Integrand size = 251, antiderivative size = 31 \[ \int \frac {-16 e^{3+x^2}+e^{x^2} \left (-2 e^6+16 e^3 x^2\right ) \log \left (x^2\right )+\left (-4 e^3 x+e^6 (1+x)\right ) \log ^2\left (x^2\right )+\left (16 e^{x^2}+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )+\left (e^3 (-2-2 x)+4 x\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )}{-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )} \, dx=2+x \left (e^3-\log \left (\left (1+x-\frac {2 e^{x^2}}{\log \left (x^2\right )}\right )^2\right )\right )^2 \] Output:
2+x*(exp(3)-ln((x-2*exp(x^2)/ln(x^2)+1)^2))^2
\[ \int \frac {-16 e^{3+x^2}+e^{x^2} \left (-2 e^6+16 e^3 x^2\right ) \log \left (x^2\right )+\left (-4 e^3 x+e^6 (1+x)\right ) \log ^2\left (x^2\right )+\left (16 e^{x^2}+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )+\left (e^3 (-2-2 x)+4 x\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )}{-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )} \, dx=\int \frac {-16 e^{3+x^2}+e^{x^2} \left (-2 e^6+16 e^3 x^2\right ) \log \left (x^2\right )+\left (-4 e^3 x+e^6 (1+x)\right ) \log ^2\left (x^2\right )+\left (16 e^{x^2}+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )+\left (e^3 (-2-2 x)+4 x\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )}{-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )} \, dx \] Input:
Integrate[(-16*E^(3 + x^2) + E^x^2*(-2*E^6 + 16*E^3*x^2)*Log[x^2] + (-4*E^ 3*x + E^6*(1 + x))*Log[x^2]^2 + (16*E^x^2 + E^x^2*(4*E^3 - 16*x^2)*Log[x^2 ] + (E^3*(-2 - 2*x) + 4*x)*Log[x^2]^2)*Log[(4*E^(2*x^2) + E^x^2*(-4 - 4*x) *Log[x^2] + (1 + 2*x + x^2)*Log[x^2]^2)/Log[x^2]^2] + (-2*E^x^2*Log[x^2] + (1 + x)*Log[x^2]^2)*Log[(4*E^(2*x^2) + E^x^2*(-4 - 4*x)*Log[x^2] + (1 + 2 *x + x^2)*Log[x^2]^2)/Log[x^2]^2]^2)/(-2*E^x^2*Log[x^2] + (1 + x)*Log[x^2] ^2),x]
Output:
Integrate[(-16*E^(3 + x^2) + E^x^2*(-2*E^6 + 16*E^3*x^2)*Log[x^2] + (-4*E^ 3*x + E^6*(1 + x))*Log[x^2]^2 + (16*E^x^2 + E^x^2*(4*E^3 - 16*x^2)*Log[x^2 ] + (E^3*(-2 - 2*x) + 4*x)*Log[x^2]^2)*Log[(4*E^(2*x^2) + E^x^2*(-4 - 4*x) *Log[x^2] + (1 + 2*x + x^2)*Log[x^2]^2)/Log[x^2]^2] + (-2*E^x^2*Log[x^2] + (1 + x)*Log[x^2]^2)*Log[(4*E^(2*x^2) + E^x^2*(-4 - 4*x)*Log[x^2] + (1 + 2 *x + x^2)*Log[x^2]^2)/Log[x^2]^2]^2)/(-2*E^x^2*Log[x^2] + (1 + x)*Log[x^2] ^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.
