Integrand size = 163, antiderivative size = 27 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 (1-x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^2} \]
Time = 0.64 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 (-1+x)^2 \left (1+x+e^{e^{25 x^2}} x\right )^2} \]
Integrate[(-16*x + E^E^(25*x^2)*(8 - 16*x + E^(25*x^2)*(400*x^2 - 400*x^3) ))/(-25 + 75*x^2 - 75*x^4 + 25*x^6 + E^(3*E^(25*x^2))*(-25*x^3 + 75*x^4 - 75*x^5 + 25*x^6) + E^(2*E^(25*x^2))*(-75*x^2 + 150*x^3 - 150*x^5 + 75*x^6) + E^E^(25*x^2)*(-75*x + 75*x^2 + 150*x^3 - 150*x^4 - 75*x^5 + 75*x^6)),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.
\(\displaystyle \int \frac {e^{e^{25 x^2}} \left (e^{25 x^2} \left (400 x^2-400 x^3\right )-16 x+8\right )-16 x}{25 x^6-75 x^4+75 x^2+e^{2 e^{25 x^2}} \left (75 x^6-150 x^5+150 x^3-75 x^2\right )+e^{3 e^{25 x^2}} \left (25 x^6-75 x^5+75 x^4-25 x^3\right )+e^{e^{25 x^2}} \left (75 x^6-75 x^5-150 x^4+150 x^3+75 x^2-75 x\right )-25} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {8 \left (50 e^{25 x^2+e^{25 x^2}} (x-1) x^2+e^{e^{25 x^2}} (2 x-1)+2 x\right )}{25 (1-x)^3 \left (e^{e^{25 x^2}} x+x+1\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {8}{25} \int -\frac {50 e^{25 x^2+e^{25 x^2}} (1-x) x^2-2 x+e^{e^{25 x^2}} (1-2 x)}{(1-x)^3 \left (e^{e^{25 x^2}} x+x+1\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {8}{25} \int \frac {50 e^{25 x^2+e^{25 x^2}} (1-x) x^2-2 x+e^{e^{25 x^2}} (1-2 x)}{(1-x)^3 \left (e^{e^{25 x^2}} x+x+1\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {8}{25} \int \left (\frac {50 e^{25 x^2+e^{25 x^2}} x^2}{(x-1)^2 \left (e^{e^{25 x^2}} x+x+1\right )^3}+\frac {2 x}{(x-1)^3 \left (e^{e^{25 x^2}} x+x+1\right )^3}+\frac {e^{e^{25 x^2}} (2 x-1)}{(x-1)^3 \left (e^{e^{25 x^2}} x+x+1\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {8}{25} \left (50 \int \frac {e^{25 x^2+e^{25 x^2}}}{\left (e^{e^{25 x^2}} x+x+1\right )^3}dx+2 \int \frac {1}{(x-1)^3 \left (e^{e^{25 x^2}} x+x+1\right )^3}dx+\int \frac {e^{e^{25 x^2}}}{(x-1)^3 \left (e^{e^{25 x^2}} x+x+1\right )^3}dx+2 \int \frac {1}{(x-1)^2 \left (e^{e^{25 x^2}} x+x+1\right )^3}dx+2 \int \frac {e^{e^{25 x^2}}}{(x-1)^2 \left (e^{e^{25 x^2}} x+x+1\right )^3}dx+50 \int \frac {e^{25 x^2+e^{25 x^2}}}{(x-1)^2 \left (e^{e^{25 x^2}} x+x+1\right )^3}dx+100 \int \frac {e^{25 x^2+e^{25 x^2}}}{(x-1) \left (e^{e^{25 x^2}} x+x+1\right )^3}dx\right )\) |
Int[(-16*x + E^E^(25*x^2)*(8 - 16*x + E^(25*x^2)*(400*x^2 - 400*x^3)))/(-2 5 + 75*x^2 - 75*x^4 + 25*x^6 + E^(3*E^(25*x^2))*(-25*x^3 + 75*x^4 - 75*x^5 + 25*x^6) + E^(2*E^(25*x^2))*(-75*x^2 + 150*x^3 - 150*x^5 + 75*x^6) + E^E ^(25*x^2)*(-75*x + 75*x^2 + 150*x^3 - 150*x^4 - 75*x^5 + 75*x^6)),x]
3.21.12.3.1 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 = 1.77 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {4}{25 \left (x^{2}-2 x +1\right ) \left (x +1+x \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}}\right )^{2}}\) | \(27\) |
