Integrand size = 89, antiderivative size = 25 \[ \int \frac {e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} \left (-4 x^2+\left (12 x^2-4 x^3\right ) \log (x)+4 x^3 \log ^2(x)\right )}{20 \log ^2(x)+20 x \log ^3(x)+5 x^2 \log ^4(x)} \, dx=e^{3+\frac {2 x^2}{5 \log (x) \left (\frac {2}{x}+\log (x)\right )}} \]
Time = 3.95 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} \left (-4 x^2+\left (12 x^2-4 x^3\right ) \log (x)+4 x^3 \log ^2(x)\right )}{20 \log ^2(x)+20 x \log ^3(x)+5 x^2 \log ^4(x)} \, dx=e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} \]
Integrate[(E^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(10*Log[x] + 5*x*Log[x]^ 2))*(-4*x^2 + (12*x^2 - 4*x^3)*Log[x] + 4*x^3*Log[x]^2))/(20*Log[x]^2 + 20 *x*Log[x]^3 + 5*x^2*Log[x]^4),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.
\(\displaystyle \int \frac {\left (4 x^3 \log ^2(x)-4 x^2+\left (12 x^2-4 x^3\right ) \log (x)\right ) \exp \left (\frac {2 x^3+15 x \log ^2(x)+30 \log (x)}{5 x \log ^2(x)+10 \log (x)}\right )}{5 x^2 \log ^4(x)+20 x \log ^3(x)+20 \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {4 x^2 \left (x \log ^2(x)-x \log (x)+3 \log (x)-1\right ) \exp \left (\frac {2 x^3+15 x \log ^2(x)+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right )}{5 \log ^2(x) (x \log (x)+2)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{5} \int -\frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x}{5 \log (x) (x \log (x)+2)}+\frac {6}{x \log (x)+2}\right ) x^2 \left (-x \log ^2(x)+x \log (x)-3 \log (x)+1\right )}{\log ^2(x) (x \log (x)+2)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {4}{5} \int \frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x}{5 \log (x) (x \log (x)+2)}+\frac {6}{x \log (x)+2}\right ) x^2 \left (-x \log ^2(x)+x \log (x)-3 \log (x)+1\right )}{\log ^2(x) (x \log (x)+2)^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\frac {4}{5} \int \frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) x^2 \left (-x \log ^2(x)+x \log (x)-3 \log (x)+1\right )}{\log ^2(x) (x \log (x)+2)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {4}{5} \int \left (\frac {3 \exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) x^3}{4 (x \log (x)+2)}-\frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) (x-2) x^3}{4 (x \log (x)+2)^2}-\frac {3 \exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) x^2}{4 \log (x)}+\frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) x^2}{4 \log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4}{5} \left (\frac {1}{2} \int \frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) x^3}{(x \log (x)+2)^2}dx+\frac {3}{4} \int \frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) x^3}{x \log (x)+2}dx-\frac {1}{4} \int \frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) x^4}{(x \log (x)+2)^2}dx+\frac {1}{4} \int \frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) x^2}{\log ^2(x)}dx-\frac {3}{4} \int \frac {\exp \left (\frac {2 x^3+15 \log ^2(x) x+30 \log (x)}{5 \log (x) (x \log (x)+2)}\right ) x^2}{\log (x)}dx\right )\) |
Int[(E^((2*x^3 + 30*Log[x] + 15*x*Log[x]^2)/(10*Log[x] + 5*x*Log[x]^2))*(- 4*x^2 + (12*x^2 - 4*x^3)*Log[x] + 4*x^3*Log[x]^2))/(20*Log[x]^2 + 20*x*Log [x]^3 + 5*x^2*Log[x]^4),x]
3.22.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 1.49 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32
method | result | size |
risch | \({\mathrm e}^{\frac {15 x \ln \left (x \right )^{2}+30 \ln \left (x \right )+2 x^{3}}{5 \ln \left (x \right ) \left (x \ln \left (x \right )+2\right )}}\) | \(33\) |