| \(\displaystyle \int \frac {-16 e^{x^2+3}+\left (e^6 (x+1)-4 e^3 x\right ) \log ^2\left (x^2\right )+\left ((x+1) \log ^2\left (x^2\right )-2 e^{x^2} \log \left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+\left (x^2+2 x+1\right ) \log ^2\left (x^2\right )+e^{x^2} (-4 x-4) \log \left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (16 e^{x^2}+\left (e^3 (-2 x-2)+4 x\right ) \log ^2\left (x^2\right )+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+\left (x^2+2 x+1\right ) \log ^2\left (x^2\right )+e^{x^2} (-4 x-4) \log \left (x^2\right )}{\log ^2\left (x^2\right )}\right )+e^{x^2} \left (16 e^3 x^2-2 e^6\right ) \log \left (x^2\right )}{(x+1) \log ^2\left (x^2\right )-2 e^{x^2} \log \left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (e^3-\log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )\right ) \left (16 e^{x^2}+4 \left (1-\frac {e^3}{4}\right ) x \log ^2\left (x^2\right )+x \log ^2\left (x^2\right ) \log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )-e^3 \log ^2\left (x^2\right )+\log ^2\left (x^2\right ) \log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )-2 e^{x^2} \log \left (x^2\right ) \log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )-16 e^{x^2} x^2 \log \left (x^2\right )+2 e^{x^2+3} \log \left (x^2\right )\right )}{\log \left (x^2\right ) \left (2 e^{x^2}-x \log \left (x^2\right )-\log \left (x^2\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (e^3-\log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )\right ) \left (-\log \left (x^2\right ) \log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )-8 x^2 \log \left (x^2\right )+e^3 \log \left (x^2\right )+8\right )}{\log \left (x^2\right )}-\frac {4 \left (2 x^2 \log \left (x^2\right )-x \log \left (x^2\right )+2 x^3 \log \left (x^2\right )-2 x-2\right ) \left (e^3-\log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )\right )}{2 e^{x^2}-x \log \left (x^2\right )-\log \left (x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -8 \int \frac {\log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )}{\log \left (x^2\right )}dx-8 \int \frac {\log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )}{-x \log \left (x^2\right )-\log \left (x^2\right )+2 e^{x^2}}dx+8 \int \frac {x \log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )}{x \log \left (x^2\right )+\log \left (x^2\right )-2 e^{x^2}}dx+4 \int \frac {x \log \left (x^2\right ) \log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )}{x \log \left (x^2\right )+\log \left (x^2\right )-2 e^{x^2}}dx-8 \int \frac {x^2 \log \left (x^2\right ) \log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )}{x \log \left (x^2\right )+\log \left (x^2\right )-2 e^{x^2}}dx+\int \log ^2\left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )dx-\frac {32}{3} \int \frac {x^2}{x \log \left (x^2\right )+\log \left (x^2\right )-2 e^{x^2}}dx+\frac {32}{3} \int \frac {x^5 \log \left (x^2\right )}{x \log \left (x^2\right )+\log \left (x^2\right )-2 e^{x^2}}dx+\frac {32}{3} \int \frac {x^4 \log \left (x^2\right )}{x \log \left (x^2\right )+\log \left (x^2\right )-2 e^{x^2}}dx-8 \int \frac {x^3 \log \left (x^2\right ) \log \left (\frac {\left ((x+1) \log \left (x^2\right )-2 e^{x^2}\right )^2}{\log ^2\left (x^2\right )}\right )}{x \log \left (x^2\right )+\log \left (x^2\right )-2 e^{x^2}}dx-\frac {32}{3} \int \frac {x^3}{x \log \left (x^2\right )+\log \left (x^2\right )-2 e^{x^2}}dx-\frac {16}{3} \int \frac {x^3 \log \left (x^2\right )}{x \log \left (x^2\right )+\log \left (x^2\right )-2 e^{x^2}}dx+\frac {16 x^3 \operatorname {ExpIntegralEi}\left (\frac {3 \log \left (x^2\right )}{2}\right )}{3 \left (x^2\right )^{3/2}}-\frac {32 x^5}{15}-2 e^3 x \log \left (\frac {\left (2 e^{x^2}-(x+1) \log \left (x^2\right )\right )^2}{\log ^2\left (x^2\right )}\right )+\frac {8}{3} x^3 \log \left (\frac {\left (2 e^{x^2}-(x+1) \log \left (x^2\right )\right )^2}{\log ^2\left (x^2\right )}\right )+e^6 x\) |
Input:
Int[(-16*E^(3 + x^2) + E^x^2*(-2*E^6 + 16*E^3*x^2)*Log[x^2] + (-4*E^3*x + E^6*(1 + x))*Log[x^2]^2 + (16*E^x^2 + E^x^2*(4*E^3 - 16*x^2)*Log[x^2] + (E ^3*(-2 - 2*x) + 4*x)*Log[x^2]^2)*Log[(4*E^(2*x^2) + E^x^2*(-4 - 4*x)*Log[x ^2] + (1 + 2*x + x^2)*Log[x^2]^2)/Log[x^2]^2] + (-2*E^x^2*Log[x^2] + (1 + x)*Log[x^2]^2)*Log[(4*E^(2*x^2) + E^x^2*(-4 - 4*x)*Log[x^2] + (1 + 2*x + x ^2)*Log[x^2]^2)/Log[x^2]^2]^2)/(-2*E^x^2*Log[x^2] + (1 + x)*Log[x^2]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(29)=58\).
