Integrand size = 68, antiderivative size = 23 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=1-\log ^2\left (2 e^4+\frac {x}{e^3+\log (x)}\right ) \] Output:
1-ln(2*exp(4)+x/(ln(x)+exp(3)))^2
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-\log ^2\left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right ) \] Input:
Integrate[((2 - 2*E^3 - 2*Log[x])*Log[(2*E^7 + x + 2*E^4*Log[x])/(E^3 + Lo g[x])])/(2*E^10 + E^3*x + (4*E^7 + x)*Log[x] + 2*E^4*Log[x]^2),x]
Output:
-Log[(2*E^7 + x + 2*E^4*Log[x])/(E^3 + Log[x])]^2
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.
| \(\displaystyle \int \frac {\left (-2 \log (x)-2 e^3+2\right ) \log \left (\frac {x+2 e^4 \log (x)+2 e^7}{\log (x)+e^3}\right )}{e^3 x+2 e^4 \log ^2(x)+\left (x+4 e^7\right ) \log (x)+2 e^{10}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 \left (-\log (x)-e^3+1\right ) \log \left (\frac {x+2 e^4 \log (x)+2 e^7}{\log (x)+e^3}\right )}{\left (\log (x)+e^3\right ) \left (x+2 e^4 \log (x)+2 e^7\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\left (-\log (x)-e^3+1\right ) \log \left (\frac {x+2 e^4 \log (x)+2 e^7}{\log (x)+e^3}\right )}{\left (\log (x)+e^3\right ) \left (x+2 e^4 \log (x)+2 e^7\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {\left (1-e^3\right ) \log \left (\frac {x+2 e^4 \log (x)+2 e^7}{\log (x)+e^3}\right )}{\left (\log (x)+e^3\right ) \left (x+2 e^4 \log (x)+2 e^7\right )}-\frac {\log (x) \log \left (\frac {x+2 e^4 \log (x)+2 e^7}{\log (x)+e^3}\right )}{\left (\log (x)+e^3\right ) \left (x+2 e^4 \log (x)+2 e^7\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\left (1-e^3\right ) \int \frac {\log \left (\frac {x+2 e^4 \log (x)+2 e^7}{\log (x)+e^3}\right )}{\left (\log (x)+e^3\right ) \left (x+2 e^4 \log (x)+2 e^7\right )}dx-\int \frac {\log (x) \log \left (\frac {x+2 e^4 \log (x)+2 e^7}{\log (x)+e^3}\right )}{\left (\log (x)+e^3\right ) \left (x+2 e^4 \log (x)+2 e^7\right )}dx\right )\) |
Input:
Int[((2 - 2*E^3 - 2*Log[x])*Log[(2*E^7 + x + 2*E^4*Log[x])/(E^3 + Log[x])] )/(2*E^10 + E^3*x + (4*E^7 + x)*Log[x] + 2*E^4*Log[x]^2),x]
Output:
$Aborted
Time = 1.59 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(-\ln \left (\frac {2 \,{\mathrm e}^{4} \ln \left (x \right )+2 \,{\mathrm e}^{3} {\mathrm e}^{4}+x}{\ln \left (x \right )+{\mathrm e}^{3}}\right )^{2}\) | \(28\) |
| norman | \(-\ln \left (\frac {2 \,{\mathrm e}^{4} \ln \left (x \right )+2 \,{\mathrm e}^{3} {\mathrm e}^{4}+x}{\ln \left (x \right )+{\mathrm e}^{3}}\right )^{2}\) | \(28\) |
| risch | \(-\ln \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )^{2}+2 \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) \ln \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )-\ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right )^{2}-i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{2}+i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{2}-i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{3}}\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )-i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{3}}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{2}+i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{3}+i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{3}}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{2}+i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{3}}\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )-i \pi \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4} \ln \left (x \right )+{\mathrm e}^{7}+\frac {x}{2}\right )}{\ln \left (x \right )+{\mathrm e}^{3}}\right )}^{3}+2 \ln \left (2\right ) \ln \left (\ln \left (x \right )+{\mathrm e}^{3}\right )-2 \ln \left (2\right ) \ln \left (\ln \left (x \right )+{\mathrm e}^{3}+\frac {x \,{\mathrm e}^{-4}}{2}\right )\) | \(465\) |
Input:
int((-2*ln(x)-2*exp(3)+2)*ln((2*exp(4)*ln(x)+2*exp(3)*exp(4)+x)/(ln(x)+exp (3)))/(2*exp(4)*ln(x)^2+(4*exp(3)*exp(4)+x)*ln(x)+2*exp(3)^2*exp(4)+x*exp( 3)),x,method=_RETURNVERBOSE)
Output:
-ln((2*exp(4)*ln(x)+2*exp(3)*exp(4)+x)/(ln(x)+exp(3)))^2
Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-\log \left (\frac {2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}}{e^{3} + \log \left (x\right )}\right )^{2} \] Input:
integrate((-2*log(x)-2*exp(3)+2)*log((2*exp(4)*log(x)+2*exp(3)*exp(4)+x)/( log(x)+exp(3)))/(2*exp(4)*log(x)^2+(4*exp(3)*exp(4)+x)*log(x)+2*exp(3)^2*e xp(4)+x*exp(3)),x, algorithm="fricas")
Output:
-log((2*e^4*log(x) + x + 2*e^7)/(e^3 + log(x)))^2
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=- \log {\left (\frac {x + 2 e^{4} \log {\left (x \right )} + 2 e^{7}}{\log {\left (x \right )} + e^{3}} \right )}^{2} \] Input:
integrate((-2*ln(x)-2*exp(3)+2)*ln((2*exp(4)*ln(x)+2*exp(3)*exp(4)+x)/(ln( x)+exp(3)))/(2*exp(4)*ln(x)**2+(4*exp(3)*exp(4)+x)*ln(x)+2*exp(3)**2*exp(4 )+x*exp(3)),x)
Output:
-log((x + 2*exp(4)*log(x) + 2*exp(7))/(log(x) + exp(3)))**2
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (21) = 42\).
