Integrand size = 105, antiderivative size = 27 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {1}{3} e^{\frac {(2+x)^2 (x+\log (x))}{x^2}} (x-\log (x))^2 \]
Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {1}{3} e^{4+\frac {4}{x}+x} x^{\frac {(2+x)^2}{x^2}} (x-\log (x))^2 \]
Integrate[(E^((4*x + 4*x^2 + x^3 + (4 + 4*x + x^2)*Log[x])/x^2)*(4*x^2 - 2 *x^3 + 3*x^4 + x^5 + (-8*x - 6*x^2 - 8*x^3 - 2*x^4)*Log[x] + (4 + 16*x + 9 *x^2 + x^3)*Log[x]^2 + (-8 - 4*x)*Log[x]^3))/(3*x^3),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.
\(\displaystyle \int \frac {e^{\frac {x^3+4 x^2+\left (x^2+4 x+4\right ) \log (x)+4 x}{x^2}} \left (x^5+3 x^4-2 x^3+4 x^2+\left (x^3+9 x^2+16 x+4\right ) \log ^2(x)+\left (-2 x^4-8 x^3-6 x^2-8 x\right ) \log (x)+(-4 x-8) \log ^3(x)\right )}{3 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int e^{\frac {x^3+4 x^2+4 x}{x^2}} x^{\frac {x^2+4 x+4}{x^2}-3} \left (x^5+3 x^4-2 x^3+4 x^2-4 (x+2) \log ^3(x)+\left (x^3+9 x^2+16 x+4\right ) \log ^2(x)-2 \left (x^4+4 x^3+3 x^2+4 x\right ) \log (x)\right )dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {1}{3} \int e^{x+4+\frac {4}{x}} x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \left (x^5+3 x^4-2 x^3+4 x^2-4 (x+2) \log ^3(x)+\left (x^3+9 x^2+16 x+4\right ) \log ^2(x)-2 \left (x^4+4 x^3+3 x^2+4 x\right ) \log (x)\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{3} \int \left (-4 e^{x+4+\frac {4}{x}} (x+2) \log ^3(x) x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}}+e^{x+4+\frac {4}{x}} \left (x^3+9 x^2+16 x+4\right ) \log ^2(x) x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}}-2 e^{x+4+\frac {4}{x}} \left (x^3+4 x^2+3 x+4\right ) \log (x) x^{1-\frac {2 \left (x^2-2 x-2\right )}{x^2}}+4 e^{x+4+\frac {4}{x}} x^{2-\frac {2 \left (x^2-2 x-2\right )}{x^2}}-2 e^{x+4+\frac {4}{x}} x^{3-\frac {2 \left (x^2-2 x-2\right )}{x^2}}+3 e^{x+4+\frac {4}{x}} x^{4-\frac {2 \left (x^2-2 x-2\right )}{x^2}}+e^{x+4+\frac {4}{x}} x^{5-\frac {2 \left (x^2-2 x-2\right )}{x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\int e^{x+4+\frac {4}{x}} x^{5-\frac {2 \left (x^2-2 x-2\right )}{x^2}}dx+8 \int \frac {\int e^{x+4+\frac {4}{x}} x^{-1+\frac {4}{x}+\frac {4}{x^2}}dx}{x}dx-8 \int e^{x+4+\frac {4}{x}} x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \log ^3(x)dx-4 \int e^{x+4+\frac {4}{x}} x^{1-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \log ^3(x)dx+4 \int e^{x+4+\frac {4}{x}} x^{-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \log ^2(x)dx+16 \int e^{x+4+\frac {4}{x}} x^{1-\frac {2 \left (x^2-2 x-2\right )}{x^2}} \log ^2(x)dx-8 \log (x) \int e^{x+4+\frac {4}{x}} x^{1-\frac {2 \left (x^2-2 x-2\right )}{x^2}}dx+4 \int e^{x+4+\frac {4}{x}} x^{\frac {4 (x+1)}{x^2}}dx-2 \int e^{x+4+\frac {4}{x}} x^{\frac {(x+2)^2}{x^2}}dx+3 \int e^{x+4+\frac {4}{x}} x^{\frac {2 \left (x^2+2 x+2\right )}{x^2}}dx+6 \int \frac {\int e^{x+4+\frac {4}{x}} x^{\frac {4 (x+1)}{x^2}}dx}{x}dx+8 \int \frac {\int e^{x+4+\frac {4}{x}} x^{\frac {(x+2)^2}{x^2}}dx}{x}dx+2 \int \frac {\int e^{x+4+\frac {4}{x}} x^{\frac {2 \left (x^2+2 x+2\right )}{x^2}}dx}{x}dx+9 \int e^{x+4+\frac {4}{x}} x^{\frac {4 (x+1)}{x^2}} \log ^2(x)dx+\int e^{x+4+\frac {4}{x}} x^{\frac {(x+2)^2}{x^2}} \log ^2(x)dx-6 \log (x) \int e^{x+4+\frac {4}{x}} x^{\frac {4 (x+1)}{x^2}}dx-8 \log (x) \int e^{x+4+\frac {4}{x}} x^{\frac {(x+2)^2}{x^2}}dx-2 \log (x) \int e^{x+4+\frac {4}{x}} x^{\frac {2 \left (x^2+2 x+2\right )}{x^2}}dx\right )\) |
Int[(E^((4*x + 4*x^2 + x^3 + (4 + 4*x + x^2)*Log[x])/x^2)*(4*x^2 - 2*x^3 + 3*x^4 + x^5 + (-8*x - 6*x^2 - 8*x^3 - 2*x^4)*Log[x] + (4 + 16*x + 9*x^2 + x^3)*Log[x]^2 + (-8 - 4*x)*Log[x]^3))/(3*x^3),x]