parallelrisch | \(\frac {4}{25 \left ({\mathrm e}^{2 \,{\mathrm e}^{25 x^{2}}} x^{4}+2 \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}} x^{4}-2 \,{\mathrm e}^{2 \,{\mathrm e}^{25 x^{2}}} x^{3}+x^{4}-2 \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}} x^{3}+{\mathrm e}^{2 \,{\mathrm e}^{25 x^{2}}} x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}} x^{2}-2 x^{2}+2 x \,{\mathrm e}^{{\mathrm e}^{25 x^{2}}}+1\right )}\) | \(101\) |
int((((-400*x^3+400*x^2)*exp(25*x^2)-16*x+8)*exp(exp(25*x^2))-16*x)/((25*x ^6-75*x^5+75*x^4-25*x^3)*exp(exp(25*x^2))^3+(75*x^6-150*x^5+150*x^3-75*x^2 )*exp(exp(25*x^2))^2+(75*x^6-75*x^5-150*x^4+150*x^3+75*x^2-75*x)*exp(exp(2 5*x^2))+25*x^6-75*x^4+75*x^2-25),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 \, {\left (x^{4} - 2 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} + 2 \, {\left (x^{4} - x^{3} - x^{2} + x\right )} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + 1\right )}} \]
integrate((((-400*x^3+400*x^2)*exp(25*x^2)-16*x+8)*exp(exp(25*x^2))-16*x)/ ((25*x^6-75*x^5+75*x^4-25*x^3)*exp(exp(25*x^2))^3+(75*x^6-150*x^5+150*x^3- 75*x^2)*exp(exp(25*x^2))^2+(75*x^6-75*x^5-150*x^4+150*x^3+75*x^2-75*x)*exp (exp(25*x^2))+25*x^6-75*x^4+75*x^2-25),x, algorithm=\
4/25/(x^4 - 2*x^2 + (x^4 - 2*x^3 + x^2)*e^(2*e^(25*x^2)) + 2*(x^4 - x^3 - x^2 + x)*e^(e^(25*x^2)) + 1)
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 x^{4} - 50 x^{2} + \left (25 x^{4} - 50 x^{3} + 25 x^{2}\right ) e^{2 e^{25 x^{2}}} + \left (50 x^{4} - 50 x^{3} - 50 x^{2} + 50 x\right ) e^{e^{25 x^{2}}} + 25} \]
integrate((((-400*x**3+400*x**2)*exp(25*x**2)-16*x+8)*exp(exp(25*x**2))-16 *x)/((25*x**6-75*x**5+75*x**4-25*x**3)*exp(exp(25*x**2))**3+(75*x**6-150*x **5+150*x**3-75*x**2)*exp(exp(25*x**2))**2+(75*x**6-75*x**5-150*x**4+150*x **3+75*x**2-75*x)*exp(exp(25*x**2))+25*x**6-75*x**4+75*x**2-25),x)
4/(25*x**4 - 50*x**2 + (25*x**4 - 50*x**3 + 25*x**2)*exp(2*exp(25*x**2)) + (50*x**4 - 50*x**3 - 50*x**2 + 50*x)*exp(exp(25*x**2)) + 25)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 \, {\left (x^{4} - 2 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} + 2 \, {\left (x^{4} - x^{3} - x^{2} + x\right )} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + 1\right )}} \]
integrate((((-400*x^3+400*x^2)*exp(25*x^2)-16*x+8)*exp(exp(25*x^2))-16*x)/ ((25*x^6-75*x^5+75*x^4-25*x^3)*exp(exp(25*x^2))^3+(75*x^6-150*x^5+150*x^3- 75*x^2)*exp(exp(25*x^2))^2+(75*x^6-75*x^5-150*x^4+150*x^3+75*x^2-75*x)*exp (exp(25*x^2))+25*x^6-75*x^4+75*x^2-25),x, algorithm=\
4/25/(x^4 - 2*x^2 + (x^4 - 2*x^3 + x^2)*e^(2*e^(25*x^2)) + 2*(x^4 - x^3 - x^2 + x)*e^(e^(25*x^2)) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (21) = 42\).