Time = 38.69 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52
| method | result | size |
| parallelrisch | \(x \,{\mathrm e}^{6}-2 \,{\mathrm e}^{3} x \ln \left (\frac {\left (x^{2}+2 x +1\right ) \ln \left (x^{2}\right )^{2}+\left (-4-4 x \right ) {\mathrm e}^{x^{2}} \ln \left (x^{2}\right )+4 \,{\mathrm e}^{2 x^{2}}}{\ln \left (x^{2}\right )^{2}}\right )+{\ln \left (\frac {\left (x^{2}+2 x +1\right ) \ln \left (x^{2}\right )^{2}+\left (-4-4 x \right ) {\mathrm e}^{x^{2}} \ln \left (x^{2}\right )+4 \,{\mathrm e}^{2 x^{2}}}{\ln \left (x^{2}\right )^{2}}\right )}^{2} x\) | \(109\) |
| risch | \(\text {Expression too large to display}\) | \(22794\) |
Input:
int((((1+x)*ln(x^2)^2-2*exp(x^2)*ln(x^2))*ln(((x^2+2*x+1)*ln(x^2)^2+(-4-4* x)*exp(x^2)*ln(x^2)+4*exp(x^2)^2)/ln(x^2)^2)^2+(((-2-2*x)*exp(3)+4*x)*ln(x ^2)^2+(4*exp(3)-16*x^2)*exp(x^2)*ln(x^2)+16*exp(x^2))*ln(((x^2+2*x+1)*ln(x ^2)^2+(-4-4*x)*exp(x^2)*ln(x^2)+4*exp(x^2)^2)/ln(x^2)^2)+((1+x)*exp(3)^2-4 *x*exp(3))*ln(x^2)^2+(-2*exp(3)^2+16*x^2*exp(3))*exp(x^2)*ln(x^2)-16*exp(x ^2)*exp(3))/((1+x)*ln(x^2)^2-2*exp(x^2)*ln(x^2)),x,method=_RETURNVERBOSE)
Output:
x*exp(3)^2-2*exp(3)*x*ln(((x^2+2*x+1)*ln(x^2)^2+(-4-4*x)*exp(x^2)*ln(x^2)+ 4*exp(x^2)^2)/ln(x^2)^2)+ln(((x^2+2*x+1)*ln(x^2)^2+(-4-4*x)*exp(x^2)*ln(x^ 2)+4*exp(x^2)^2)/ln(x^2)^2)^2*x
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (29) = 58\).
