Integrand size = 111, antiderivative size = 30 \[ \int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx=\left (-e^2+e^x+x\right ) \left (x-\log \left (x+\frac {1}{4} (2 x+x \log (x))\right )\right ) \] Output:
(x-ln(1/4*x*ln(x)+3/2*x))*(exp(x)+x-exp(2))
Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx=-e^2 x+e^x x+x^2+e^2 \log (x)+e^2 \log (6+\log (x))-\left (e^x+x\right ) \log \left (\frac {1}{4} x (6+\log (x))\right ) \] Input:
Integrate[(E^2*(7 - 6*x) - 7*x + 12*x^2 + E^x*(-7 + 6*x + 6*x^2) + (E^2*(1 - x) - x + 2*x^2 + E^x*(-1 + x + x^2))*Log[x] + (-6*x - 6*E^x*x + (-x - E ^x*x)*Log[x])*Log[(6*x + x*Log[x])/4])/(6*x + x*Log[x]),x]
Output:
-(E^2*x) + E^x*x + x^2 + E^2*Log[x] + E^2*Log[6 + Log[x]] - (E^x + x)*Log[ (x*(6 + Log[x]))/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.
\(\displaystyle \int \frac {12 x^2+e^x \left (6 x^2+6 x-7\right )+\left (2 x^2+e^x \left (x^2+x-1\right )-x+e^2 (1-x)\right ) \log (x)-7 x+e^2 (7-6 x)+\left (-6 e^x x-6 x+\left (-e^x x-x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx\) |
\(\Big \downarrow \) 3041 |
\(\displaystyle \int \frac {12 x^2+e^x \left (6 x^2+6 x-7\right )+\left (2 x^2+e^x \left (x^2+x-1\right )-x+e^2 (1-x)\right ) \log (x)-7 x+e^2 (7-6 x)+\left (-6 e^x x-6 x+\left (-e^x x-x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{x (\log (x)+6)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^x \left (6 x^2+x^2 \log (x)+6 x+x \log (x)-x \log (x) \log \left (\frac {1}{4} x (\log (x)+6)\right )-6 x \log \left (\frac {1}{4} x (\log (x)+6)\right )-\log (x)-7\right )}{x (\log (x)+6)}+\frac {2 x \log (x)}{\log (x)+6}+\frac {12 x}{\log (x)+6}-\frac {\log (x) \log \left (\frac {1}{4} x (\log (x)+6)\right )}{\log (x)+6}-\frac {6 \log \left (\frac {1}{4} x (\log (x)+6)\right )}{\log (x)+6}-\frac {\log (x)}{\log (x)+6}-\frac {7}{\log (x)+6}-\frac {e^2 (6 x-7)}{x (\log (x)+6)}-\frac {e^2 (x-1) \log (x)}{x (\log (x)+6)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -6 \int \frac {\log \left (\frac {1}{4} x (\log (x)+6)\right )}{\log (x)+6}dx-\int \frac {\log (x) \log \left (\frac {1}{4} x (\log (x)+6)\right )}{\log (x)+6}dx-\frac {\operatorname {ExpIntegralEi}(\log (x)+6)}{e^6}+\frac {12 \operatorname {ExpIntegralEi}(2 (\log (x)+6))}{e^{12}}+\frac {2 \log (x) \operatorname {ExpIntegralEi}(2 (\log (x)+6))}{e^{12}}-\frac {2 (\log (x)+6) \operatorname {ExpIntegralEi}(2 (\log (x)+6))}{e^{12}}+x^2+\frac {e^x \left (6 x^2+x^2 \log (x)-x \log (x) \log \left (\frac {1}{4} x (\log (x)+6)\right )-6 x \log \left (\frac {1}{4} x (\log (x)+6)\right )\right )}{x (\log (x)+6)}-e^2 x-x+e^2 \log (x)+e^2 \log (\log (x)+6)\) |
Input:
Int[(E^2*(7 - 6*x) - 7*x + 12*x^2 + E^x*(-7 + 6*x + 6*x^2) + (E^2*(1 - x) - x + 2*x^2 + E^x*(-1 + x + x^2))*Log[x] + (-6*x - 6*E^x*x + (-x - E^x*x)* Log[x])*Log[(6*x + x*Log[x])/4])/(6*x + x*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(23)=46\).
