Integrand size = 42, antiderivative size = 19 \[ \int \frac {e^{9+x^2+\log (x) \log \left (x^5 \log ^2(4)\right )} \left (2 x^2+5 \log (x)+\log \left (x^5 \log ^2(4)\right )\right )}{x} \, dx=e^{9+x^2+\log (x) \log \left (x^5 \log ^2(4)\right )} \] Output:
exp(ln(x)*ln(4*x^5*ln(2)^2)+x^2+9)
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{9+x^2+\log (x) \log \left (x^5 \log ^2(4)\right )} \left (2 x^2+5 \log (x)+\log \left (x^5 \log ^2(4)\right )\right )}{x} \, dx=e^{9+x^2} x^{\log \left (x^5\right )+2 \log (\log (4))} \] Input:
Integrate[(E^(9 + x^2 + Log[x]*Log[x^5*Log[4]^2])*(2*x^2 + 5*Log[x] + Log[ x^5*Log[4]^2]))/x,x]
Output:
E^(9 + x^2)*x^(Log[x^5] + 2*Log[Log[4]])
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.
\(\displaystyle \int \frac {e^{\log (x) \log \left (x^5 \log ^2(4)\right )+x^2+9} \left (\log \left (x^5 \log ^2(4)\right )+2 x^2+5 \log (x)\right )}{x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{\log (x) \log \left (x^5 \log ^2(4)\right )+x^2+9} \left (2 x^2+5 \log (x)\right )}{x}+\frac {e^{\log (x) \log \left (x^5 \log ^2(4)\right )+x^2+9} \log \left (x^5 \log ^2(4)\right )}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int e^{x^2+\log (x) \log \left (x^5 \log ^2(4)\right )+9} xdx+5 \int \frac {e^{x^2+\log (x) \log \left (x^5 \log ^2(4)\right )+9} \log (x)}{x}dx+\int \frac {e^{x^2+\log (x) \log \left (x^5 \log ^2(4)\right )+9} \log \left (x^5 \log ^2(4)\right )}{x}dx\) |
Input:
Int[(E^(9 + x^2 + Log[x]*Log[x^5*Log[4]^2])*(2*x^2 + 5*Log[x] + Log[x^5*Lo g[4]^2]))/x,x]
Output:
$Aborted
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \({\mathrm e}^{\ln \left (x \right ) \ln \left (4 x^{5} \ln \left (2\right )^{2}\right )+x^{2}+9}\) | \(20\) |
default | \({\mathrm e}^{\ln \left (x \right ) \ln \left (4 x^{5} \ln \left (2\right )^{2}\right )+x^{2}+9}\) | \(20\) |
parallelrisch | \({\mathrm e}^{\ln \left (x \right ) \ln \left (4 x^{5} \ln \left (2\right )^{2}\right )+x^{2}+9}\) | \(20\) |
risch | \(x^{-\frac {i \pi \,\operatorname {csgn}\left (i x^{5}\right )}{2}+\frac {5 i \pi \,\operatorname {csgn}\left (i x \right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{4}\right ) \operatorname {csgn}\left (i x^{5}\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )}{2}+2 \ln \left (\ln \left (2\right )\right )+2 \ln \left (2\right )+5 \ln \left (x \right )} {\mathrm e}^{x^{2}+9}\) | \(124\) |
Input:
int((ln(4*x^5*ln(2)^2)+5*ln(x)+2*x^2)*exp(ln(x)*ln(4*x^5*ln(2)^2)+x^2+9)/x ,x,method=_RETURNVERBOSE)
Output:
exp(ln(x)*ln(4*x^5*ln(2)^2)+x^2+9)
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {e^{9+x^2+\log (x) \log \left (x^5 \log ^2(4)\right )} \left (2 x^2+5 \log (x)+\log \left (x^5 \log ^2(4)\right )\right )}{x} \, dx=e^{\left (x^{2} + \log \left (4 \, \log \left (2\right )^{2}\right ) \log \left (x\right ) + 5 \, \log \left (x\right )^{2} + 9\right )} \] Input:
integrate((log(4*x^5*log(2)^2)+5*log(x)+2*x^2)*exp(log(x)*log(4*x^5*log(2) ^2)+x^2+9)/x,x, algorithm="fricas")
Output:
e^(x^2 + log(4*log(2)^2)*log(x) + 5*log(x)^2 + 9)
