Integrand size = 96, antiderivative size = 22 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=\left (-1+\frac {1}{x}\right ) \log \left (e^x \left (-3-\frac {14 x}{3}\right ) x+\log (x)\right ) \] Output:
ln(ln(x)+exp(x)*(-3-14/3*x)*x)*(-1+1/x)
Time = 0.59 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=-\log \left (9 e^x x+14 e^x x^2-3 \log (x)\right )+\frac {\log \left (-\frac {1}{3} e^x x (9+14 x)+\log (x)\right )}{x} \] Input:
Integrate[(3 - 3*x + E^x*(-9*x - 28*x^2 + 23*x^3 + 14*x^4) + (E^x*(9*x + 1 4*x^2) - 3*Log[x])*Log[(E^x*(-9*x - 14*x^2) + 3*Log[x])/3])/(E^x*(-9*x^3 - 14*x^4) + 3*x^2*Log[x]),x]
Output:
-Log[9*E^x*x + 14*E^x*x^2 - 3*Log[x]] + Log[-1/3*(E^x*x*(9 + 14*x)) + Log[ x]]/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.
\(\displaystyle \int \frac {\left (e^x \left (14 x^2+9 x\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-14 x^2-9 x\right )+3 \log (x)\right )\right )+e^x \left (14 x^4+23 x^3-28 x^2-9 x\right )-3 x+3}{3 x^2 \log (x)+e^x \left (-14 x^4-9 x^3\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-14 x^3-23 x^2+28 x-14 x \log \left (\log (x)-\frac {1}{3} e^x x (14 x+9)\right )-9 \log \left (\log (x)-\frac {1}{3} e^x x (14 x+9)\right )+9}{x^2 (14 x+9)}-\frac {3 (x-1) \left (14 x^2 \log (x)-14 x+37 x \log (x)+9 \log (x)-9\right )}{x^2 (14 x+9) \left (14 e^x x^2+9 e^x x-3 \log (x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \int \frac {1}{x^2 \left (14 e^x x^2+9 e^x x-3 \log (x)\right )}dx+3 \int \frac {1}{x \left (14 e^x x^2+9 e^x x-3 \log (x)\right )}dx-3 \int \frac {\log (x)}{14 e^x x^2+9 e^x x-3 \log (x)}dx+3 \int \frac {\log (x)}{x^2 \left (14 e^x x^2+9 e^x x-3 \log (x)\right )}dx+\frac {14}{3} \int \frac {\log (x)}{x \left (14 e^x x^2+9 e^x x-3 \log (x)\right )}dx-\frac {322}{3} \int \frac {\log (x)}{(14 x+9) \left (14 e^x x^2+9 e^x x-3 \log (x)\right )}dx-\int \frac {\log \left (\log (x)-\frac {1}{3} e^x x (14 x+9)\right )}{x^2}dx-x-\frac {1}{x}+\frac {14 \log (x)}{9}-\frac {23}{9} \log (14 x+9)\) |
Input:
Int[(3 - 3*x + E^x*(-9*x - 28*x^2 + 23*x^3 + 14*x^4) + (E^x*(9*x + 14*x^2) - 3*Log[x])*Log[(E^x*(-9*x - 14*x^2) + 3*Log[x])/3])/(E^x*(-9*x^3 - 14*x^ 4) + 3*x^2*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(40\) vs. \(2(19)=38\).
Time = 2.92 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86
method | result | size |
risch | \(\frac {\ln \left (\ln \left (x \right )+\frac {\left (-14 x^{2}-9 x \right ) {\mathrm e}^{x}}{3}\right )}{x}-\ln \left (\ln \left (x \right )-\frac {14 \,{\mathrm e}^{x} x^{2}}{3}-3 \,{\mathrm e}^{x} x \right )\) | \(41\) |
parallelrisch | \(\frac {420 \ln \left ({\mathrm e}^{x} x^{2}+\frac {9 \,{\mathrm e}^{x} x}{14}-\frac {3 \ln \left (x \right )}{14}\right ) x -1176 \ln \left (\ln \left (x \right )+\frac {\left (-14 x^{2}-9 x \right ) {\mathrm e}^{x}}{3}\right ) x +756 \ln \left (\ln \left (x \right )+\frac {\left (-14 x^{2}-9 x \right ) {\mathrm e}^{x}}{3}\right )}{756 x}\) | \(66\) |
Input:
int(((-3*ln(x)+(14*x^2+9*x)*exp(x))*ln(ln(x)+1/3*(-14*x^2-9*x)*exp(x))+(14 *x^4+23*x^3-28*x^2-9*x)*exp(x)-3*x+3)/(3*x^2*ln(x)+(-14*x^4-9*x^3)*exp(x)) ,x,method=_RETURNVERBOSE)
Output:
1/x*ln(ln(x)+1/3*(-14*x^2-9*x)*exp(x))-ln(ln(x)-14/3*exp(x)*x^2-3*exp(x)*x )
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=-\frac {{\left (x - 1\right )} \log \left (-\frac {1}{3} \, {\left (14 \, x^{2} + 9 \, x\right )} e^{x} + \log \left (x\right )\right )}{x} \] Input:
integrate(((-3*log(x)+(14*x^2+9*x)*exp(x))*log(log(x)+1/3*(-14*x^2-9*x)*ex p(x))+(14*x^4+23*x^3-28*x^2-9*x)*exp(x)-3*x+3)/(3*x^2*log(x)+(-14*x^4-9*x^ 3)*exp(x)),x, algorithm="fricas")
Output:
-(x - 1)*log(-1/3*(14*x^2 + 9*x)*e^x + log(x))/x
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.76 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=- \log {\left (14 x^{2} + 9 x \right )} - \log {\left (e^{x} - \frac {3 \log {\left (x \right )}}{14 x^{2} + 9 x} \right )} + \frac {\log {\left (\left (- \frac {14 x^{2}}{3} - 3 x\right ) e^{x} + \log {\left (x \right )} \right )}}{x} \] Input:
integrate(((-3*ln(x)+(14*x**2+9*x)*exp(x))*ln(ln(x)+1/3*(-14*x**2-9*x)*exp (x))+(14*x**4+23*x**3-28*x**2-9*x)*exp(x)-3*x+3)/(3*x**2*ln(x)+(-14*x**4-9 *x**3)*exp(x)),x)
Output:
-log(14*x**2 + 9*x) - log(exp(x) - 3*log(x)/(14*x**2 + 9*x)) + log((-14*x* *2/3 - 3*x)*exp(x) + log(x))/x
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (20) = 40\).
