Integrand size = 179, antiderivative size = 36 \[ \int \frac {\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )+e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}} \left (4-4 e^4+e^8+\left (4+4 e^4-e^8\right ) \log (x)+\left (-4+2 e^4+\left (4-2 e^4\right ) \log (x)\right ) \log \left (4 x^2\right )+(1-\log (x)) \log ^2\left (4 x^2\right )\right )}{\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )} \, dx=x \left (2+\frac {1-x}{x}-\frac {e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}}}{\log (x)}\right ) \] Output:
((1-x)/x-exp(4/(ln(4*x^2)+exp(4)-2))/ln(x)+2)*x
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )+e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}} \left (4-4 e^4+e^8+\left (4+4 e^4-e^8\right ) \log (x)+\left (-4+2 e^4+\left (4-2 e^4\right ) \log (x)\right ) \log \left (4 x^2\right )+(1-\log (x)) \log ^2\left (4 x^2\right )\right )}{\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )} \, dx=x-\frac {e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}} x}{\log (x)} \] Input:
Integrate[((4 - 4*E^4 + E^8)*Log[x]^2 + (-4 + 2*E^4)*Log[x]^2*Log[4*x^2] + Log[x]^2*Log[4*x^2]^2 + E^(4/(-2 + E^4 + Log[4*x^2]))*(4 - 4*E^4 + E^8 + (4 + 4*E^4 - E^8)*Log[x] + (-4 + 2*E^4 + (4 - 2*E^4)*Log[x])*Log[4*x^2] + (1 - Log[x])*Log[4*x^2]^2))/((4 - 4*E^4 + E^8)*Log[x]^2 + (-4 + 2*E^4)*Log [x]^2*Log[4*x^2] + Log[x]^2*Log[4*x^2]^2),x]
Output:
x - (E^(4/(-2 + E^4 + Log[4*x^2]))*x)/Log[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.
| \(\displaystyle \int \frac {\log ^2\left (4 x^2\right ) \log ^2(x)+\left (2 e^4-4\right ) \log \left (4 x^2\right ) \log ^2(x)+e^{\frac {4}{\log \left (4 x^2\right )+e^4-2}} \left ((1-\log (x)) \log ^2\left (4 x^2\right )+\left (\left (4-2 e^4\right ) \log (x)+2 e^4-4\right ) \log \left (4 x^2\right )+\left (4+4 e^4-e^8\right ) \log (x)+e^8-4 e^4+4\right )+\left (4-4 e^4+e^8\right ) \log ^2(x)}{\log ^2\left (4 x^2\right ) \log ^2(x)+\left (2 e^4-4\right ) \log \left (4 x^2\right ) \log ^2(x)+\left (4-4 e^4+e^8\right ) \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\log ^2\left (4 x^2\right ) \log ^2(x)+\left (2 e^4-4\right ) \log \left (4 x^2\right ) \log ^2(x)+e^{\frac {4}{\log \left (4 x^2\right )+e^4-2}} \left ((1-\log (x)) \log ^2\left (4 x^2\right )+\left (\left (4-2 e^4\right ) \log (x)+2 e^4-4\right ) \log \left (4 x^2\right )+\left (4+4 e^4-e^8\right ) \log (x)+e^8-4 e^4+4\right )+\left (4-4 e^4+e^8\right ) \log ^2(x)}{\log ^2(x) \left (2 \left (1-\frac {e^4}{2}\right )-\log \left (4 x^2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {4}{\log \left (4 x^2\right )-2 \left (1-\frac {e^4}{2}\right )}} \left (-\log (x) \log ^2\left (4 x^2\right )+\log ^2\left (4 x^2\right )+4 \left (1-\frac {e^4}{2}\right ) \log (x) \log \left (4 x^2\right )-4 \left (1-\frac {e^4}{2}\right ) \log \left (4 x^2\right )+4 \left (1+e^4-\frac {e^8}{4}\right ) \log (x)+4 \left (1+\frac {1}{4} e^4 \left (e^4-4\right )\right )\right )}{\log ^2(x) \left (2 \left (1-\frac {e^4}{2}\right )-\log \left (4 