Time = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.87 \[ \int \frac {-16 e^{3+x^2}+e^{x^2} \left (-2 e^6+16 e^3 x^2\right ) \log \left (x^2\right )+\left (-4 e^3 x+e^6 (1+x)\right ) \log ^2\left (x^2\right )+\left (16 e^{x^2}+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )+\left (e^3 (-2-2 x)+4 x\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )}{-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )} \, dx=-2 \, x e^{3} \log \left (\frac {{\left ({\left (x^{2} + 2 \, x + 1\right )} e^{6} \log \left (x^{2}\right )^{2} - 4 \, {\left (x + 1\right )} e^{\left (x^{2} + 6\right )} \log \left (x^{2}\right ) + 4 \, e^{\left (2 \, x^{2} + 6\right )}\right )} e^{\left (-6\right )}}{\log \left (x^{2}\right )^{2}}\right ) + x \log \left (\frac {{\left ({\left (x^{2} + 2 \, x + 1\right )} e^{6} \log \left (x^{2}\right )^{2} - 4 \, {\left (x + 1\right )} e^{\left (x^{2} + 6\right )} \log \left (x^{2}\right ) + 4 \, e^{\left (2 \, x^{2} + 6\right )}\right )} e^{\left (-6\right )}}{\log \left (x^{2}\right )^{2}}\right )^{2} + x e^{6} \] Input:
integrate((((1+x)*log(x^2)^2-2*exp(x^2)*log(x^2))*log(((x^2+2*x+1)*log(x^2 )^2+(-4-4*x)*exp(x^2)*log(x^2)+4*exp(x^2)^2)/log(x^2)^2)^2+(((-2-2*x)*exp( 3)+4*x)*log(x^2)^2+(4*exp(3)-16*x^2)*exp(x^2)*log(x^2)+16*exp(x^2))*log((( x^2+2*x+1)*log(x^2)^2+(-4-4*x)*exp(x^2)*log(x^2)+4*exp(x^2)^2)/log(x^2)^2) +((1+x)*exp(3)^2-4*x*exp(3))*log(x^2)^2+(-2*exp(3)^2+16*x^2*exp(3))*exp(x^ 2)*log(x^2)-16*exp(x^2)*exp(3))/((1+x)*log(x^2)^2-2*exp(x^2)*log(x^2)),x, algorithm="fricas")
Output:
-2*x*e^3*log(((x^2 + 2*x + 1)*e^6*log(x^2)^2 - 4*(x + 1)*e^(x^2 + 6)*log(x ^2) + 4*e^(2*x^2 + 6))*e^(-6)/log(x^2)^2) + x*log(((x^2 + 2*x + 1)*e^6*log (x^2)^2 - 4*(x + 1)*e^(x^2 + 6)*log(x^2) + 4*e^(2*x^2 + 6))*e^(-6)/log(x^2 )^2)^2 + x*e^6
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (26) = 52\).
Time = 0.76 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.61 \[ \int \frac {-16 e^{3+x^2}+e^{x^2} \left (-2 e^6+16 e^3 x^2\right ) \log \left (x^2\right )+\left (-4 e^3 x+e^6 (1+x)\right ) \log ^2\left (x^2\right )+\left (16 e^{x^2}+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )+\left (e^3 (-2-2 x)+4 x\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )}{-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )} \, dx=x \log {\left (\frac {\left (- 4 x - 4\right ) e^{x^{2}} \log {\left (x^{2} \right )} + \left (x^{2} + 2 x + 1\right ) \log {\left (x^{2} \right )}^{2} + 4 e^{2 x^{2}}}{\log {\left (x^{2} \right )}^{2}} \right )}^{2} - 2 x e^{3} \log {\left (\frac {\left (- 4 x - 4\right ) e^{x^{2}} \log {\left (x^{2} \right )} + \left (x^{2} + 2 x + 1\right ) \log {\left (x^{2} \right )}^{2} + 4 e^{2 x^{2}}}{\log {\left (x^{2} \right )}^{2}} \right )} + x e^{6} \] Input:
