Integrand size = 174, antiderivative size = 28 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=\left (e^{2 x-2 \left (-1-e^x\right ) x \left (x+\log ^2(x)\right )}-x\right )^2 \]
Time = 8.94 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=\left (e^{2 x \left (1+x+e^x x+\left (1+e^x\right ) \log ^2(x)\right )}-x\right )^2 \]
Integrate[2*x + E^(4*x + 4*x^2 + 4*E^x*x^2 + 4*(x + E^x*x)*Log[x]^2)*(4 + 8*x + E^x*(8*x + 4*x^2) + (8 + 8*E^x)*Log[x] + (4 + E^x*(4 + 4*x))*Log[x]^ 2) + E^(2*x + 2*x^2 + 2*E^x*x^2 + 2*(x + E^x*x)*Log[x]^2)*(-2 - 4*x - 8*x^ 2 + E^x*(-8*x^2 - 4*x^3) + (-8*x - 8*E^x*x)*Log[x] + (-4*x + E^x*(-4*x - 4 *x^2))*Log[x]^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.
| \(\displaystyle \int \left (\left (e^x \left (4 x^2+8 x\right )+8 x+\left (e^x (4 x+4)+4\right ) \log ^2(x)+\left (8 e^x+8\right ) \log (x)+4\right ) \exp \left (4 e^x x^2+4 x^2+4 x+4 \left (e^x x+x\right ) \log ^2(x)\right )+\left (-8 x^2+\left (e^x \left (-4 x^2-4 x\right )-4 x\right ) \log ^2(x)+e^x \left (-4 x^3-8 x^2\right )-4 x+\left (-8 e^x x-8 x\right ) \log (x)-2\right ) \exp \left (2 e^x x^2+2 x^2+2 x+2 \left (e^x x+x\right ) \log ^2(x)\right )+2 x\right ) \, dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \int \exp \left (4 \left (e^x \log ^2(x)+\log ^2(x)+e^x x+x+1\right ) x+x\right ) x^2dx+8 \int \exp \left (4 \left (e^x \log ^2(x)+\log ^2(x)+e^x x+x+1\right ) x+x\right ) xdx+8 \int \exp \left (4 \left (e^x \log ^2(x)+\log ^2(x)+e^x x+x+1\right ) x+x\right ) \log (x)dx+4 \int \exp \left (4 \left (e^x \log ^2(x)+\log ^2(x)+e^x x+x+1\right ) x+x\right ) \log ^2(x)dx+4 \int \exp \left (4 \left (e^x \log ^2(x)+\log ^2(x)+e^x x+x+1\right ) x+x\right ) x \log ^2(x)dx+4 \int e^{4 x \left (e^x \log ^2(x)+\log ^2(x)+e^x x+x+1\right )}dx+8 \int e^{4 x \left (e^x \log ^2(x)+\log ^2(x)+e^x x+x+1\right )} xdx+8 \int e^{4 x \left (e^x \log ^2(x)+\log ^2(x)+e^x x+x+1\right )} \log (x)dx+4 \int e^{4 x \left (e^x \log ^2(x)+\log ^2(x)+e^x x+x+1\right )} \log ^2(x)dx-\frac {2 \left (2 x^2+\left (e^x \left (x^2+x\right )+x\right ) \log ^2(x)+e^x \left (x^3+2 x^2\right )+x+2 \left (e^x x+x\right ) \log (x)\right ) \exp \left (2 e^x x^2+2 x^2+2 x+2 \left (e^x x+x\right ) \log ^2(x)\right )}{e^x x^2+2 e^x x+2 x+\left (e^x x+e^x+1\right ) \log ^2(x)+\frac {2 \left (e^x x+x\right ) \log (x)}{x}+1}+x^2\) |
Int[2*x + E^(4*x + 4*x^2 + 4*E^x*x^2 + 4*(x + E^x*x)*Log[x]^2)*(4 + 8*x + E^x*(8*x + 4*x^2) + (8 + 8*E^x)*Log[x] + (4 + E^x*(4 + 4*x))*Log[x]^2) + E ^(2*x + 2*x^2 + 2*E^x*x^2 + 2*(x + E^x*x)*Log[x]^2)*(-2 - 4*x - 8*x^2 + E^ x*(-8*x^2 - 4*x^3) + (-8*x - 8*E^x*x)*Log[x] + (-4*x + E^x*(-4*x - 4*x^2)) *Log[x]^2),x]
3.15.81.3.1 Defintions of rubi rules used