Time = 2.39 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60
method | result | size |
parallelrisch | \(-{\mathrm e}^{2} x +\ln \left (\frac {x \left (\ln \left (x \right )+6\right )}{4}\right ) {\mathrm e}^{2}+x^{2}+{\mathrm e}^{x} x -\ln \left (\frac {x \left (\ln \left (x \right )+6\right )}{4}\right ) x -{\mathrm e}^{x} \ln \left (\frac {x \left (\ln \left (x \right )+6\right )}{4}\right )\) | \(48\) |
risch | \(2 \,{\mathrm e}^{x} \ln \left (2\right )+\frac {i \pi x \operatorname {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{3}}{2}+\frac {i {\mathrm e}^{x} \pi \operatorname {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{3}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{2}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+6\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{2}}{2}+{\mathrm e}^{2} \ln \left (x \right )-{\mathrm e}^{x} \ln \left (x \right )+2 x \ln \left (2\right )+{\mathrm e}^{x} x -x \ln \left (x \right )-{\mathrm e}^{2} x +x^{2}-\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{2}}{2}-\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+6\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )^{2}}{2}+{\mathrm e}^{2} \ln \left (\ln \left (x \right )+6\right )+\left (-{\mathrm e}^{x}-x \right ) \ln \left (\ln \left (x \right )+6\right )+\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+6\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )}{2}+\frac {i \pi x \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+6\right )\right ) \operatorname {csgn}\left (i x \left (\ln \left (x \right )+6\right )\right )}{2}\) | \(243\) |
Input:
int((((-exp(x)*x-x)*ln(x)-6*exp(x)*x-6*x)*ln(1/4*x*ln(x)+3/2*x)+((x^2+x-1) *exp(x)+(1-x)*exp(2)+2*x^2-x)*ln(x)+(6*x^2+6*x-7)*exp(x)+(-6*x+7)*exp(2)+1 2*x^2-7*x)/(x*ln(x)+6*x),x,method=_RETURNVERBOSE)
Output:
-exp(2)*x+ln(1/4*x*(ln(x)+6))*exp(2)+x^2+exp(x)*x-ln(1/4*x*(ln(x)+6))*x-ex p(x)*ln(1/4*x*(ln(x)+6))
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx=x^{2} - x e^{2} + x e^{x} - {\left (x - e^{2} + e^{x}\right )} \log \left (\frac {1}{4} \, x \log \left (x\right ) + \frac {3}{2} \, x\right ) \] Input:
integrate((((-exp(x)*x-x)*log(x)-6*exp(x)*x-6*x)*log(1/4*x*log(x)+3/2*x)+( (x^2+x-1)*exp(x)+(1-x)*exp(2)+2*x^2-x)*log(x)+(6*x^2+6*x-7)*exp(x)+(-6*x+7 )*exp(2)+12*x^2-7*x)/(x*log(x)+6*x),x, algorithm="fricas")
Output:
x^2 - x*e^2 + x*e^x - (x - e^2 + e^x)*log(1/4*x*log(x) + 3/2*x)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).
Time = 7.52 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx=x^{2} - x \log {\left (\frac {x \log {\left (x \right )}}{4} + \frac {3 x}{2} \right )} - x e^{2} + \left (x - \log {\left (\frac {x \log {\left (x \right )}}{4} + \frac {3 x}{2} \right )}\right ) e^{x} + e^{2} \log {\left (x \right )} + e^{2} \log {\left (\log {\left (x \right )} + 6 \right )} \] Input:
integrate((((-exp(x)*x-x)*ln(x)-6*exp(x)*x-6*x)*ln(1/4*x*ln(x)+3/2*x)+((x* *2+x-1)*exp(x)+(1-x)*exp(2)+2*x**2-x)*ln(x)+(6*x**2+6*x-7)*exp(x)+(-6*x+7) *exp(2)+12*x**2-7*x)/(x*ln(x)+6*x),x)
Output:
x**2 - x*log(x*log(x)/4 + 3*x/2) - x*exp(2) + (x - log(x*log(x)/4 + 3*x/2) )*exp(x) + exp(2)*log(x) + exp(2)*log(log(x) + 6)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).