x^2\right )\right )^2}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {e^{\frac {4}{\log \left (4 x^2\right )-2 \left (1-\frac {e^4}{2}\right )}}}{\log ^2(x)}dx-\int \frac {e^{\frac {4}{\log \left (4 x^2\right )-2 \left (1-\frac {e^4}{2}\right )}}}{\log (x)}dx+8 \int \frac {e^{\frac {4}{\log \left (4 x^2\right )-2 \left (1-\frac {e^4}{2}\right )}}}{\log (x) \left (2 \left (1-\frac {e^4}{2}\right )-\log \left (4 x^2\right )\right )^2}dx+x\) |
Input:
Int[((4 - 4*E^4 + E^8)*Log[x]^2 + (-4 + 2*E^4)*Log[x]^2*Log[4*x^2] + Log[x ]^2*Log[4*x^2]^2 + E^(4/(-2 + E^4 + Log[4*x^2]))*(4 - 4*E^4 + E^8 + (4 + 4 *E^4 - E^8)*Log[x] + (-4 + 2*E^4 + (4 - 2*E^4)*Log[x])*Log[4*x^2] + (1 - L og[x])*Log[4*x^2]^2))/((4 - 4*E^4 + E^8)*Log[x]^2 + (-4 + 2*E^4)*Log[x]^2* Log[4*x^2] + Log[x]^2*Log[4*x^2]^2),x]
Output:
$Aborted
Time = 5.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81
| method | result | size |
| parallelrisch | \(\frac {x \ln \left (x \right )-x \,{\mathrm e}^{\frac {4}{\ln \left (4 x^{2}\right )+{\mathrm e}^{4}-2}}}{\ln \left (x \right )}\) | \(29\) |
| default | \(x +\frac {\left (2-\ln \left (4 x^{2}\right )+2 \ln \left (x \right )-{\mathrm e}^{4}\right ) x \,{\mathrm e}^{\frac {4}{\ln \left (4 x^{2}\right )+{\mathrm e}^{4}-2}}-2 x \ln \left (x \right ) {\mathrm e}^{\frac {4}{\ln \left (4 x^{2}\right )+{\mathrm e}^{4}-2}}}{\left (\ln \left (4 x^{2}\right )+{\mathrm e}^{4}-2\right ) \ln \left (x \right )}\) | \(76\) |
| parts | \(x +\frac {\left (2-\ln \left (4 x^{2}\right )+2 \ln \left (x \right )-{\mathrm e}^{4}\right ) x \,{\mathrm e}^{\frac {4}{\ln \left (4 x^{2}\right )+{\mathrm e}^{4}-2}}-2 x \ln \left (x \right ) {\mathrm e}^{\frac {4}{\ln \left (4 x^{2}\right )+{\mathrm e}^{4}-2}}}{\left (\ln \left (4 x^{2}\right )+{\mathrm e}^{4}-2\right ) \ln \left (x \right )}\) | \(76\) |
| risch | \(x -\frac {x \,{\mathrm e}^{\frac {8}{-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+4 \ln \left (2\right )+2 \,{\mathrm e}^{4}+4 \ln \left (x \right )-4}}}{\ln \left (x \right )}\) | \(78\) |
Input:
int((((1-ln(x))*ln(4*x^2)^2+((-2*exp(4)+4)*ln(x)+2*exp(4)-4)*ln(4*x^2)+(-e xp(4)^2+4*exp(4)+4)*ln(x)+exp(4)^2-4*exp(4)+4)*exp(4/(ln(4*x^2)+exp(4)-2)) +ln(x)^2*ln(4*x^2)^2+(2*exp(4)-4)*ln(x)^2*ln(4*x^2)+(exp(4)^2-4*exp(4)+4)* ln(x)^2)/(ln(x)^2*ln(4*x^2)^2+(2*exp(4)-4)*ln(x)^2*ln(4*x^2)+(exp(4)^2-4*e xp(4)+4)*ln(x)^2),x,method=_RETURNVERBOSE)
Output:
(x*ln(x)-x*exp(4/(ln(4*x^2)+exp(4)-2)))/ln(x)
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )+e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}} \left (4-4 e^4+e^8+\left (4+4 e^4-e^8\right ) \log (x)+\left (-4+2 e^4+\left (4-2 e^4\right ) \log (x)\right ) \log \left (4 x^2\right )+(1-\log (x)) \log ^2\left (4 x^2\right )\right )}{\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )} \, dx=-\frac {x e^{\left (\frac {4}{e^{4} + 2 \, \log \left (2\right ) + 2 \, \log \left (x\right ) - 2}\right )} - x \log \left (x\right )}{\log \left (x\right )} \] Input:
integrate((((1-log(x))*log(4*x^2)^2+((-2*exp(4)+4)*log(x)+2*exp(4)-4)*log( 4*x^2)+(-exp(4)^2+4*exp(4)+4)*log(x)+exp(4)^2-4*exp(4)+4)*exp(4/(log(4*x^2 )+exp(4)-2))+log(x)^2*log(4*x^2)^2+(2*exp(4)-4)*log(x)^2*log(4*x^2)+(exp(4 )^2-4*exp(4)+4)*log(x)^2)/(log(x)^2*log(4*x^2)^2+(2*exp(4)-4)*log(x)^2*log (4*x^2)+(exp(4)^2-4*exp(4)+4)*log(x)^2),x, algorithm="fricas")
Output:
-(x*e^(4/(e^4 + 2*log(2) + 2*log(x) - 2)) - x*log(x))/log(x)
Exception generated. \[ \int \frac {\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )+e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}} \left (4-4 e^4+e^8+\left (4+4 e^4-e^8\right ) \log (x)+\left (-4+2 e^4+\left (4-2 e^4\right ) \log (x)\right ) \log \left (4 x^2\right )+(1-\log (x)) \log ^2\left (4 x^2\right )\right )}{\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((1-ln(x))*ln(4*x**2)**2+((-2*exp(4)+4)*ln(x)+2*exp(4)-4)*ln(4* x**2)+(-exp(4)**2+4*exp(4)+4)*ln(x)+exp(4)**2-4*exp(4)+4)*exp(4/(ln(4*x**2 )+exp(4)-2))+ln(x)**2*ln(4*x**2)**2+(2*exp(4)-4)*ln(x)**2*ln(4*x**2)+(exp( 4)**2-4*exp(4)+4)*ln(x)**2)/(ln(x)**2*ln(4*x**2)**2+(2*exp(4)-4)*ln(x)**2* ln(4*x**2)+(exp(4)**2-4*exp(4)+4)*ln(x)**2),x)
Output:
Exception raised: TypeError >> '>' not supported between instances of 'Pol y' and 'int'
\[ \int \frac {\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )+e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}} \left (4-4 e^4+e^8+\left (4+4 e^4-e^8\right ) \log (x)+\left (-4+2 e^4+\left (4-2 e^4\right ) \log (x)\right ) \log \left (4 x^2\right )+(1-\log (x)) \log ^2\left (4 x^2\right )\right )}{\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )} \, dx=\int { \frac {2 \, {\left (e^{4} - 2\right )} \log \left (4 \, x^{2}\right ) \log \left (x\right )^{2} + \log \left (4 \, x^{2}\right )^{2} \log \left (x\right )^{2} + {\left (e^{8} - 4 \, e^{4} + 4\right )} \log \left (x\right )^{2} - {\left ({\left (\log \left (x\right ) - 1\right )} \log \left (4 \, x^{2}\right )^{2} + 2 \, {\left ({\left (e^{4} - 2\right )} \log \left (x\right ) - e^{4} + 2\right )} \log \left (4 \, x^{2}\right ) + {\left (e^{8} - 4 \, e^{4} - 4\right )} \log \left (x\right ) - e^{8} + 4 \, e^{4} - 4\right )} e^{\left (\frac {4}{e^{4} + \log \left (4 \, x^{2}\right ) - 2}\right )}}{2 \, {\left (e^{4} - 2\right )} \log \left (4 \, x^{2}\right ) \log \left (x\right )^{2} + \log \left (4 \, x^{2}\right )^{2} \log \left (x\right )^{2} + {\left (e^{8} - 4 \, e^{4} + 4\right )} \log \left (x\right )^{2}} \,d x } \] Input:
integrate((((1-log(x))*log(4*x^2)^2+((-2*exp(4)+4)*log(x)+2*exp(4)-4)*log( 4*x^2)+(-exp(4)^2+4*exp(4)+4)*log(x)+exp(4)^2-4*exp(4)+4)*exp(4/(log(4*x^2 )+exp(4)-2))+log(x)^2*log(4*x^2)^2+(2*exp(4)-4)*log(x)^2*log(4*x^2)+(exp(4 )^2-4*exp(4)+4)*log(x)^2)/(log(x)^2*log(4*x^2)^2+(2*exp(4)-4)*log(x)^2*log (4*x^2)+(exp(4)^2-4*exp(4)+4)*log(x)^2),x, algorithm="maxima")
Output:
x - integrate((4*(e^4 + 2*log(2) - 3)*log(x)^2 + 4*log(x)^3 - 4*(log(2) - 1)*e^4 - 4*log(2)^2 + (4*(log(2) - 2)*e^4 + 4*log(2)^2 + e^8 - 16*log(2) + 4)*log(x) - e^8 + 8*log(2) - 4)*e^(4/(e^4 + 2*log(2) + 2*log(x) - 2))/(4* (e^4 + 2*log(2) - 2)*log(x)^3 + 4*log(x)^4 + (4*(log(2) - 1)*e^4 + 4*log(2 )^2 + e^8 - 8*log(2) + 4)*log(x)^2), x)
