Integrand size = 165, antiderivative size = 26 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4}{5+\left (2+e^{12+x^2}\right )^2+\frac {5}{x^2}-\log (x)} \]
Time = 0.73 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4 x^2}{5+\left (9+4 e^{12+x^2}+e^{24+2 x^2}\right ) x^2-x^2 \log (x)} \]
Integrate[(40*x + 4*x^3 - 32*E^(12 + x^2)*x^5 - 16*E^(24 + 2*x^2)*x^5)/(25 + 90*x^2 + 81*x^4 + 8*E^(36 + 3*x^2)*x^4 + E^(48 + 4*x^2)*x^4 + E^(24 + 2 *x^2)*(10*x^2 + 34*x^4) + E^(12 + x^2)*(40*x^2 + 72*x^4) + (-10*x^2 - 18*x ^4 - 8*E^(12 + x^2)*x^4 - 2*E^(24 + 2*x^2)*x^4)*Log[x] + x^4*Log[x]^2),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.
\(\displaystyle \int \frac {4 x^3-32 e^{x^2+12} x^5-16 e^{2 x^2+24} x^5+40 x}{81 x^4+x^4 \log ^2(x)+90 x^2+8 e^{3 x^2+36} x^4+e^{4 x^2+48} x^4+e^{2 x^2+24} \left (34 x^4+10 x^2\right )+e^{x^2+12} \left (72 x^4+40 x^2\right )+\left (-18 x^4-10 x^2-8 e^{x^2+12} x^4-2 e^{2 x^2+24} x^4\right ) \log (x)+25} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {4 x \left (x^2-4 e^{x^2+12} \left (e^{x^2+12}+2\right ) x^4+10\right )}{\left (\left (4 e^{x^2+12}+e^{2 x^2+24}+9\right ) x^2-x^2 \log (x)+5\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \int \frac {x \left (-4 e^{x^2+12} \left (2+e^{x^2+12}\right ) x^4+x^2+10\right )}{\left (\left (9+4 e^{x^2+12}+e^{2 x^2+24}\right ) x^2-\log (x) x^2+5\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 4 \int \left (\frac {x \left (8 e^{x^2+12} x^4-4 \log (x) x^4+36 x^4+21 x^2+10\right )}{\left (4 e^{x^2+12} x^2+e^{2 x^2+24} x^2-\log (x) x^2+9 x^2+5\right )^2}-\frac {4 x^3}{4 e^{x^2+12} x^2+e^{2 x^2+24} x^2-\log (x) x^2+9 x^2+5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (10 \int \frac {x}{\left (4 e^{x^2+12} x^2+e^{2 x^2+24} x^2-\log (x) x^2+9 x^2+5\right )^2}dx+36 \int \frac {x^5}{\left (4 e^{x^2+12} x^2+e^{2 x^2+24} x^2-\log (x) x^2+9 x^2+5\right )^2}dx+8 \int \frac {e^{x^2+12} x^5}{\left (4 e^{x^2+12} x^2+e^{2 x^2+24} x^2-\log (x) x^2+9 x^2+5\right )^2}dx-4 \int \frac {x^5 \log (x)}{\left (-4 e^{x^2+12} x^2-e^{2 x^2+24} x^2+\log (x) x^2-9 x^2-5\right )^2}dx+21 \int \frac {x^3}{\left (4 e^{x^2+12} x^2+e^{2 x^2+24} x^2-\log (x) x^2+9 x^2+5\right )^2}dx-4 \int \frac {x^3}{4 e^{x^2+12} x^2+e^{2 x^2+24} x^2-\log (x) x^2+9 x^2+5}dx\right )\) |
Int[(40*x + 4*x^3 - 32*E^(12 + x^2)*x^5 - 16*E^(24 + 2*x^2)*x^5)/(25 + 90* x^2 + 81*x^4 + 8*E^(36 + 3*x^2)*x^4 + E^(48 + 4*x^2)*x^4 + E^(24 + 2*x^2)* (10*x^2 + 34*x^4) + E^(12 + x^2)*(40*x^2 + 72*x^4) + (-10*x^2 - 18*x^4 - 8 *E^(12 + x^2)*x^4 - 2*E^(24 + 2*x^2)*x^4)*Log[x] + x^4*Log[x]^2),x]
3.24.10.3.1 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 = 0.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73
method | result | size |
risch | \(\frac {4 x^{2}}{{\mathrm e}^{2 x^{2}+24} x^{2}+4 \,{\mathrm e}^{x^{2}+12} x^{2}-x^{2} \ln \left (x \right )+9 x^{2}+5}\) | \(45\) |
parallelrisch | \(-\frac {4 x^{2}}{-{\mathrm e}^{2 x^{2}+24} x^{2}+x^{2} \ln \left (x \right )-4 \,{\mathrm e}^{x^{2}+12} x^{2}-9 x^{2}-5}\) | \(45\) |