Time = 0.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.03 \[ \int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx=x^{2} - x {\left (e^{2} - 2 \, \log \left (2\right )\right )} + {\left (x + 2 \, \log \left (2\right ) - \log \left (x\right )\right )} e^{x} - {\left (x - e^{2}\right )} \log \left (x\right ) - {\left (x + 6 \, e^{2} + e^{x}\right )} \log \left (\log \left (x\right ) + 6\right ) + 7 \, e^{2} \log \left (\log \left (x\right ) + 6\right ) \] Input:
integrate((((-exp(x)*x-x)*log(x)-6*exp(x)*x-6*x)*log(1/4*x*log(x)+3/2*x)+( (x^2+x-1)*exp(x)+(1-x)*exp(2)+2*x^2-x)*log(x)+(6*x^2+6*x-7)*exp(x)+(-6*x+7 )*exp(2)+12*x^2-7*x)/(x*log(x)+6*x),x, algorithm="maxima")
Output:
x^2 - x*(e^2 - 2*log(2)) + (x + 2*log(2) - log(x))*e^x - (x - e^2)*log(x) - (x + 6*e^2 + e^x)*log(log(x) + 6) + 7*e^2*log(log(x) + 6)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (23) = 46\).
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.17 \[ \int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx=x^{2} - x e^{2} + x e^{x} + 2 \, x \log \left (2\right ) + 2 \, e^{x} \log \left (2\right ) - x \log \left (x\right ) + e^{2} \log \left (x\right ) - e^{x} \log \left (x\right ) - x \log \left (\log \left (x\right ) + 6\right ) + e^{2} \log \left (\log \left (x\right ) + 6\right ) - e^{x} \log \left (\log \left (x\right ) + 6\right ) \] Input:
integrate((((-exp(x)*x-x)*log(x)-6*exp(x)*x-6*x)*log(1/4*x*log(x)+3/2*x)+( (x^2+x-1)*exp(x)+(1-x)*exp(2)+2*x^2-x)*log(x)+(6*x^2+6*x-7)*exp(x)+(-6*x+7 )*exp(2)+12*x^2-7*x)/(x*log(x)+6*x),x, algorithm="giac")
Output:
x^2 - x*e^2 + x*e^x + 2*x*log(2) + 2*e^x*log(2) - x*log(x) + e^2*log(x) - e^x*log(x) - x*log(log(x) + 6) + e^2*log(log(x) + 6) - e^x*log(log(x) + 6)
Time = 1.75 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx={\mathrm {e}}^2\,\ln \left (\ln \left (x\right )+6\right )-\ln \left (\frac {3\,x}{2}+\frac {x\,\ln \left (x\right )}{4}\right )\,\left (x+{\mathrm {e}}^x\right )-x\,{\mathrm {e}}^2+{\mathrm {e}}^2\,\ln \left (x\right )+x\,{\mathrm {e}}^x+x^2 \] Input:
int(-(7*x + log((3*x)/2 + (x*log(x))/4)*(6*x + 6*x*exp(x) + log(x)*(x + x* exp(x))) - exp(x)*(6*x + 6*x^2 - 7) - 12*x^2 + log(x)*(x + exp(2)*(x - 1) - 2*x^2 - exp(x)*(x + x^2 - 1)) + exp(2)*(6*x - 7))/(6*x + x*log(x)),x)
Output:
exp(2)*log(log(x) + 6) - log((3*x)/2 + (x*log(x))/4)*(x + exp(x)) - x*exp( 2) + exp(2)*log(x) + x*exp(x) + x^2
Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {e^2 (7-6 x)-7 x+12 x^2+e^x \left (-7+6 x+6 x^2\right )+\left (e^2 (1-x)-x+2 x^2+e^x \left (-1+x+x^2\right )\right ) \log (x)+\left (-6 x-6 e^x x+\left (-x-e^x x\right ) \log (x)\right ) \log \left (\frac {1}{4} (6 x+x \log (x))\right )}{6 x+x \log (x)} \, dx=-e^{x} \mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x}{4}+\frac {3 x}{2}\right )+e^{x} x +\mathrm {log}\left (\mathrm {log}\left (x \right )+6\right ) e^{2}-\mathrm {log}\left (\frac {\mathrm {log}\left (x \right ) x}{4}+\frac {3 x}{2}\right ) x +\mathrm {log}\left (x \right ) e^{2}-e^{2} x +x^{2} \] Input:
int((((-exp(x)*x-x)*log(x)-6*exp(x)*x-6*x)*log(1/4*x*log(x)+3/2*x)+((x^2+x -1)*exp(x)+(1-x)*exp(2)+2*x^2-x)*log(x)+(6*x^2+6*x-7)*exp(x)+(-6*x+7)*exp( 2)+12*x^2-7*x)/(x*log(x)+6*x),x)
Output:
- e**x*log((log(x)*x + 6*x)/4) + e**x*x + log(log(x) + 6)*e**2 - log((log (x)*x + 6*x)/4)*x + log(x)*e**2 - e**2*x + x**2