integrate((((1+x)*ln(x**2)**2-2*exp(x**2)*ln(x**2))*ln(((x**2+2*x+1)*ln(x* *2)**2+(-4-4*x)*exp(x**2)*ln(x**2)+4*exp(x**2)**2)/ln(x**2)**2)**2+(((-2-2 *x)*exp(3)+4*x)*ln(x**2)**2+(4*exp(3)-16*x**2)*exp(x**2)*ln(x**2)+16*exp(x **2))*ln(((x**2+2*x+1)*ln(x**2)**2+(-4-4*x)*exp(x**2)*ln(x**2)+4*exp(x**2) **2)/ln(x**2)**2)+((1+x)*exp(3)**2-4*x*exp(3))*ln(x**2)**2+(-2*exp(3)**2+1 6*x**2*exp(3))*exp(x**2)*ln(x**2)-16*exp(x**2)*exp(3))/((1+x)*ln(x**2)**2- 2*exp(x**2)*ln(x**2)),x)
Output:
x*log(((-4*x - 4)*exp(x**2)*log(x**2) + (x**2 + 2*x + 1)*log(x**2)**2 + 4* exp(2*x**2))/log(x**2)**2)**2 - 2*x*exp(3)*log(((-4*x - 4)*exp(x**2)*log(x **2) + (x**2 + 2*x + 1)*log(x**2)**2 + 4*exp(2*x**2))/log(x**2)**2) + x*ex p(6)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {-16 e^{3+x^2}+e^{x^2} \left (-2 e^6+16 e^3 x^2\right ) \log \left (x^2\right )+\left (-4 e^3 x+e^6 (1+x)\right ) \log ^2\left (x^2\right )+\left (16 e^{x^2}+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )+\left (e^3 (-2-2 x)+4 x\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )}{-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )} \, dx=4 \, x \log \left (-{\left (x + 1\right )} \log \left (x\right ) + e^{\left (x^{2}\right )}\right )^{2} + 4 \, x e^{3} \log \left (\log \left (x\right )\right ) + 4 \, x \log \left (\log \left (x\right )\right )^{2} + x e^{6} - 4 \, {\left (x e^{3} + 2 \, x \log \left (\log \left (x\right )\right )\right )} \log \left (-{\left (x + 1\right )} \log \left (x\right ) + e^{\left (x^{2}\right )}\right ) \] Input:
integrate((((1+x)*log(x^2)^2-2*exp(x^2)*log(x^2))*log(((x^2+2*x+1)*log(x^2 )^2+(-4-4*x)*exp(x^2)*log(x^2)+4*exp(x^2)^2)/log(x^2)^2)^2+(((-2-2*x)*exp( 3)+4*x)*log(x^2)^2+(4*exp(3)-16*x^2)*exp(x^2)*log(x^2)+16*exp(x^2))*log((( x^2+2*x+1)*log(x^2)^2+(-4-4*x)*exp(x^2)*log(x^2)+4*exp(x^2)^2)/log(x^2)^2) +((1+x)*exp(3)^2-4*x*exp(3))*log(x^2)^2+(-2*exp(3)^2+16*x^2*exp(3))*exp(x^ 2)*log(x^2)-16*exp(x^2)*exp(3))/((1+x)*log(x^2)^2-2*exp(x^2)*log(x^2)),x, algorithm="maxima")
Output:
4*x*log(-(x + 1)*log(x) + e^(x^2))^2 + 4*x*e^3*log(log(x)) + 4*x*log(log(x ))^2 + x*e^6 - 4*(x*e^3 + 2*x*log(log(x)))*log(-(x + 1)*log(x) + e^(x^2))
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (29) = 58\).
Time = 3.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 6.94 \[ \int \frac {-16 e^{3+x^2}+e^{x^2} \left (-2 e^6+16 e^3 x^2\right ) \log \left (x^2\right )+\left (-4 e^3 x+e^6 (1+x)\right ) \log ^2\left (x^2\right )+\left (16 e^{x^2}+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )+\left (e^3 (-2-2 x)+4 x\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )}{-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )} \, dx=-2 \, x e^{3} \log \left (x^{2} \log \left (x^{2}\right )^{2} - 4 \, x e^{\left (x^{2}\right )} \log \left (x^{2}\right ) + 2 \, x \log \left (x^{2}\right )^{2} - 4 \, e^{\left (x^{2}\right )} \log \left (x^{2}\right ) + \log \left (x^{2}\right )^{2} + 4 \, e^{\left (2 \, x^{2}\right )}\right ) + x \log \left (x^{2} \log \left (x^{2}\right )^{2} - 4 \, x e^{\left (x^{2}\right )} \log \left (x^{2}\right ) + 2 \, x \log \left (x^{2}\right )^{2} - 4 \, e^{\left (x^{2}\right )} \log \left (x^{2}\right ) + \log \left (x^{2}\right )^{2} + 4 \, e^{\left (2 \, x^{2}\right )}\right )^{2} + 2 \, x e^{3} \log \left (\log \left (x^{2}\right )^{2}\right ) - 2 \, x \log \left (x^{2} \log \left (x^{2}\right )^{2} - 4 \, x e^{\left (x^{2}\right )} \log \left (x^{2}\right ) + 2 \, x \log \left (x^{2}\right )^{2} - 4 \, e^{\left (x^{2}\right )} \log \left (x^{2}\right ) + \log \left (x^{2}\right )^{2} + 4 \, e^{\left (2 \, x^{2}\right )}\right ) \log \left (\log \left (x^{2}\right )^{2}\right ) + x \log \left (\log \left (x^{2}\right )^{2}\right )^{2} + x e^{6} \] Input:
integrate((((1+x)*log(x^2)^2-2*exp(x^2)*log(x^2))*log(((x^2+2*x+1)*log(x^2 )^2+(-4-4*x)*exp(x^2)*log(x^2)+4*exp(x^2)^2)/log(x^2)^2)^2+(((-2-2*x)*exp( 3)+4*x)*log(x^2)^2+(4*exp(3)-16*x^2)*exp(x^2)*log(x^2)+16*exp(x^2))*log((( x^2+2*x+1)*log(x^2)^2+(-4-4*x)*exp(x^2)*log(x^2)+4*exp(x^2)^2)/log(x^2)^2) +((1+x)*exp(3)^2-4*x*exp(3))*log(x^2)^2+(-2*exp(3)^2+16*x^2*exp(3))*exp(x^ 2)*log(x^2)-16*exp(x^2)*exp(3))/((1+x)*log(x^2)^2-2*exp(x^2)*log(x^2)),x, algorithm="giac")
Output:
-2*x*e^3*log(x^2*log(x^2)^2 - 4*x*e^(x^2)*log(x^2) + 2*x*log(x^2)^2 - 4*e^ (x^2)*log(x^2) + log(x^2)^2 + 4*e^(2*x^2)) + x*log(x^2*log(x^2)^2 - 4*x*e^ (x^2)*log(x^2) + 2*x*log(x^2)^2 - 4*e^(x^2)*log(x^2) + log(x^2)^2 + 4*e^(2 *x^2))^2 + 2*x*e^3*log(log(x^2)^2) - 2*x*log(x^2*log(x^2)^2 - 4*x*e^(x^2)* log(x^2) + 2*x*log(x^2)^2 - 4*e^(x^2)*log(x^2) + log(x^2)^2 + 4*e^(2*x^2)) *log(log(x^2)^2) + x*log(log(x^2)^2)^2 + x*e^6
Time = 3.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {-16 e^{3+x^2}+e^{x^2} \left (-2 e^6+16 e^3 x^2\right ) \log \left (x^2\right )+\left (-4 e^3 x+e^6 (1+x)\right ) \log ^2\left (x^2\right )+\left (16 e^{x^2}+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )+\left (e^3 (-2-2 x)+4 x\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )}{-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )} \, dx=x\,{\left ({\mathrm {e}}^3-\ln \left (\frac {\left (x^2+2\,x+1\right )\,{\ln \left (x^2\right )}^2-{\mathrm {e}}^{x^2}\,\left (4\,x+4\right )\,\ln \left (x^2\right )+4\,{\mathrm {e}}^{2\,x^2}}{{\ln \left (x^2\right )}^2}\right )\right )}^2 \] Input:
