Integrand size = 145, antiderivative size = 31 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3}{12-e^{3+x}+e^{2+2 x}-\frac {\log ^2\left (x^2\right )}{x^2}} \]
Time = 0.68 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3 x^2}{\left (12-e^{3+x}+e^{2+2 x}\right ) x^2-\log ^2\left (x^2\right )} \]
Integrate[(3*E^(3 + x)*x^4 - 6*E^(2 + 2*x)*x^4 + 12*x*Log[x^2] - 6*x*Log[x ^2]^2)/(144*x^4 - 24*E^(3 + x)*x^4 + E^(6 + 2*x)*x^4 + E^(4 + 4*x)*x^4 + E ^(2 + 2*x)*(24*x^4 - 2*E^(3 + x)*x^4) + (-24*x^2 + 2*E^(3 + x)*x^2 - 2*E^( 2 + 2*x)*x^2)*Log[x^2]^2 + Log[x^2]^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.
| \(\displaystyle \int \frac {3 e^{x+3} x^4-6 e^{2 x+2} x^4-6 x \log ^2\left (x^2\right )+12 x \log \left (x^2\right )}{-24 e^{x+3} x^4+e^{2 x+6} x^4+e^{4 x+4} x^4+144 x^4+e^{2 x+2} \left (24 x^4-2 e^{x+3} x^4\right )+\log ^4\left (x^2\right )+\left (2 e^{x+3} x^2-2 e^{2 x+2} x^2-24 x^2\right ) \log ^2\left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {3 x \left (-e^{x+2} \left (2 e^x-e\right ) x^3-2 \log ^2\left (x^2\right )+4 \log \left (x^2\right )\right )}{\left (\left (-e^{x+3}+e^{2 x+2}+12\right ) x^2-\log ^2\left (x^2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \int \frac {x \left (e^{x+2} \left (e-2 e^x\right ) x^3-2 \log ^2\left (x^2\right )+4 \log \left (x^2\right )\right )}{\left (\left (12-e^{x+3}+e^{2 x+2}\right ) x^2-\log ^2\left (x^2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 \int \left (-\frac {2 x^2}{-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )}-\frac {\left (e^{x+3} x^3-24 x^3+2 \log ^2\left (x^2\right ) x+2 \log ^2\left (x^2\right )-4 \log \left (x^2\right )\right ) x}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (4 \int \frac {x \log \left (x^2\right )}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx-2 \int \frac {x \log ^2\left (x^2\right )}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx-2 \int \frac {x^2 \log ^2\left (x^2\right )}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx-2 \int \frac {x^2}{-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )}dx+24 \int \frac {x^4}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx-\int \frac {e^{x+3} x^4}{\left (-e^{x+3} x^2+e^{2 x+2} x^2+12 x^2-\log ^2\left (x^2\right )\right )^2}dx\right )\) |
Int[(3*E^(3 + x)*x^4 - 6*E^(2 + 2*x)*x^4 + 12*x*Log[x^2] - 6*x*Log[x^2]^2) /(144*x^4 - 24*E^(3 + x)*x^4 + E^(6 + 2*x)*x^4 + E^(4 + 4*x)*x^4 + E^(2 + 2*x)*(24*x^4 - 2*E^(3 + x)*x^4) + (-24*x^2 + 2*E^(3 + x)*x^2 - 2*E^(2 + 2* x)*x^2)*Log[x^2]^2 + Log[x^2]^4),x]
3.6.11.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.32 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32
| method | result | size |
| parallelrisch | \(\frac {3 x^{2}}{x^{2} {\mathrm e}^{2+2 x}-x^{2} {\mathrm e}^{3+x}+12 x^{2}-\ln \left (x^{2}\right )^{2}}\) | \(41\) |
| risch | \(-\frac {12 x^{2}}{-\operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4} \pi ^{2}+4 \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3} \pi ^{2}-6 \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2} \pi ^{2}+4 \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right ) \pi ^{2}-\operatorname {csgn}\left (i x^{2}\right )^{6} \pi ^{2}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+16 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-8 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 x^{2} {\mathrm e}^{3+x}-4 x^{2} {\mathrm e}^{2+2 x}-48 x^{2}+16 \ln \left (x \right )^{2}}\) | \(191\) |
