Integrand size = 120, antiderivative size = 33 \[ \int \frac {150 x^2-500 x^4+100 e^x x^4+30 \log (x)+\left (-15-100 x^2+20 e^x x^2\right ) \log ^2(x)+\left (-5+e^x\right ) \log ^4(x)}{-150 x^3+500 x^4+100 e^x x^4-500 x^5+\left (-15 x+100 x^2+20 e^x x^2-100 x^3\right ) \log ^2(x)+\left (5+e^x-5 x\right ) \log ^4(x)} \, dx=\log \left (-1+x+\frac {1}{5} \left (-e^x+\frac {3}{2 x+\frac {\log ^2(x)}{5 x}}\right )\right ) \]
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.79 \[ \int \frac {150 x^2-500 x^4+100 e^x x^4+30 \log (x)+\left (-15-100 x^2+20 e^x x^2\right ) \log ^2(x)+\left (-5+e^x\right ) \log ^4(x)}{-150 x^3+500 x^4+100 e^x x^4-500 x^5+\left (-15 x+100 x^2+20 e^x x^2-100 x^3\right ) \log ^2(x)+\left (5+e^x-5 x\right ) \log ^4(x)} \, dx=-\log \left (10 x^2+\log ^2(x)\right )+\log \left (15 x-50 x^2-10 e^x x^2+50 x^3-5 \log ^2(x)-e^x \log ^2(x)+5 x \log ^2(x)\right ) \]
Integrate[(150*x^2 - 500*x^4 + 100*E^x*x^4 + 30*Log[x] + (-15 - 100*x^2 + 20*E^x*x^2)*Log[x]^2 + (-5 + E^x)*Log[x]^4)/(-150*x^3 + 500*x^4 + 100*E^x* x^4 - 500*x^5 + (-15*x + 100*x^2 + 20*E^x*x^2 - 100*x^3)*Log[x]^2 + (5 + E ^x - 5*x)*Log[x]^4),x]
-Log[10*x^2 + Log[x]^2] + Log[15*x - 50*x^2 - 10*E^x*x^2 + 50*x^3 - 5*Log[ x]^2 - E^x*Log[x]^2 + 5*x*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.
\(\displaystyle \int \frac {100 e^x x^4-500 x^4+150 x^2+\left (20 e^x x^2-100 x^2-15\right ) \log ^2(x)+\left (e^x-5\right ) \log ^4(x)+30 \log (x)}{-500 x^5+100 e^x x^4+500 x^4-150 x^3+\left (-100 x^3+20 e^x x^2+100 x^2-15 x\right ) \log ^2(x)+\left (-5 x+e^x+5\right ) \log ^4(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-100 e^x x^4+500 x^4-150 x^2-\left (20 e^x x^2-100 x^2-15\right ) \log ^2(x)-\left (e^x-5\right ) \log ^4(x)-30 \log (x)}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+15 x+5 x \log ^2(x)-e^x \log ^2(x)-5 \log ^2(x)\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (1-\frac {5 \left (100 x^5-200 x^4+30 x^3+20 x^3 \log ^2(x)+30 x^2-40 x^2 \log ^2(x)+x \log ^4(x)-2 \log ^4(x)+3 x \log ^2(x)-3 \log ^2(x)+6 \log (x)\right )}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+15 x+5 x \log ^2(x)-e^x \log ^2(x)-5 \log ^2(x)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -150 \int \frac {x^2}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx-150 \int \frac {x^3}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx-30 \int \frac {\log (x)}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx+15 \int \frac {\log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx-15 \int \frac {x \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx+200 \int \frac {x^2 \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx-100 \int \frac {x^3 \log ^2(x)}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx+10 \int \frac {\log ^4(x)}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx-5 \int \frac {x \log ^4(x)}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx-500 \int \frac {x^5}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx+1000 \int \frac {x^4}{\left (10 x^2+\log ^2(x)\right ) \left (50 x^3-10 e^x x^2-50 x^2+5 \log ^2(x) x+15 x-e^x \log ^2(x)-5 \log ^2(x)\right )}dx+x\) |
Int[(150*x^2 - 500*x^4 + 100*E^x*x^4 + 30*Log[x] + (-15 - 100*x^2 + 20*E^x *x^2)*Log[x]^2 + (-5 + E^x)*Log[x]^4)/(-150*x^3 + 500*x^4 + 100*E^x*x^4 - 500*x^5 + (-15*x + 100*x^2 + 20*E^x*x^2 - 100*x^3)*Log[x]^2 + (5 + E^x - 5 *x)*Log[x]^4),x]
3.9.82.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).