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \frac {e^{9+x^2+\log (x) \log \left (x^5 \log ^2(4)\right )} \left (2 x^2+5 \log (x)+\log \left (x^5 \log ^2(4)\right )\right )}{x} \, dx=e^{x^{2} + \left (5 \log {\left (x \right )} + \log {\left (4 \log {\left (2 \right )}^{2} \right )}\right ) \log {\left (x \right )} + 9} \] Input:
integrate((ln(4*x**5*ln(2)**2)+5*ln(x)+2*x**2)*exp(ln(x)*ln(4*x**5*ln(2)** 2)+x**2+9)/x,x)
Output:
exp(x**2 + (5*log(x) + log(4*log(2)**2))*log(x) + 9)
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int \frac {e^{9+x^2+\log (x) \log \left (x^5 \log ^2(4)\right )} \left (2 x^2+5 \log (x)+\log \left (x^5 \log ^2(4)\right )\right )}{x} \, dx=e^{\left (x^{2} + 2 \, \log \left (2\right ) \log \left (x\right ) + 5 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) \log \left (\log \left (2\right )\right ) + 9\right )} \] Input:
integrate((log(4*x^5*log(2)^2)+5*log(x)+2*x^2)*exp(log(x)*log(4*x^5*log(2) ^2)+x^2+9)/x,x, algorithm="maxima")
Output:
e^(x^2 + 2*log(2)*log(x) + 5*log(x)^2 + 2*log(x)*log(log(2)) + 9)
Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{9+x^2+\log (x) \log \left (x^5 \log ^2(4)\right )} \left (2 x^2+5 \log (x)+\log \left (x^5 \log ^2(4)\right )\right )}{x} \, dx=e^{\left (x^{2} + \log \left (4 \, x^{5} \log \left (2\right )^{2}\right ) \log \left (x\right ) + 9\right )} \] Input:
integrate((log(4*x^5*log(2)^2)+5*log(x)+2*x^2)*exp(log(x)*log(4*x^5*log(2) ^2)+x^2+9)/x,x, algorithm="giac")
Output:
e^(x^2 + log(4*x^5*log(2)^2)*log(x) + 9)
Time = 2.98 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37 \[ \int \frac {e^{9+x^2+\log (x) \log \left (x^5 \log ^2(4)\right )} \left (2 x^2+5 \log (x)+\log \left (x^5 \log ^2(4)\right )\right )}{x} \, dx=x^{2\,\ln \left (\ln \left (2\right )\right )}\,x^{\ln \left (x^5\right )}\,x^{2\,\ln \left (2\right )}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^9 \] Input:
int((exp(log(4*x^5*log(2)^2)*log(x) + x^2 + 9)*(log(4*x^5*log(2)^2) + 5*lo g(x) + 2*x^2))/x,x)
Output:
x^(2*log(log(2)))*x^log(x^5)*x^(2*log(2))*exp(x^2)*exp(9)
\[ \int \frac {e^{9+x^2+\log (x) \log \left (x^5 \log ^2(4)\right )} \left (2 x^2+5 \log (x)+\log \left (x^5 \log ^2(4)\right )\right )}{x} \, dx=e^{9} \left (\int \frac {x^{5 \,\mathrm {log}\left (x \right )} e^{x^{2}} \mathrm {log}\left (2\right )^{2 \,\mathrm {log}\left (x \right )} 4^{\mathrm {log}\left (x \right )} \mathrm {log}\left (4 \mathrm {log}\left (2\right )^{2} x^{5}\right )}{x}d x +5 \left (\int \frac {x^{5 \,\mathrm {log}\left (x \right )} e^{x^{2}} \mathrm {log}\left (2\right )^{2 \,\mathrm {log}\left (x \right )} 4^{\mathrm {log}\left (x \right )} \mathrm {log}\left (x \right )}{x}d x \right )+2 \left (\int x^{5 \,\mathrm {log}\left (x \right )} e^{x^{2}} \mathrm {log}\left (2\right )^{2 \,\mathrm {log}\left (x \right )} 4^{\mathrm {log}\left (x \right )} x d x \right )\right ) \] Input:
int((log(4*x^5*log(2)^2)+5*log(x)+2*x^2)*exp(log(x)*log(4*x^5*log(2)^2)+x^ 2+9)/x,x)
Output:
e**9*(int((x**(5*log(x))*e**(x**2)*log(2)**(2*log(x))*4**log(x)*log(4*log( 2)**2*x**5))/x,x) + 5*int((x**(5*log(x))*e**(x**2)*log(2)**(2*log(x))*4**l og(x)*log(x))/x,x) + 2*int(x**(5*log(x))*e**(x**2)*log(2)**(2*log(x))*4**l og(x)*x,x))