Time = 0.65 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86
| method | result | size |
| risch | \({\mathrm e}^{4 x \left ({\mathrm e}^{x} \ln \left (x \right )^{2}+\ln \left (x \right )^{2}+{\mathrm e}^{x} x +x +1\right )}-2 \,{\mathrm e}^{2 x \left ({\mathrm e}^{x} \ln \left (x \right )^{2}+\ln \left (x \right )^{2}+{\mathrm e}^{x} x +x +1\right )} x +x^{2}\) | \(52\) |
| parallelrisch | \({\mathrm e}^{4 \left ({\mathrm e}^{x} x +x \right ) \ln \left (x \right )^{2}+4 \,{\mathrm e}^{x} x^{2}+4 x^{2}+4 x}-2 x \,{\mathrm e}^{2 \left ({\mathrm e}^{x} x +x \right ) \ln \left (x \right )^{2}+2 \,{\mathrm e}^{x} x^{2}+2 x^{2}+2 x}+x^{2}\) | \(58\) |
int((((4+4*x)*exp(x)+4)*ln(x)^2+(8*exp(x)+8)*ln(x)+(4*x^2+8*x)*exp(x)+8*x+ 4)*exp((exp(x)*x+x)*ln(x)^2+exp(x)*x^2+x^2+x)^4+(((-4*x^2-4*x)*exp(x)-4*x) *ln(x)^2+(-8*exp(x)*x-8*x)*ln(x)+(-4*x^3-8*x^2)*exp(x)-8*x^2-4*x-2)*exp((e xp(x)*x+x)*ln(x)^2+exp(x)*x^2+x^2+x)^2+2*x,x,method=_RETURNVERBOSE)
exp(4*x*(exp(x)*ln(x)^2+ln(x)^2+exp(x)*x+x+1))-2*exp(2*x*(exp(x)*ln(x)^2+l n(x)^2+exp(x)*x+x+1))*x+x^2
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=x^{2} - 2 \, x e^{\left (2 \, x^{2} e^{x} + 2 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + 2 \, x^{2} + 2 \, x\right )} + e^{\left (4 \, x^{2} e^{x} + 4 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + 4 \, x^{2} + 4 \, x\right )} \]
integrate((((4+4*x)*exp(x)+4)*log(x)^2+(8*exp(x)+8)*log(x)+(4*x^2+8*x)*exp (x)+8*x+4)*exp((exp(x)*x+x)*log(x)^2+exp(x)*x^2+x^2+x)^4+(((-4*x^2-4*x)*ex p(x)-4*x)*log(x)^2+(-8*exp(x)*x-8*x)*log(x)+(-4*x^3-8*x^2)*exp(x)-8*x^2-4* x-2)*exp((exp(x)*x+x)*log(x)^2+exp(x)*x^2+x^2+x)^2+2*x,x, algorithm=\
x^2 - 2*x*e^(2*x^2*e^x + 2*(x*e^x + x)*log(x)^2 + 2*x^2 + 2*x) + e^(4*x^2* e^x + 4*(x*e^x + x)*log(x)^2 + 4*x^2 + 4*x)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).
Time = 1.96 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.50 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=x^{2} - 2 x e^{2 x^{2} e^{x} + 2 x^{2} + 2 x + 2 \left (x e^{x} + x\right ) \log {\left (x \right )}^{2}} + e^{4 x^{2} e^{x} + 4 x^{2} + 4 x + 4 \left (x e^{x} + x\right ) \log {\left (x \right )}^{2}} \]
integrate((((4+4*x)*exp(x)+4)*ln(x)**2+(8*exp(x)+8)*ln(x)+(4*x**2+8*x)*exp (x)+8*x+4)*exp((exp(x)*x+x)*ln(x)**2+exp(x)*x**2+x**2+x)**4+(((-4*x**2-4*x )*exp(x)-4*x)*ln(x)**2+(-8*exp(x)*x-8*x)*ln(x)+(-4*x**3-8*x**2)*exp(x)-8*x **2-4*x-2)*exp((exp(x)*x+x)*ln(x)**2+exp(x)*x**2+x**2+x)**2+2*x,x)
x**2 - 2*x*exp(2*x**2*exp(x) + 2*x**2 + 2*x + 2*(x*exp(x) + x)*log(x)**2) + exp(4*x**2*exp(x) + 4*x**2 + 4*x + 4*(x*exp(x) + x)*log(x)**2)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (24) = 48\).