3.26.84.3.1 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]]
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11
method | result | size |
risch | \(\frac {\left (\ln \left (x \right )^{2}-2 x \ln \left (x \right )+x^{2}\right ) {\mathrm e}^{\frac {\left (2+x \right )^{2} \left (x +\ln \left (x \right )\right )}{x^{2}}}}{3}\) | \(30\) |
parallelrisch | \(\frac {x^{2} {\mathrm e}^{\frac {\left (x^{2}+4 x +4\right ) \ln \left (x \right )+x^{3}+4 x^{2}+4 x}{x^{2}}}}{3}-\frac {2 \,{\mathrm e}^{\frac {\left (x^{2}+4 x +4\right ) \ln \left (x \right )+x^{3}+4 x^{2}+4 x}{x^{2}}} x \ln \left (x \right )}{3}+\frac {\ln \left (x \right )^{2} {\mathrm e}^{\frac {\left (x^{2}+4 x +4\right ) \ln \left (x \right )+x^{3}+4 x^{2}+4 x}{x^{2}}}}{3}\) | \(102\) |
int(1/3*((-4*x-8)*ln(x)^3+(x^3+9*x^2+16*x+4)*ln(x)^2+(-2*x^4-8*x^3-6*x^2-8 *x)*ln(x)+x^5+3*x^4-2*x^3+4*x^2)*exp(((x^2+4*x+4)*ln(x)+x^3+4*x^2+4*x)/x^2 )/x^3,x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {1}{3} \, {\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )} e^{\left (\frac {x^{3} + 4 \, x^{2} + {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right ) + 4 \, x}{x^{2}}\right )} \]
integrate(1/3*((-4*x-8)*log(x)^3+(x^3+9*x^2+16*x+4)*log(x)^2+(-2*x^4-8*x^3 -6*x^2-8*x)*log(x)+x^5+3*x^4-2*x^3+4*x^2)*exp(((x^2+4*x+4)*log(x)+x^3+4*x^ 2+4*x)/x^2)/x^3,x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {\left (x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}\right ) e^{\frac {x^{3} + 4 x^{2} + 4 x + \left (x^{2} + 4 x + 4\right ) \log {\left (x \right )}}{x^{2}}}}{3} \]
integrate(1/3*((-4*x-8)*ln(x)**3+(x**3+9*x**2+16*x+4)*ln(x)**2+(-2*x**4-8* x**3-6*x**2-8*x)*ln(x)+x**5+3*x**4-2*x**3+4*x**2)*exp(((x**2+4*x+4)*ln(x)+ x**3+4*x**2+4*x)/x**2)/x**3,x)
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\frac {1}{3} \, {\left (x^{3} e^{4} - 2 \, x^{2} e^{4} \log \left (x\right ) + x e^{4} \log \left (x\right )^{2}\right )} e^{\left (x + \frac {4 \, \log \left (x\right )}{x} + \frac {4}{x} + \frac {4 \, \log \left (x\right )}{x^{2}}\right )} \]
integrate(1/3*((-4*x-8)*log(x)^3+(x^3+9*x^2+16*x+4)*log(x)^2+(-2*x^4-8*x^3 -6*x^2-8*x)*log(x)+x^5+3*x^4-2*x^3+4*x^2)*exp(((x^2+4*x+4)*log(x)+x^3+4*x^ 2+4*x)/x^2)/x^3,x, algorithm=\
\[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=\int { \frac {{\left (x^{5} + 3 \, x^{4} - 4 \, {\left (x + 2\right )} \log \left (x\right )^{3} - 2 \, x^{3} + {\left (x^{3} + 9 \, x^{2} + 16 \, x + 4\right )} \log \left (x\right )^{2} + 4 \, x^{2} - 2 \, {\left (x^{4} + 4 \, x^{3} + 3 \, x^{2} + 4 \, x\right )} \log \left (x\right )\right )} e^{\left (\frac {x^{3} + 4 \, x^{2} + {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right ) + 4 \, x}{x^{2}}\right )}}{3 \, x^{3}} \,d x } \]
integrate(1/3*((-4*x-8)*log(x)^3+(x^3+9*x^2+16*x+4)*log(x)^2+(-2*x^4-8*x^3 -6*x^2-8*x)*log(x)+x^5+3*x^4-2*x^3+4*x^2)*exp(((x^2+4*x+4)*log(x)+x^3+4*x^ 2+4*x)/x^2)/x^3,x, algorithm=\
integrate(1/3*(x^5 + 3*x^4 - 4*(x + 2)*log(x)^3 - 2*x^3 + (x^3 + 9*x^2 + 1 6*x + 4)*log(x)^2 + 4*x^2 - 2*(x^4 + 4*x^3 + 3*x^2 + 4*x)*log(x))*e^((x^3 + 4*x^2 + (x^2 + 4*x + 4)*log(x) + 4*x)/x^2)/x^3, x)
Time = 11.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\frac {4 x+4 x^2+x^3+\left (4+4 x+x^2\right ) \log (x)}{x^2}} \left (4 x^2-2 x^3+3 x^4+x^5+\left (-8 x-6 x^2-8 x^3-2 x^4\right ) \log (x)+\left (4+16 x+9 x^2+x^3\right ) \log ^2(x)+(-8-4 x) \log ^3(x)\right )}{3 x^3} \, dx=x\,x^{4/x}\,x^{\frac {4}{x^2}}\,{\mathrm {e}}^{x+\frac {4}{x}+4}\,\left (\frac {x^2}{3}-\frac {2\,x\,\ln \left (x\right )}{3}+\frac {{\ln \left (x\right )}^2}{3}\right ) \]