parallelrisch | \({\mathrm e}^{\frac {15 x \ln \left (x \right )^{2}+30 \ln \left (x \right )+2 x^{3}}{5 \ln \left (x \right ) \left (x \ln \left (x \right )+2\right )}}\) | \(33\) |
int((4*x^3*ln(x)^2+(-4*x^3+12*x^2)*ln(x)-4*x^2)*exp((15*x*ln(x)^2+30*ln(x) +2*x^3)/(5*x*ln(x)^2+10*ln(x)))/(5*x^2*ln(x)^4+20*x*ln(x)^3+20*ln(x)^2),x, method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} \left (-4 x^2+\left (12 x^2-4 x^3\right ) \log (x)+4 x^3 \log ^2(x)\right )}{20 \log ^2(x)+20 x \log ^3(x)+5 x^2 \log ^4(x)} \, dx=e^{\left (\frac {2 \, x^{3} + 15 \, x \log \left (x\right )^{2} + 30 \, \log \left (x\right )}{5 \, {\left (x \log \left (x\right )^{2} + 2 \, \log \left (x\right )\right )}}\right )} \]
integrate((4*x^3*log(x)^2+(-4*x^3+12*x^2)*log(x)-4*x^2)*exp((15*x*log(x)^2 +30*log(x)+2*x^3)/(5*x*log(x)^2+10*log(x)))/(5*x^2*log(x)^4+20*x*log(x)^3+ 20*log(x)^2),x, algorithm=\
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} \left (-4 x^2+\left (12 x^2-4 x^3\right ) \log (x)+4 x^3 \log ^2(x)\right )}{20 \log ^2(x)+20 x \log ^3(x)+5 x^2 \log ^4(x)} \, dx=e^{\frac {2 x^{3} + 15 x \log {\left (x \right )}^{2} + 30 \log {\left (x \right )}}{5 x \log {\left (x \right )}^{2} + 10 \log {\left (x \right )}}} \]
integrate((4*x**3*ln(x)**2+(-4*x**3+12*x**2)*ln(x)-4*x**2)*exp((15*x*ln(x) **2+30*ln(x)+2*x**3)/(5*x*ln(x)**2+10*ln(x)))/(5*x**2*ln(x)**4+20*x*ln(x)* *3+20*ln(x)**2),x)
Time = 0.35 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} \left (-4 x^2+\left (12 x^2-4 x^3\right ) \log (x)+4 x^3 \log ^2(x)\right )}{20 \log ^2(x)+20 x \log ^3(x)+5 x^2 \log ^4(x)} \, dx=e^{\left (\frac {2 \, x^{2}}{5 \, \log \left (x\right )^{2}} - \frac {16}{5 \, {\left (x \log \left (x\right )^{5} + 2 \, \log \left (x\right )^{4}\right )}} - \frac {4 \, x}{5 \, \log \left (x\right )^{3}} + \frac {8}{5 \, \log \left (x\right )^{4}} + 3\right )} \]
integrate((4*x^3*log(x)^2+(-4*x^3+12*x^2)*log(x)-4*x^2)*exp((15*x*log(x)^2 +30*log(x)+2*x^3)/(5*x*log(x)^2+10*log(x)))/(5*x^2*log(x)^4+20*x*log(x)^3+ 20*log(x)^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (22) = 44\).
Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.28 \[ \int \frac {e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} \left (-4 x^2+\left (12 x^2-4 x^3\right ) \log (x)+4 x^3 \log ^2(x)\right )}{20 \log ^2(x)+20 x \log ^3(x)+5 x^2 \log ^4(x)} \, dx=e^{\left (\frac {2 \, x^{3}}{5 \, {\left (x \log \left (x\right )^{2} + 2 \, \log \left (x\right )\right )}} + \frac {3 \, x \log \left (x\right )^{2}}{x \log \left (x\right )^{2} + 2 \, \log \left (x\right )} + \frac {6 \, \log \left (x\right )}{x \log \left (x\right )^{2} + 2 \, \log \left (x\right )}\right )} \]
integrate((4*x^3*log(x)^2+(-4*x^3+12*x^2)*log(x)-4*x^2)*exp((15*x*log(x)^2 +30*log(x)+2*x^3)/(5*x*log(x)^2+10*log(x)))/(5*x^2*log(x)^4+20*x*log(x)^3+ 20*log(x)^2),x, algorithm=\
e^(2/5*x^3/(x*log(x)^2 + 2*log(x)) + 3*x*log(x)^2/(x*log(x)^2 + 2*log(x)) + 6*log(x)/(x*log(x)^2 + 2*log(x)))
Time = 11.85 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {e^{\frac {2 x^3+30 \log (x)+15 x \log ^2(x)}{10 \log (x)+5 x \log ^2(x)}} \left (-4 x^2+\left (12 x^2-4 x^3\right ) \log (x)+4 x^3 \log ^2(x)\right )}{20 \log ^2(x)+20 x \log ^3(x)+5 x^2 \log ^4(x)} \, dx=x^{\frac {3\,x}{x\,\ln \left (x\right )+2}}\,x^{\frac {6}{x\,{\ln \left (x\right )}^2+2\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {2\,x^3}{5\,x\,{\ln \left (x\right )}^2+10\,\ln \left (x\right )}} \]
int((exp((30*log(x) + 15*x*log(x)^2 + 2*x^3)/(10*log(x) + 5*x*log(x)^2))*( log(x)*(12*x^2 - 4*x^3) + 4*x^3*log(x)^2 - 4*x^2))/(20*x*log(x)^3 + 20*log (x)^2 + 5*x^2*log(x)^4),x)