int((-16*x^5*exp(x^2+12)^2-32*x^5*exp(x^2+12)+4*x^3+40*x)/(x^4*ln(x)^2+(-2 *x^4*exp(x^2+12)^2-8*x^4*exp(x^2+12)-18*x^4-10*x^2)*ln(x)+x^4*exp(x^2+12)^ 4+8*x^4*exp(x^2+12)^3+(34*x^4+10*x^2)*exp(x^2+12)^2+(72*x^4+40*x^2)*exp(x^ 2+12)+81*x^4+90*x^2+25),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4 \, x^{2}}{x^{2} e^{\left (2 \, x^{2} + 24\right )} + 4 \, x^{2} e^{\left (x^{2} + 12\right )} - x^{2} \log \left (x\right ) + 9 \, x^{2} + 5} \]
integrate((-16*x^5*exp(x^2+12)^2-32*x^5*exp(x^2+12)+4*x^3+40*x)/(x^4*log(x )^2+(-2*x^4*exp(x^2+12)^2-8*x^4*exp(x^2+12)-18*x^4-10*x^2)*log(x)+x^4*exp( x^2+12)^4+8*x^4*exp(x^2+12)^3+(34*x^4+10*x^2)*exp(x^2+12)^2+(72*x^4+40*x^2 )*exp(x^2+12)+81*x^4+90*x^2+25),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4 x^{2}}{4 x^{2} e^{x^{2} + 12} + x^{2} e^{2 x^{2} + 24} - x^{2} \log {\left (x \right )} + 9 x^{2} + 5} \]
integrate((-16*x**5*exp(x**2+12)**2-32*x**5*exp(x**2+12)+4*x**3+40*x)/(x** 4*ln(x)**2+(-2*x**4*exp(x**2+12)**2-8*x**4*exp(x**2+12)-18*x**4-10*x**2)*l n(x)+x**4*exp(x**2+12)**4+8*x**4*exp(x**2+12)**3+(34*x**4+10*x**2)*exp(x** 2+12)**2+(72*x**4+40*x**2)*exp(x**2+12)+81*x**4+90*x**2+25),x)
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {4 \, x^{2}}{x^{2} e^{\left (2 \, x^{2} + 24\right )} + 4 \, x^{2} e^{\left (x^{2} + 12\right )} - x^{2} \log \left (x\right ) + 9 \, x^{2} + 5} \]
integrate((-16*x^5*exp(x^2+12)^2-32*x^5*exp(x^2+12)+4*x^3+40*x)/(x^4*log(x )^2+(-2*x^4*exp(x^2+12)^2-8*x^4*exp(x^2+12)-18*x^4-10*x^2)*log(x)+x^4*exp( x^2+12)^4+8*x^4*exp(x^2+12)^3+(34*x^4+10*x^2)*exp(x^2+12)^2+(72*x^4+40*x^2 )*exp(x^2+12)+81*x^4+90*x^2+25),x, algorithm=\
Time = 0.89 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\frac {8 \, x^{2}}{x^{2} e^{\left (2 \, x^{2} + 24\right )} + 4 \, x^{2} e^{\left (x^{2} + 12\right )} - x^{2} \log \left (x\right ) + 9 \, x^{2} + 5} \]
integrate((-16*x^5*exp(x^2+12)^2-32*x^5*exp(x^2+12)+4*x^3+40*x)/(x^4*log(x )^2+(-2*x^4*exp(x^2+12)^2-8*x^4*exp(x^2+12)-18*x^4-10*x^2)*log(x)+x^4*exp( x^2+12)^4+8*x^4*exp(x^2+12)^3+(34*x^4+10*x^2)*exp(x^2+12)^2+(72*x^4+40*x^2 )*exp(x^2+12)+81*x^4+90*x^2+25),x, algorithm=\
Timed out. \[ \int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx=\int \frac {40\,x-16\,x^5\,{\mathrm {e}}^{2\,x^2+24}-32\,x^5\,{\mathrm {e}}^{x^2+12}+4\,x^3}{{\mathrm {e}}^{2\,x^2+24}\,\left (34\,x^4+10\,x^2\right )-\ln \left (x\right )\,\left (2\,x^4\,{\mathrm {e}}^{2\,x^2+24}+8\,x^4\,{\mathrm {e}}^{x^2+12}+10\,x^2+18\,x^4\right )+8\,x^4\,{\mathrm {e}}^{3\,x^2+36}+x^4\,{\mathrm {e}}^{4\,x^2+48}+x^4\,{\ln \left (x\right )}^2+{\mathrm {e}}^{x^2+12}\,\left (72\,x^4+40\,x^2\right )+90\,x^2+81\,x^4+25} \,d x \]
int((40*x - 16*x^5*exp(2*x^2 + 24) - 32*x^5*exp(x^2 + 12) + 4*x^3)/(exp(2* x^2 + 24)*(10*x^2 + 34*x^4) - log(x)*(2*x^4*exp(2*x^2 + 24) + 8*x^4*exp(x^ 2 + 12) + 10*x^2 + 18*x^4) + 8*x^4*exp(3*x^2 + 36) + x^4*exp(4*x^2 + 48) + x^4*log(x)^2 + exp(x^2 + 12)*(40*x^2 + 72*x^4) + 90*x^2 + 81*x^4 + 25),x)
int((40*x - 16*x^5*exp(2*x^2 + 24) - 32*x^5*exp(x^2 + 12) + 4*x^3)/(exp(2* x^2 + 24)*(10*x^2 + 34*x^4) - log(x)*(2*x^4*exp(2*x^2 + 24) + 8*x^4*exp(x^ 2 + 12) + 10*x^2 + 18*x^4) + 8*x^4*exp(3*x^2 + 36) + x^4*exp(4*x^2 + 48) + x^4*log(x)^2 + exp(x^2 + 12)*(40*x^2 + 72*x^4) + 90*x^2 + 81*x^4 + 25), x )