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.36 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=-\frac {\log \left (3\right ) - \log \left (-{\left (14 \, x^{2} + 9 \, x\right )} e^{x} + 3 \, \log \left (x\right )\right )}{x} - \log \left (14 \, x + 9\right ) - \log \left (x\right ) - \log \left (\frac {{\left (14 \, x^{2} + 9 \, x\right )} e^{x} - 3 \, \log \left (x\right )}{14 \, x^{2} + 9 \, x}\right ) \] Input:
integrate(((-3*log(x)+(14*x^2+9*x)*exp(x))*log(log(x)+1/3*(-14*x^2-9*x)*ex p(x))+(14*x^4+23*x^3-28*x^2-9*x)*exp(x)-3*x+3)/(3*x^2*log(x)+(-14*x^4-9*x^ 3)*exp(x)),x, algorithm="maxima")
Output:
-(log(3) - log(-(14*x^2 + 9*x)*e^x + 3*log(x)))/x - log(14*x + 9) - log(x) - log(((14*x^2 + 9*x)*e^x - 3*log(x))/(14*x^2 + 9*x))
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=-\frac {x \log \left (14 \, x^{2} e^{x} + 9 \, x e^{x} - 3 \, \log \left (x\right )\right ) + \log \left (3\right ) - \log \left (-14 \, x^{2} e^{x} - 9 \, x e^{x} + 3 \, \log \left (x\right )\right )}{x} \] Input:
integrate(((-3*log(x)+(14*x^2+9*x)*exp(x))*log(log(x)+1/3*(-14*x^2-9*x)*ex p(x))+(14*x^4+23*x^3-28*x^2-9*x)*exp(x)-3*x+3)/(3*x^2*log(x)+(-14*x^4-9*x^ 3)*exp(x)),x, algorithm="giac")
Output:
-(x*log(14*x^2*e^x + 9*x*e^x - 3*log(x)) + log(3) - log(-14*x^2*e^x - 9*x* e^x + 3*log(x)))/x
Timed out. \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=\int \frac {3\,x+\ln \left (\ln \left (x\right )-\frac {{\mathrm {e}}^x\,\left (14\,x^2+9\,x\right )}{3}\right )\,\left (3\,\ln \left (x\right )-{\mathrm {e}}^x\,\left (14\,x^2+9\,x\right )\right )+{\mathrm {e}}^x\,\left (-14\,x^4-23\,x^3+28\,x^2+9\,x\right )-3}{{\mathrm {e}}^x\,\left (14\,x^4+9\,x^3\right )-3\,x^2\,\ln \left (x\right )} \,d x \] Input:
int((3*x + log(log(x) - (exp(x)*(9*x + 14*x^2))/3)*(3*log(x) - exp(x)*(9*x + 14*x^2)) + exp(x)*(9*x + 28*x^2 - 23*x^3 - 14*x^4) - 3)/(exp(x)*(9*x^3 + 14*x^4) - 3*x^2*log(x)),x)
Output:
int((3*x + log(log(x) - (exp(x)*(9*x + 14*x^2))/3)*(3*log(x) - exp(x)*(9*x + 14*x^2)) + exp(x)*(9*x + 28*x^2 - 23*x^3 - 14*x^4) - 3)/(exp(x)*(9*x^3 + 14*x^4) - 3*x^2*log(x)), x)
Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {3-3 x+e^x \left (-9 x-28 x^2+23 x^3+14 x^4\right )+\left (e^x \left (9 x+14 x^2\right )-3 \log (x)\right ) \log \left (\frac {1}{3} \left (e^x \left (-9 x-14 x^2\right )+3 \log (x)\right )\right )}{e^x \left (-9 x^3-14 x^4\right )+3 x^2 \log (x)} \, dx=\frac {\mathrm {log}\left (-\frac {14 e^{x} x^{2}}{3}-3 e^{x} x +\mathrm {log}\left (x \right )\right ) \left (1-x \right )}{x} \] Input:
int(((-3*log(x)+(14*x^2+9*x)*exp(x))*log(log(x)+1/3*(-14*x^2-9*x)*exp(x))+ (14*x^4+23*x^3-28*x^2-9*x)*exp(x)-3*x+3)/(3*x^2*log(x)+(-14*x^4-9*x^3)*exp (x)),x)
Output:
(log(( - 14*e**x*x**2 - 9*e**x*x + 3*log(x))/3)*( - x + 1))/x