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.78 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-2 \, {\left (\log \left (2\right ) + \log \left (e^{3} + \log \left (x\right )\right ) + 4\right )} \log \left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right ) + \log \left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right )^{2} - 2 \, {\left (\log \left (\frac {1}{2} \, {\left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right )} e^{\left (-4\right )}\right ) - \log \left (e^{3} + \log \left (x\right )\right )\right )} \log \left (\frac {2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}}{e^{3} + \log \left (x\right )}\right ) + 2 \, {\left (\log \left (2\right ) + 4\right )} \log \left (e^{3} + \log \left (x\right )\right ) + \log \left (e^{3} + \log \left (x\right )\right )^{2} \] Input:
integrate((-2*log(x)-2*exp(3)+2)*log((2*exp(4)*log(x)+2*exp(3)*exp(4)+x)/( log(x)+exp(3)))/(2*exp(4)*log(x)^2+(4*exp(3)*exp(4)+x)*log(x)+2*exp(3)^2*e xp(4)+x*exp(3)),x, algorithm="maxima")
Output:
-2*(log(2) + log(e^3 + log(x)) + 4)*log(2*e^4*log(x) + x + 2*e^7) + log(2* e^4*log(x) + x + 2*e^7)^2 - 2*(log(1/2*(2*e^4*log(x) + x + 2*e^7)*e^(-4)) - log(e^3 + log(x)))*log((2*e^4*log(x) + x + 2*e^7)/(e^3 + log(x))) + 2*(l og(2) + 4)*log(e^3 + log(x)) + log(e^3 + log(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (21) = 42\).
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-\log \left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right )^{2} + 2 \, \log \left (2 \, e^{4} \log \left (x\right ) + x + 2 \, e^{7}\right ) \log \left (e^{3} + \log \left (x\right )\right ) - \log \left (e^{3} + \log \left (x\right )\right )^{2} \] Input:
integrate((-2*log(x)-2*exp(3)+2)*log((2*exp(4)*log(x)+2*exp(3)*exp(4)+x)/( log(x)+exp(3)))/(2*exp(4)*log(x)^2+(4*exp(3)*exp(4)+x)*log(x)+2*exp(3)^2*e xp(4)+x*exp(3)),x, algorithm="giac")
Output:
-log(2*e^4*log(x) + x + 2*e^7)^2 + 2*log(2*e^4*log(x) + x + 2*e^7)*log(e^3 + log(x)) - log(e^3 + log(x))^2
Time = 6.60 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-{\ln \left (\frac {x+2\,{\mathrm {e}}^7+2\,{\mathrm {e}}^4\,\ln \left (x\right )}{{\mathrm {e}}^3+\ln \left (x\right )}\right )}^2 \] Input:
int(-(log((x + 2*exp(7) + 2*exp(4)*log(x))/(exp(3) + log(x)))*(2*exp(3) + 2*log(x) - 2))/(2*exp(10) + 2*exp(4)*log(x)^2 + x*exp(3) + log(x)*(x + 4*e xp(7))),x)
Output:
-log((x + 2*exp(7) + 2*exp(4)*log(x))/(exp(3) + log(x)))^2
Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {\left (2-2 e^3-2 \log (x)\right ) \log \left (\frac {2 e^7+x+2 e^4 \log (x)}{e^3+\log (x)}\right )}{2 e^{10}+e^3 x+\left (4 e^7+x\right ) \log (x)+2 e^4 \log ^2(x)} \, dx=-\mathrm {log}\left (\frac {2 \,\mathrm {log}\left (x \right ) e^{4}+2 e^{7}+x}{\mathrm {log}\left (x \right )+e^{3}}\right )^{2} \] Input:
int((-2*log(x)-2*exp(3)+2)*log((2*exp(4)*log(x)+2*exp(3)*exp(4)+x)/(log(x) +exp(3)))/(2*exp(4)*log(x)^2+(4*exp(3)*exp(4)+x)*log(x)+2*exp(3)^2*exp(4)+ x*exp(3)),x)
Output:
- log((2*log(x)*e**4 + 2*e**7 + x)/(log(x) + e**3))**2