Exception generated. \[ \int \frac {\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )+e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}} \left (4-4 e^4+e^8+\left (4+4 e^4-e^8\right ) \log (x)+\left (-4+2 e^4+\left (4-2 e^4\right ) \log (x)\right ) \log \left (4 x^2\right )+(1-\log (x)) \log ^2\left (4 x^2\right )\right )}{\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((1-log(x))*log(4*x^2)^2+((-2*exp(4)+4)*log(x)+2*exp(4)-4)*log( 4*x^2)+(-exp(4)^2+4*exp(4)+4)*log(x)+exp(4)^2-4*exp(4)+4)*exp(4/(log(4*x^2 )+exp(4)-2))+log(x)^2*log(4*x^2)^2+(2*exp(4)-4)*log(x)^2*log(4*x^2)+(exp(4 )^2-4*exp(4)+4)*log(x)^2)/(log(x)^2*log(4*x^2)^2+(2*exp(4)-4)*log(x)^2*log (4*x^2)+(exp(4)^2-4*exp(4)+4)*log(x)^2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-4194304,[1,7,3,0,1]%%%}+%%%{-4194304,[1,7,2,1,1]%%%}+%%%{ 25165824,
Time = 3.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \frac {\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )+e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}} \left (4-4 e^4+e^8+\left (4+4 e^4-e^8\right ) \log (x)+\left (-4+2 e^4+\left (4-2 e^4\right ) \log (x)\right ) \log \left (4 x^2\right )+(1-\log (x)) \log ^2\left (4 x^2\right )\right )}{\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )} \, dx=x-\frac {x\,{\mathrm {e}}^{\frac {4}{{\mathrm {e}}^4+\ln \left (4\,x^2\right )-2}}}{\ln \left (x\right )} \] Input:
int((log(x)^2*(exp(8) - 4*exp(4) + 4) - exp(4/(exp(4) + log(4*x^2) - 2))*( 4*exp(4) - exp(8) - log(x)*(4*exp(4) - exp(8) + 4) + log(4*x^2)*(log(x)*(2 *exp(4) - 4) - 2*exp(4) + 4) + log(4*x^2)^2*(log(x) - 1) - 4) + log(4*x^2) ^2*log(x)^2 + log(4*x^2)*log(x)^2*(2*exp(4) - 4))/(log(x)^2*(exp(8) - 4*ex p(4) + 4) + log(4*x^2)^2*log(x)^2 + log(4*x^2)*log(x)^2*(2*exp(4) - 4)),x)
Output:
x - (x*exp(4/(exp(4) + log(4*x^2) - 2)))/log(x)
Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \frac {\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )+e^{\frac {4}{-2+e^4+\log \left (4 x^2\right )}} \left (4-4 e^4+e^8+\left (4+4 e^4-e^8\right ) \log (x)+\left (-4+2 e^4+\left (4-2 e^4\right ) \log (x)\right ) \log \left (4 x^2\right )+(1-\log (x)) \log ^2\left (4 x^2\right )\right )}{\left (4-4 e^4+e^8\right ) \log ^2(x)+\left (-4+2 e^4\right ) \log ^2(x) \log \left (4 x^2\right )+\log ^2(x) \log ^2\left (4 x^2\right )} \, dx=\frac {x \left (-e^{\frac {4}{\mathrm {log}\left (4 x^{2}\right )+e^{4}-2}}+\mathrm {log}\left (x \right )\right )}{\mathrm {log}\left (x \right )} \] Input:
int((((1-log(x))*log(4*x^2)^2+((-2*exp(4)+4)*log(x)+2*exp(4)-4)*log(4*x^2) +(-exp(4)^2+4*exp(4)+4)*log(x)+exp(4)^2-4*exp(4)+4)*exp(4/(log(4*x^2)+exp( 4)-2))+log(x)^2*log(4*x^2)^2+(2*exp(4)-4)*log(x)^2*log(4*x^2)+(exp(4)^2-4* exp(4)+4)*log(x)^2)/(log(x)^2*log(4*x^2)^2+(2*exp(4)-4)*log(x)^2*log(4*x^2 )+(exp(4)^2-4*exp(4)+4)*log(x)^2),x)
Output:
(x*( - e**(4/(log(4*x**2) + e**4 - 2)) + log(x)))/log(x)