Integrand size = 94, antiderivative size = 22 \[ \int \frac {1+\left (x+2 e^{x^2} x^2\right ) \log (x)+(5+2 x) \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx=x \left (5+x+\log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )\right ) \] Output:
x*(5+x+ln(4*ln(4*ln(x)*exp(exp(x^2)+x))))
Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1+\left (x+2 e^{x^2} x^2\right ) \log (x)+(5+2 x) \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx=5 x+x^2+x \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right ) \] Input:
Integrate[(1 + (x + 2*E^x^2*x^2)*Log[x] + (5 + 2*x)*Log[x]*Log[4*E^(E^x^2 + x)*Log[x]] + Log[x]*Log[4*E^(E^x^2 + x)*Log[x]]*Log[4*Log[4*E^(E^x^2 + x )*Log[x]]])/(Log[x]*Log[4*E^(E^x^2 + x)*Log[x]]),x]
Output:
5*x + x^2 + x*Log[4*Log[4*E^(E^x^2 + x)*Log[x]]]
Time = 1.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {7293, 2009}
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.
\(\displaystyle \int \frac {\left (2 e^{x^2} x^2+x\right ) \log (x)+(2 x+5) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log (x)+\log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right ) \log (x)+1}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 e^{x^2} x^2}{\log \left (4 e^{e^{x^2}+x} \log (x)\right )}+\frac {2 x \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log (x)+5 \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log (x)+\log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right ) \log (x)+x \log (x)+1}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^2+x \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )+5 x\) |
Input:
Int[(1 + (x + 2*E^x^2*x^2)*Log[x] + (5 + 2*x)*Log[x]*Log[4*E^(E^x^2 + x)*L og[x]] + Log[x]*Log[4*E^(E^x^2 + x)*Log[x]]*Log[4*Log[4*E^(E^x^2 + x)*Log[ x]]])/(Log[x]*Log[4*E^(E^x^2 + x)*Log[x]]),x]
Output:
5*x + x^2 + x*Log[4*Log[4*E^(E^x^2 + x)*Log[x]]]
Time = 3.40 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(x^{2}+\ln \left (4 \ln \left (4 \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right )\right ) x +5 x\) | \(25\) |
risch | \(x^{2}+x \ln \left (8 \ln \left (2\right )+4 \ln \left (\ln \left (x \right )\right )+4 \ln \left ({\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right )-2 i \pi \,\operatorname {csgn}\left (i \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right ) \left (-\operatorname {csgn}\left (i \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (i \ln \left (x \right ) {\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{{\mathrm e}^{x^{2}}+x}\right )\right )\right )+5 x\) | \(97\) |
Input:
int((ln(x)*ln(4*ln(x)*exp(exp(x^2)+x))*ln(4*ln(4*ln(x)*exp(exp(x^2)+x)))+( 5+2*x)*ln(x)*ln(4*ln(x)*exp(exp(x^2)+x))+(2*x^2*exp(x^2)+x)*ln(x)+1)/ln(x) /ln(4*ln(x)*exp(exp(x^2)+x)),x,method=_RETURNVERBOSE)
Output:
x^2+ln(4*ln(4*ln(x)*exp(exp(x^2)+x)))*x+5*x
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1+\left (x+2 e^{x^2} x^2\right ) \log (x)+(5+2 x) \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx=x^{2} + x \log \left (4 \, \log \left (4 \, e^{\left (x + e^{\left (x^{2}\right )}\right )} \log \left (x\right )\right )\right ) + 5 \, x \] Input:
integrate((log(x)*log(4*log(x)*exp(exp(x^2)+x))*log(4*log(4*log(x)*exp(exp (x^2)+x)))+(5+2*x)*log(x)*log(4*log(x)*exp(exp(x^2)+x))+(2*exp(x^2)*x^2+x) *log(x)+1)/log(x)/log(4*log(x)*exp(exp(x^2)+x)),x, algorithm="fricas")
Output:
x^2 + x*log(4*log(4*e^(x + e^(x^2))*log(x))) + 5*x