Time = 0.41 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=x^{2} - 2 \, x e^{\left (2 \, x e^{x} \log \left (x\right )^{2} + 2 \, x^{2} e^{x} + 2 \, x \log \left (x\right )^{2} + 2 \, x^{2} + 2 \, x\right )} + e^{\left (4 \, x e^{x} \log \left (x\right )^{2} + 4 \, x^{2} e^{x} + 4 \, x \log \left (x\right )^{2} + 4 \, x^{2} + 4 \, x\right )} \]
integrate((((4+4*x)*exp(x)+4)*log(x)^2+(8*exp(x)+8)*log(x)+(4*x^2+8*x)*exp (x)+8*x+4)*exp((exp(x)*x+x)*log(x)^2+exp(x)*x^2+x^2+x)^4+(((-4*x^2-4*x)*ex p(x)-4*x)*log(x)^2+(-8*exp(x)*x-8*x)*log(x)+(-4*x^3-8*x^2)*exp(x)-8*x^2-4* x-2)*exp((exp(x)*x+x)*log(x)^2+exp(x)*x^2+x^2+x)^2+2*x,x, algorithm=\
x^2 - 2*x*e^(2*x*e^x*log(x)^2 + 2*x^2*e^x + 2*x*log(x)^2 + 2*x^2 + 2*x) + e^(4*x*e^x*log(x)^2 + 4*x^2*e^x + 4*x*log(x)^2 + 4*x^2 + 4*x)
\[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=\int { 4 \, {\left ({\left ({\left (x + 1\right )} e^{x} + 1\right )} \log \left (x\right )^{2} + {\left (x^{2} + 2 \, x\right )} e^{x} + 2 \, {\left (e^{x} + 1\right )} \log \left (x\right ) + 2 \, x + 1\right )} e^{\left (4 \, x^{2} e^{x} + 4 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + 4 \, x^{2} + 4 \, x\right )} - 2 \, {\left (2 \, {\left ({\left (x^{2} + x\right )} e^{x} + x\right )} \log \left (x\right )^{2} + 4 \, x^{2} + 2 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{x} + 4 \, {\left (x e^{x} + x\right )} \log \left (x\right ) + 2 \, x + 1\right )} e^{\left (2 \, x^{2} e^{x} + 2 \, {\left (x e^{x} + x\right )} \log \left (x\right )^{2} + 2 \, x^{2} + 2 \, x\right )} + 2 \, x \,d x } \]
integrate((((4+4*x)*exp(x)+4)*log(x)^2+(8*exp(x)+8)*log(x)+(4*x^2+8*x)*exp (x)+8*x+4)*exp((exp(x)*x+x)*log(x)^2+exp(x)*x^2+x^2+x)^4+(((-4*x^2-4*x)*ex p(x)-4*x)*log(x)^2+(-8*exp(x)*x-8*x)*log(x)+(-4*x^3-8*x^2)*exp(x)-8*x^2-4* x-2)*exp((exp(x)*x+x)*log(x)^2+exp(x)*x^2+x^2+x)^2+2*x,x, algorithm=\
integrate(4*(((x + 1)*e^x + 1)*log(x)^2 + (x^2 + 2*x)*e^x + 2*(e^x + 1)*lo g(x) + 2*x + 1)*e^(4*x^2*e^x + 4*(x*e^x + x)*log(x)^2 + 4*x^2 + 4*x) - 2*( 2*((x^2 + x)*e^x + x)*log(x)^2 + 4*x^2 + 2*(x^3 + 2*x^2)*e^x + 4*(x*e^x + x)*log(x) + 2*x + 1)*e^(2*x^2*e^x + 2*(x*e^x + x)*log(x)^2 + 2*x^2 + 2*x) + 2*x, x)
Time = 10.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \left (2 x+e^{4 x+4 x^2+4 e^x x^2+4 \left (x+e^x x\right ) \log ^2(x)} \left (4+8 x+e^x \left (8 x+4 x^2\right )+\left (8+8 e^x\right ) \log (x)+\left (4+e^x (4+4 x)\right ) \log ^2(x)\right )+e^{2 x+2 x^2+2 e^x x^2+2 \left (x+e^x x\right ) \log ^2(x)} \left (-2-4 x-8 x^2+e^x \left (-8 x^2-4 x^3\right )+\left (-8 x-8 e^x x\right ) \log (x)+\left (-4 x+e^x \left (-4 x-4 x^2\right )\right ) \log ^2(x)\right )\right ) \, dx=x^2+{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{4\,x\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{4\,x^2}-2\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{2\,x\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{2\,x^2} \]
int(2*x - exp(2*x + 2*x^2*exp(x) + 2*log(x)^2*(x + x*exp(x)) + 2*x^2)*(4*x + exp(x)*(8*x^2 + 4*x^3) + log(x)*(8*x + 8*x*exp(x)) + log(x)^2*(4*x + ex p(x)*(4*x + 4*x^2)) + 8*x^2 + 2) + exp(4*x + 4*x^2*exp(x) + 4*log(x)^2*(x + x*exp(x)) + 4*x^2)*(8*x + log(x)^2*(exp(x)*(4*x + 4) + 4) + exp(x)*(8*x + 4*x^2) + log(x)*(8*exp(x) + 8) + 4),x)