Time = 0.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70
method | result | size |
parallelrisch | \(-\ln \left (x^{2}+\frac {\ln \left (x \right )^{2}}{10}\right )+\ln \left (x^{3}-\frac {{\mathrm e}^{x} x^{2}}{5}+\frac {x \ln \left (x \right )^{2}}{10}-\frac {{\mathrm e}^{x} \ln \left (x \right )^{2}}{50}-x^{2}-\frac {\ln \left (x \right )^{2}}{10}+\frac {3 x}{10}\right )\) | \(56\) |
risch | \(\ln \left ({\mathrm e}^{x}-5 x +5\right )+\ln \left (\ln \left (x \right )^{2}+\frac {5 x \left (10 x^{2}-2 \,{\mathrm e}^{x} x -10 x +3\right )}{5 x -{\mathrm e}^{x}-5}\right )-\ln \left (10 x^{2}+\ln \left (x \right )^{2}\right )\) | \(58\) |
int(((exp(x)-5)*ln(x)^4+(20*exp(x)*x^2-100*x^2-15)*ln(x)^2+30*ln(x)+100*ex p(x)*x^4-500*x^4+150*x^2)/((exp(x)-5*x+5)*ln(x)^4+(20*exp(x)*x^2-100*x^3+1 00*x^2-15*x)*ln(x)^2+100*exp(x)*x^4-500*x^5+500*x^4-150*x^3),x,method=_RET URNVERBOSE)
-ln(x^2+1/10*ln(x)^2)+ln(x^3-1/5*exp(x)*x^2+1/10*x*ln(x)^2-1/50*exp(x)*ln( x)^2-x^2-1/10*ln(x)^2+3/10*x)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).
Time = 0.30 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {150 x^2-500 x^4+100 e^x x^4+30 \log (x)+\left (-15-100 x^2+20 e^x x^2\right ) \log ^2(x)+\left (-5+e^x\right ) \log ^4(x)}{-150 x^3+500 x^4+100 e^x x^4-500 x^5+\left (-15 x+100 x^2+20 e^x x^2-100 x^3\right ) \log ^2(x)+\left (5+e^x-5 x\right ) \log ^4(x)} \, dx=-\log \left (10 \, x^{2} + \log \left (x\right )^{2}\right ) + \log \left (-5 \, x + e^{x} + 5\right ) + \log \left (\frac {50 \, x^{3} - 10 \, x^{2} e^{x} + {\left (5 \, x - e^{x} - 5\right )} \log \left (x\right )^{2} - 50 \, x^{2} + 15 \, x}{5 \, x - e^{x} - 5}\right ) \]
integrate(((exp(x)-5)*log(x)^4+(20*exp(x)*x^2-100*x^2-15)*log(x)^2+30*log( x)+100*exp(x)*x^4-500*x^4+150*x^2)/((exp(x)-5*x+5)*log(x)^4+(20*exp(x)*x^2 -100*x^3+100*x^2-15*x)*log(x)^2+100*exp(x)*x^4-500*x^5+500*x^4-150*x^3),x, algorithm=\
-log(10*x^2 + log(x)^2) + log(-5*x + e^x + 5) + log((50*x^3 - 10*x^2*e^x + (5*x - e^x - 5)*log(x)^2 - 50*x^2 + 15*x)/(5*x - e^x - 5))
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.47 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {150 x^2-500 x^4+100 e^x x^4+30 \log (x)+\left (-15-100 x^2+20 e^x x^2\right ) \log ^2(x)+\left (-5+e^x\right ) \log ^4(x)}{-150 x^3+500 x^4+100 e^x x^4-500 x^5+\left (-15 x+100 x^2+20 e^x x^2-100 x^3\right ) \log ^2(x)+\left (5+e^x-5 x\right ) \log ^4(x)} \, dx=\log {\left (e^{x} + \frac {- 50 x^{3} + 50 x^{2} - 5 x \log {\left (x \right )}^{2} - 15 x + 5 \log {\left (x \right )}^{2}}{10 x^{2} + \log {\left (x \right )}^{2}} \right )} \]
integrate(((exp(x)-5)*ln(x)**4+(20*exp(x)*x**2-100*x**2-15)*ln(x)**2+30*ln (x)+100*exp(x)*x**4-500*x**4+150*x**2)/((exp(x)-5*x+5)*ln(x)**4+(20*exp(x) *x**2-100*x**3+100*x**2-15*x)*ln(x)**2+100*exp(x)*x**4-500*x**5+500*x**4-1 50*x**3),x)
log(exp(x) + (-50*x**3 + 50*x**2 - 5*x*log(x)**2 - 15*x + 5*log(x)**2)/(10 *x**2 + log(x)**2))