Time = 0.36 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.70 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4}{25 \, {\left (x^{4} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} + 2 \, x^{4} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + x^{4} - 2 \, x^{3} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} - 2 \, x^{3} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + x^{2} e^{\left (2 \, e^{\left (25 \, x^{2}\right )}\right )} - 2 \, x^{2} e^{\left (e^{\left (25 \, x^{2}\right )}\right )} - 2 \, x^{2} + 2 \, x e^{\left (e^{\left (25 \, x^{2}\right )}\right )} + 1\right )}} \]
integrate((((-400*x^3+400*x^2)*exp(25*x^2)-16*x+8)*exp(exp(25*x^2))-16*x)/ ((25*x^6-75*x^5+75*x^4-25*x^3)*exp(exp(25*x^2))^3+(75*x^6-150*x^5+150*x^3- 75*x^2)*exp(exp(25*x^2))^2+(75*x^6-75*x^5-150*x^4+150*x^3+75*x^2-75*x)*exp (exp(25*x^2))+25*x^6-75*x^4+75*x^2-25),x, algorithm=\
4/25/(x^4*e^(2*e^(25*x^2)) + 2*x^4*e^(e^(25*x^2)) + x^4 - 2*x^3*e^(2*e^(25 *x^2)) - 2*x^3*e^(e^(25*x^2)) + x^2*e^(2*e^(25*x^2)) - 2*x^2*e^(e^(25*x^2) ) - 2*x^2 + 2*x*e^(e^(25*x^2)) + 1)
Time = 11.52 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.41 \[ \int \frac {-16 x+e^{e^{25 x^2}} \left (8-16 x+e^{25 x^2} \left (400 x^2-400 x^3\right )\right )}{-25+75 x^2-75 x^4+25 x^6+e^{3 e^{25 x^2}} \left (-25 x^3+75 x^4-75 x^5+25 x^6\right )+e^{2 e^{25 x^2}} \left (-75 x^2+150 x^3-150 x^5+75 x^6\right )+e^{e^{25 x^2}} \left (-75 x+75 x^2+150 x^3-150 x^4-75 x^5+75 x^6\right )} \, dx=\frac {4\,\left (x-50\,x^2\,{\mathrm {e}}^{25\,x^2}+50\,x^4\,{\mathrm {e}}^{25\,x^2}-1\right )}{25\,{\left (x-1\right )}^3\,\left (50\,x^2\,{\mathrm {e}}^{25\,x^2}+50\,x^3\,{\mathrm {e}}^{25\,x^2}+1\right )\,\left ({\left (x+1\right )}^2+x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^{25\,x^2}}+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{25\,x^2}}\,\left (x+1\right )\right )} \]
int((16*x - exp(exp(25*x^2))*(exp(25*x^2)*(400*x^2 - 400*x^3) - 16*x + 8)) /(exp(exp(25*x^2))*(75*x - 75*x^2 - 150*x^3 + 150*x^4 + 75*x^5 - 75*x^6) - 75*x^2 + 75*x^4 - 25*x^6 + exp(3*exp(25*x^2))*(25*x^3 - 75*x^4 + 75*x^5 - 25*x^6) + exp(2*exp(25*x^2))*(75*x^2 - 150*x^3 + 150*x^5 - 75*x^6) + 25), x)