^2+(2*x^2*ex p(4)+2*x^2-2*x-3)*log(3)+x^3*exp(4)+x^3-x^2-3*x),x, algorithm="giac")
Output:
log(x^3*e^(x + 4)*log(3) + x^3*e^x*log(3) - x^2*e^x*log(3) - x^2*e^4 - 3*x ^2*e^x - x*e^4*log(3) - x^2 - x*log(3) + x + log(3) + 3) - log(x^2*e^x*log (3) - x - log(3))
Timed out. \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=-\int \frac {\ln \left (3\right )\,\left (2\,x+2\,x\,{\mathrm {e}}^4\right )-{\mathrm {e}}^x\,\left ({\ln \left (3\right )}^2\,\left (2\,x^2\,{\mathrm {e}}^4+2\,x^2\right )-3\,x^2-3\,x^3+\ln \left (3\right )\,\left (2\,x^3\,{\mathrm {e}}^4+2\,x^3\right )\right )+x^2\,{\mathrm {e}}^4+x^2+{\ln \left (3\right )}^2\,\left ({\mathrm {e}}^4+1\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (x^4\,{\mathrm {e}}^4+x^4\right )+3}{3\,x-{\ln \left (3\right )}^2\,\left (x+x\,{\mathrm {e}}^4-1\right )-x^3\,{\mathrm {e}}^4+{\mathrm {e}}^x\,\left (\ln \left (3\right )\,\left (2\,x^4\,{\mathrm {e}}^4-6\,x^2-2\,x^3+2\,x^4\right )+{\ln \left (3\right )}^2\,\left (2\,x^3\,{\mathrm {e}}^4-2\,x^2+2\,x^3\right )-3\,x^3\right )-{\mathrm {e}}^{2\,x}\,\left ({\ln \left (3\right )}^2\,\left (x^5\,{\mathrm {e}}^4-x^4+x^5\right )-3\,x^4\,\ln \left (3\right )\right )+\ln \left (3\right )\,\left (2\,x-2\,x^2\,{\mathrm {e}}^4-2\,x^2+3\right )+x^2-x^3} \,d x \] Input:
int(-(log(3)*(2*x + 2*x*exp(4)) - exp(x)*(log(3)^2*(2*x^2*exp(4) + 2*x^2) - 3*x^2 - 3*x^3 + log(3)*(2*x^3*exp(4) + 2*x^3)) + x^2*exp(4) + x^2 + log( 3)^2*(exp(4) + 1) + exp(2*x)*log(3)^2*(x^4*exp(4) + x^4) + 3)/(3*x - log(3 )^2*(x + x*exp(4) - 1) - x^3*exp(4) + exp(x)*(log(3)*(2*x^4*exp(4) - 6*x^2 - 2*x^3 + 2*x^4) + log(3)^2*(2*x^3*exp(4) - 2*x^2 + 2*x^3) - 3*x^3) - exp (2*x)*(log(3)^2*(x^5*exp(4) - x^4 + x^5) - 3*x^4*log(3)) + log(3)*(2*x - 2 *x^2*exp(4) - 2*x^2 + 3) + x^2 - x^3),x)
Output:
-int((log(3)*(2*x + 2*x*exp(4)) - exp(x)*(log(3)^2*(2*x^2*exp(4) + 2*x^2) - 3*x^2 - 3*x^3 + log(3)*(2*x^3*exp(4) + 2*x^3)) + x^2*exp(4) + x^2 + log( 3)^2*(exp(4) + 1) + exp(2*x)*log(3)^2*(x^4*exp(4) + x^4) + 3)/(3*x - log(3 )^2*(x + x*exp(4) - 1) - x^3*exp(4) + exp(x)*(log(3)*(2*x^4*exp(4) - 6*x^2 - 2*x^3 + 2*x^4) + log(3)^2*(2*x^3*exp(4) - 2*x^2 + 2*x^3) - 3*x^3) - exp (2*x)*(log(3)^2*(x^5*exp(4) - x^4 + x^5) - 3*x^4*log(3)) + log(3)*(2*x - 2 *x^2*exp(4) - 2*x^2 + 3) + x^2 - x^3), x)
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.63 \[ \int \frac {3+x^2+e^4 x^2+\left (2 x+2 e^4 x\right ) \log (3)+\left (1+e^4\right ) \log ^2(3)+e^{2 x} \left (x^4+e^4 x^4\right ) \log ^2(3)+e^x \left (3 x^2+3 x^3+\left (-2 x^3-2 e^4 x^3\right ) \log (3)+\left (-2 x^2-2 e^4 x^2\right ) \log ^2(3)\right )}{-3 x-x^2+x^3+e^4 x^3+\left (-3-2 x+2 x^2+2 e^4 x^2\right ) \log (3)+\left (-1+x+e^4 x\right ) \log ^2(3)+e^x \left (3 x^3+\left (6 x^2+2 x^3-2 x^4-2 e^4 x^4\right ) \log (3)+\left (2 x^2-2 x^3-2 e^4 x^3\right ) \log ^2(3)\right )+e^{2 x} \left (-3 x^4 \log (3)+\left (-x^4+x^5+e^4 x^5\right ) \log ^2(3)\right )} \, dx=\mathrm {log}\left (e^{x} \mathrm {log}\left (3\right ) e^{4} x^{3}+e^{x} \mathrm {log}\left (3\right ) x^{3}-e^{x} \mathrm {log}\left (3\right ) x^{2}-3 e^{x} x^{2}-\mathrm {log}\left (3\right ) e^{4} x -\mathrm {log}\left (3\right ) x +\mathrm {log}\left (3\right )-e^{4} x^{2}-x^{2}+x +3\right )-\mathrm {log}\left (e^{x} \mathrm {log}\left (3\right ) x^{2}-\mathrm {log}\left (3\right )-x \right ) \] Input:
int(((x^4*exp(4)+x^4)*log(3)^2*exp(x)^2+((-2*x^2*exp(4)-2*x^2)*log(3)^2+(- 2*x^3*exp(4)-2*x^3)*log(3)+3*x^3+3*x^2)*exp(x)+(exp(4)+1)*log(3)^2+(2*x*ex p(4)+2*x)*log(3)+x^2*exp(4)+x^2+3)/(((x^5*exp(4)+x^5-x^4)*log(3)^2-3*x^4*l og(3))*exp(x)^2+((-2*x^3*exp(4)-2*x^3+2*x^2)*log(3)^2+(-2*x^4*exp(4)-2*x^4 +2*x^3+6*x^2)*log(3)+3*x^3)*exp(x)+(x*exp(4)+x-1)*log(3)^2+(2*x^2*exp(4)+2 *x^2-2*x-3)*log(3)+x^3*exp(4)+x^3-x^2-3*x),x)
Output:
log(e**x*log(3)*e**4*x**3 + e**x*log(3)*x**3 - e**x*log(3)*x**2 - 3*e**x*x **2 - log(3)*e**4*x - log(3)*x + log(3) - e**4*x**2 - x**2 + x + 3) - log( e**x*log(3)*x**2 - log(3) - x)