Timed out. \[ \int \frac {1+\left (x+2 e^{x^2} x^2\right ) \log (x)+(5+2 x) \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx=\text {Timed out} \] Input:
integrate((ln(x)*ln(4*ln(x)*exp(exp(x**2)+x))*ln(4*ln(4*ln(x)*exp(exp(x**2 )+x)))+(5+2*x)*ln(x)*ln(4*ln(x)*exp(exp(x**2)+x))+(2*exp(x**2)*x**2+x)*ln( x)+1)/ln(x)/ln(4*ln(x)*exp(exp(x**2)+x)),x)
Output:
Timed out
Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {1+\left (x+2 e^{x^2} x^2\right ) \log (x)+(5+2 x) \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx=x^{2} + 2 \, x \log \left (2\right ) + x \log \left (x + e^{\left (x^{2}\right )} + 2 \, \log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) + 5 \, x \] Input:
integrate((log(x)*log(4*log(x)*exp(exp(x^2)+x))*log(4*log(4*log(x)*exp(exp (x^2)+x)))+(5+2*x)*log(x)*log(4*log(x)*exp(exp(x^2)+x))+(2*exp(x^2)*x^2+x) *log(x)+1)/log(x)/log(4*log(x)*exp(exp(x^2)+x)),x, algorithm="maxima")
Output:
x^2 + 2*x*log(2) + x*log(x + e^(x^2) + 2*log(2) + log(log(x))) + 5*x
Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {1+\left (x+2 e^{x^2} x^2\right ) \log (x)+(5+2 x) \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx=x^{2} + 2 \, x \log \left (2\right ) + x \log \left (x + e^{\left (x^{2}\right )} + 2 \, \log \left (2\right ) + \log \left (\log \left (x\right )\right )\right ) + 5 \, x \] Input:
integrate((log(x)*log(4*log(x)*exp(exp(x^2)+x))*log(4*log(4*log(x)*exp(exp (x^2)+x)))+(5+2*x)*log(x)*log(4*log(x)*exp(exp(x^2)+x))+(2*exp(x^2)*x^2+x) *log(x)+1)/log(x)/log(4*log(x)*exp(exp(x^2)+x)),x, algorithm="giac")
Output:
x^2 + 2*x*log(2) + x*log(x + e^(x^2) + 2*log(2) + log(log(x))) + 5*x
Time = 0.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {1+\left (x+2 e^{x^2} x^2\right ) \log (x)+(5+2 x) \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx=5\,x+x\,\ln \left (4\,x+4\,{\mathrm {e}}^{x^2}+4\,\ln \left (\ln \left (x\right )\right )+\ln \left (256\right )\right )+x^2 \] Input:
int((log(x)*(x + 2*x^2*exp(x^2)) + log(4*exp(x + exp(x^2))*log(x))*log(x)* log(4*log(4*exp(x + exp(x^2))*log(x))) + log(4*exp(x + exp(x^2))*log(x))*l og(x)*(2*x + 5) + 1)/(log(4*exp(x + exp(x^2))*log(x))*log(x)),x)
Output:
5*x + x*log(4*x + 4*exp(x^2) + 4*log(log(x)) + log(256)) + x^2
\[ \int \frac {1+\left (x+2 e^{x^2} x^2\right ) \log (x)+(5+2 x) \log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )+\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right ) \log \left (4 \log \left (4 e^{e^{x^2}+x} \log (x)\right )\right )}{\log (x) \log \left (4 e^{e^{x^2}+x} \log (x)\right )} \, dx=\int \mathrm {log}\left (4 \,\mathrm {log}\left (4 e^{e^{x^{2}}+x} \mathrm {log}\left (x \right )\right )\right )d x +2 \left (\int \frac {e^{x^{2}} x^{2}}{\mathrm {log}\left (4 e^{e^{x^{2}}+x} \mathrm {log}\left (x \right )\right )}d x \right )+\int \frac {x}{\mathrm {log}\left (4 e^{e^{x^{2}}+x} \mathrm {log}\left (x \right )\right )}d x +\int \frac {1}{\mathrm {log}\left (4 e^{e^{x^{2}}+x} \mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right )}d x +x^{2}+5 x \] Input:
int((log(x)*log(4*log(x)*exp(exp(x^2)+x))*log(4*log(4*log(x)*exp(exp(x^2)+ x)))+(5+2*x)*log(x)*log(4*log(x)*exp(exp(x^2)+x))+(2*exp(x^2)*x^2+x)*log(x )+1)/log(x)/log(4*log(x)*exp(exp(x^2)+x)),x)
Output:
int(log(4*log(4*e**(e**(x**2) + x)*log(x))),x) + 2*int((e**(x**2)*x**2)/lo g(4*e**(e**(x**2) + x)*log(x)),x) + int(x/log(4*e**(e**(x**2) + x)*log(x)) ,x) + int(1/(log(4*e**(e**(x**2) + x)*log(x))*log(x)),x) + x**2 + 5*x