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \frac {150 x^2-500 x^4+100 e^x x^4+30 \log (x)+\left (-15-100 x^2+20 e^x x^2\right ) \log ^2(x)+\left (-5+e^x\right ) \log ^4(x)}{-150 x^3+500 x^4+100 e^x x^4-500 x^5+\left (-15 x+100 x^2+20 e^x x^2-100 x^3\right ) \log ^2(x)+\left (5+e^x-5 x\right ) \log ^4(x)} \, dx=\log \left (-\frac {50 \, x^{3} + 5 \, {\left (x - 1\right )} \log \left (x\right )^{2} - 50 \, x^{2} - {\left (10 \, x^{2} + \log \left (x\right )^{2}\right )} e^{x} + 15 \, x}{10 \, x^{2} + \log \left (x\right )^{2}}\right ) \]
integrate(((exp(x)-5)*log(x)^4+(20*exp(x)*x^2-100*x^2-15)*log(x)^2+30*log( x)+100*exp(x)*x^4-500*x^4+150*x^2)/((exp(x)-5*x+5)*log(x)^4+(20*exp(x)*x^2 -100*x^3+100*x^2-15*x)*log(x)^2+100*exp(x)*x^4-500*x^5+500*x^4-150*x^3),x, algorithm=\
log(-(50*x^3 + 5*(x - 1)*log(x)^2 - 50*x^2 - (10*x^2 + log(x)^2)*e^x + 15* x)/(10*x^2 + log(x)^2))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.54 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {150 x^2-500 x^4+100 e^x x^4+30 \log (x)+\left (-15-100 x^2+20 e^x x^2\right ) \log ^2(x)+\left (-5+e^x\right ) \log ^4(x)}{-150 x^3+500 x^4+100 e^x x^4-500 x^5+\left (-15 x+100 x^2+20 e^x x^2-100 x^3\right ) \log ^2(x)+\left (5+e^x-5 x\right ) \log ^4(x)} \, dx=\log \left (-50 \, x^{3} + 10 \, x^{2} e^{x} - 5 \, x \log \left (x\right )^{2} + e^{x} \log \left (x\right )^{2} + 50 \, x^{2} + 5 \, \log \left (x\right )^{2} - 15 \, x\right ) - \log \left (10 \, x^{2} + \log \left (x\right )^{2}\right ) \]
integrate(((exp(x)-5)*log(x)^4+(20*exp(x)*x^2-100*x^2-15)*log(x)^2+30*log( x)+100*exp(x)*x^4-500*x^4+150*x^2)/((exp(x)-5*x+5)*log(x)^4+(20*exp(x)*x^2 -100*x^3+100*x^2-15*x)*log(x)^2+100*exp(x)*x^4-500*x^5+500*x^4-150*x^3),x, algorithm=\
log(-50*x^3 + 10*x^2*e^x - 5*x*log(x)^2 + e^x*log(x)^2 + 50*x^2 + 5*log(x) ^2 - 15*x) - log(10*x^2 + log(x)^2)
Timed out. \[ \int \frac {150 x^2-500 x^4+100 e^x x^4+30 \log (x)+\left (-15-100 x^2+20 e^x x^2\right ) \log ^2(x)+\left (-5+e^x\right ) \log ^4(x)}{-150 x^3+500 x^4+100 e^x x^4-500 x^5+\left (-15 x+100 x^2+20 e^x x^2-100 x^3\right ) \log ^2(x)+\left (5+e^x-5 x\right ) \log ^4(x)} \, dx=\int \frac {30\,\ln \left (x\right )+100\,x^4\,{\mathrm {e}}^x-{\ln \left (x\right )}^2\,\left (100\,x^2-20\,x^2\,{\mathrm {e}}^x+15\right )+150\,x^2-500\,x^4+{\ln \left (x\right )}^4\,\left ({\mathrm {e}}^x-5\right )}{100\,x^4\,{\mathrm {e}}^x-{\ln \left (x\right )}^2\,\left (15\,x-20\,x^2\,{\mathrm {e}}^x-100\,x^2+100\,x^3\right )+{\ln \left (x\right )}^4\,\left ({\mathrm {e}}^x-5\,x+5\right )-150\,x^3+500\,x^4-500\,x^5} \,d x \]
int((30*log(x) + 100*x^4*exp(x) - log(x)^2*(100*x^2 - 20*x^2*exp(x) + 15) + 150*x^2 - 500*x^4 + log(x)^4*(exp(x) - 5))/(100*x^4*exp(x) - log(x)^2*(1 5*x - 20*x^2*exp(x) - 100*x^2 + 100*x^3) + log(x)^4*(exp(x) - 5*x + 5) - 1 50*x^3 + 500*x^4 - 500*x^5),x)