int((-6*x*ln(x^2)^2+12*x*ln(x^2)-6*x^4*exp(2+2*x)+3*x^4*exp(3+x))/(ln(x^2) ^4+(-2*x^2*exp(2+2*x)+2*x^2*exp(3+x)-24*x^2)*ln(x^2)^2+x^4*exp(2+2*x)^2+(- 2*x^4*exp(3+x)+24*x^4)*exp(2+2*x)+x^4*exp(3+x)^2-24*x^4*exp(3+x)+144*x^4), x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3 \, x^{2} e^{4}}{12 \, x^{2} e^{4} + x^{2} e^{\left (2 \, x + 6\right )} - x^{2} e^{\left (x + 7\right )} - e^{4} \log \left (x^{2}\right )^{2}} \]
integrate((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2+2*x)+3*x^4*exp(3+x))/ (log(x^2)^4+(-2*x^2*exp(2+2*x)+2*x^2*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2 +2*x)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2+2*x)+x^4*exp(3+x)^2-24*x^4*exp(3+x) +144*x^4),x, algorithm=\
Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3 x^{2} e^{4}}{- x^{2} e^{4} e^{x + 3} + x^{2} e^{2 x + 6} + 12 x^{2} e^{4} - e^{4} \log {\left (x^{2} \right )}^{2}} \]
integrate((-6*x*ln(x**2)**2+12*x*ln(x**2)-6*x**4*exp(2+2*x)+3*x**4*exp(3+x ))/(ln(x**2)**4+(-2*x**2*exp(2+2*x)+2*x**2*exp(3+x)-24*x**2)*ln(x**2)**2+x **4*exp(2+2*x)**2+(-2*x**4*exp(3+x)+24*x**4)*exp(2+2*x)+x**4*exp(3+x)**2-2 4*x**4*exp(3+x)+144*x**4),x)
3*x**2*exp(4)/(-x**2*exp(4)*exp(x + 3) + x**2*exp(2*x + 6) + 12*x**2*exp(4 ) - exp(4)*log(x**2)**2)
Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\frac {3 \, x^{2}}{x^{2} e^{\left (2 \, x + 2\right )} - x^{2} e^{\left (x + 3\right )} + 12 \, x^{2} - 4 \, \log \left (x\right )^{2}} \]
integrate((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2+2*x)+3*x^4*exp(3+x))/ (log(x^2)^4+(-2*x^2*exp(2+2*x)+2*x^2*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2 +2*x)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2+2*x)+x^4*exp(3+x)^2-24*x^4*exp(3+x) +144*x^4),x, algorithm=\
Timed out. \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\text {Timed out} \]
integrate((-6*x*log(x^2)^2+12*x*log(x^2)-6*x^4*exp(2+2*x)+3*x^4*exp(3+x))/ (log(x^2)^4+(-2*x^2*exp(2+2*x)+2*x^2*exp(3+x)-24*x^2)*log(x^2)^2+x^4*exp(2 +2*x)^2+(-2*x^4*exp(3+x)+24*x^4)*exp(2+2*x)+x^4*exp(3+x)^2-24*x^4*exp(3+x) +144*x^4),x, algorithm=\
Timed out. \[ \int \frac {3 e^{3+x} x^4-6 e^{2+2 x} x^4+12 x \log \left (x^2\right )-6 x \log ^2\left (x^2\right )}{144 x^4-24 e^{3+x} x^4+e^{6+2 x} x^4+e^{4+4 x} x^4+e^{2+2 x} \left (24 x^4-2 e^{3+x} x^4\right )+\left (-24 x^2+2 e^{3+x} x^2-2 e^{2+2 x} x^2\right ) \log ^2\left (x^2\right )+\log ^4\left (x^2\right )} \, dx=\int \frac {12\,x\,\ln \left (x^2\right )+3\,x^4\,{\mathrm {e}}^{x+3}-6\,x\,{\ln \left (x^2\right )}^2-6\,x^4\,{\mathrm {e}}^{2\,x+2}}{{\ln \left (x^2\right )}^4-24\,x^4\,{\mathrm {e}}^{x+3}-{\mathrm {e}}^{2\,x+2}\,\left (2\,x^4\,{\mathrm {e}}^{x+3}-24\,x^4\right )+x^4\,{\mathrm {e}}^{2\,x+6}+x^4\,{\mathrm {e}}^{4\,x+4}+144\,x^4-{\ln \left (x^2\right )}^2\,\left (2\,x^2\,{\mathrm {e}}^{2\,x+2}-2\,x^2\,{\mathrm {e}}^{x+3}+24\,x^2\right )} \,d x \]
int((12*x*log(x^2) + 3*x^4*exp(x + 3) - 6*x*log(x^2)^2 - 6*x^4*exp(2*x + 2 ))/(log(x^2)^4 - 24*x^4*exp(x + 3) - exp(2*x + 2)*(2*x^4*exp(x + 3) - 24*x ^4) + x^4*exp(2*x + 6) + x^4*exp(4*x + 4) + 144*x^4 - log(x^2)^2*(2*x^2*ex p(2*x + 2) - 2*x^2*exp(x + 3) + 24*x^2)),x)