int(-(16*exp(x^2)*exp(3) + log(x^2)^2*(4*x*exp(3) - exp(6)*(x + 1)) - log( (4*exp(2*x^2) + log(x^2)^2*(2*x + x^2 + 1) - log(x^2)*exp(x^2)*(4*x + 4))/ log(x^2)^2)*(16*exp(x^2) + log(x^2)^2*(4*x - exp(3)*(2*x + 2)) + log(x^2)* exp(x^2)*(4*exp(3) - 16*x^2)) - log((4*exp(2*x^2) + log(x^2)^2*(2*x + x^2 + 1) - log(x^2)*exp(x^2)*(4*x + 4))/log(x^2)^2)^2*(log(x^2)^2*(x + 1) - 2* log(x^2)*exp(x^2)) + log(x^2)*exp(x^2)*(2*exp(6) - 16*x^2*exp(3)))/(log(x^ 2)^2*(x + 1) - 2*log(x^2)*exp(x^2)),x)
Output:
x*(exp(3) - log((4*exp(2*x^2) + log(x^2)^2*(2*x + x^2 + 1) - log(x^2)*exp( x^2)*(4*x + 4))/log(x^2)^2))^2
Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.68 \[ \int \frac {-16 e^{3+x^2}+e^{x^2} \left (-2 e^6+16 e^3 x^2\right ) \log \left (x^2\right )+\left (-4 e^3 x+e^6 (1+x)\right ) \log ^2\left (x^2\right )+\left (16 e^{x^2}+e^{x^2} \left (4 e^3-16 x^2\right ) \log \left (x^2\right )+\left (e^3 (-2-2 x)+4 x\right ) \log ^2\left (x^2\right )\right ) \log \left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )+\left (-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )\right ) \log ^2\left (\frac {4 e^{2 x^2}+e^{x^2} (-4-4 x) \log \left (x^2\right )+\left (1+2 x+x^2\right ) \log ^2\left (x^2\right )}{\log ^2\left (x^2\right )}\right )}{-2 e^{x^2} \log \left (x^2\right )+(1+x) \log ^2\left (x^2\right )} \, dx=x \left (\mathrm {log}\left (\frac {4 e^{2 x^{2}}-4 e^{x^{2}} \mathrm {log}\left (x^{2}\right ) x -4 e^{x^{2}} \mathrm {log}\left (x^{2}\right )+\mathrm {log}\left (x^{2}\right )^{2} x^{2}+2 \mathrm {log}\left (x^{2}\right )^{2} x +\mathrm {log}\left (x^{2}\right )^{2}}{\mathrm {log}\left (x^{2}\right )^{2}}\right )^{2}-2 \,\mathrm {log}\left (\frac {4 e^{2 x^{2}}-4 e^{x^{2}} \mathrm {log}\left (x^{2}\right ) x -4 e^{x^{2}} \mathrm {log}\left (x^{2}\right )+\mathrm {log}\left (x^{2}\right )^{2} x^{2}+2 \mathrm {log}\left (x^{2}\right )^{2} x +\mathrm {log}\left (x^{2}\right )^{2}}{\mathrm {log}\left (x^{2}\right )^{2}}\right ) e^{3}+e^{6}\right ) \] Input:
int((((1+x)*log(x^2)^2-2*exp(x^2)*log(x^2))*log(((x^2+2*x+1)*log(x^2)^2+(- 4-4*x)*exp(x^2)*log(x^2)+4*exp(x^2)^2)/log(x^2)^2)^2+(((-2-2*x)*exp(3)+4*x )*log(x^2)^2+(4*exp(3)-16*x^2)*exp(x^2)*log(x^2)+16*exp(x^2))*log(((x^2+2* x+1)*log(x^2)^2+(-4-4*x)*exp(x^2)*log(x^2)+4*exp(x^2)^2)/log(x^2)^2)+((1+x )*exp(3)^2-4*x*exp(3))*log(x^2)^2+(-2*exp(3)^2+16*x^2*exp(3))*exp(x^2)*log (x^2)-16*exp(x^2)*exp(3))/((1+x)*log(x^2)^2-2*exp(x^2)*log(x^2)),x)
Output:
x*(log((4*e**(2*x**2) - 4*e**(x**2)*log(x**2)*x - 4*e**(x**2)*log(x**2) + log(x**2)**2*x**2 + 2*log(x**2)**2*x + log(x**2)**2)/log(x**2)**2)**2 - 2* log((4*e**(2*x**2) - 4*e**(x**2)*log(x**2)*x - 4*e**(x**2)*log(x**2) + log (x**2)**2*x**2 + 2*log(x**2)**2*x + log(x**2)**2)/log(x**2)**2)*e**3 + e** 6)