Integrand size = 92, antiderivative size = 24 \[ \int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x+e^{36 x^4} \left (288 x^3-144 e^{x^2} x^3-144 x^4\right )\right )}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx=\frac {-2+e^{x^2}+x}{5+e^{5+e^{36 x^4}}} \] Output:
(x-2+exp(x^2))/(5+exp(exp(36*x^4)+5))
Time = 0.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x+e^{36 x^4} \left (288 x^3-144 e^{x^2} x^3-144 x^4\right )\right )}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx=\frac {-2+e^{x^2}+x}{5+e^{5+e^{36 x^4}}} \] Input:
Integrate[(5 + 10*E^x^2*x + E^(5 + E^(36*x^4))*(1 + 2*E^x^2*x + E^(36*x^4) *(288*x^3 - 144*E^x^2*x^3 - 144*x^4)))/(25 + 10*E^(5 + E^(36*x^4)) + E^(10 + 2*E^(36*x^4))),x]
Output:
(-2 + E^x^2 + x)/(5 + E^(5 + E^(36*x^4)))
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.
\(\displaystyle \int \frac {10 e^{x^2} x+e^{e^{36 x^4}+5} \left (2 e^{x^2} x+e^{36 x^4} \left (-144 x^4+288 x^3-144 e^{x^2} x^3\right )+1\right )+5}{10 e^{e^{36 x^4}+5}+e^{2 e^{36 x^4}+10}+25} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {10 e^{x^2} x+e^{e^{36 x^4}+5} \left (2 e^{x^2} x+e^{36 x^4} \left (-144 x^4+288 x^3-144 e^{x^2} x^3\right )+1\right )+5}{\left (e^{e^{36 x^4}+5}+5\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{e^{36 x^4}+5}}{\left (e^{e^{36 x^4}+5}+5\right )^2}+\frac {5}{\left (e^{e^{36 x^4}+5}+5\right )^2}+\frac {10 e^{x^2} x}{\left (e^{e^{36 x^4}+5}+5\right )^2}+\frac {2 e^{e^{36 x^4}+x^2+5} x}{\left (e^{e^{36 x^4}+5}+5\right )^2}-\frac {144 e^{36 x^4+e^{36 x^4}+5} \left (e^{x^2}+x-2\right ) x^3}{\left (e^{e^{36 x^4}+5}+5\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 5 \text {Subst}\left (\int \frac {e^x}{\left (5+e^{5+e^{36 x^2}}\right )^2}dx,x,x^2\right )+\text {Subst}\left (\int \frac {e^{x+e^{36 x^2}+5}}{\left (5+e^{5+e^{36 x^2}}\right )^2}dx,x,x^2\right )-72 \text {Subst}\left (\int \frac {e^{36 x^2+x+e^{36 x^2}+5} x}{\left (5+e^{5+e^{36 x^2}}\right )^2}dx,x,x^2\right )+\int \frac {1}{5+e^{5+e^{36 x^4}}}dx-144 \int \frac {e^{36 x^4+e^{36 x^4}+5} x^4}{\left (5+e^{5+e^{36 x^4}}\right )^2}dx-\frac {2}{e^{e^{36 x^4}+5}+5}\) |
Input:
Int[(5 + 10*E^x^2*x + E^(5 + E^(36*x^4))*(1 + 2*E^x^2*x + E^(36*x^4)*(288* x^3 - 144*E^x^2*x^3 - 144*x^4)))/(25 + 10*E^(5 + E^(36*x^4)) + E^(10 + 2*E ^(36*x^4))),x]
Output:
$Aborted
Time = 0.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {x -2+{\mathrm e}^{x^{2}}}{5+{\mathrm e}^{{\mathrm e}^{36 x^{4}}+5}}\) | \(22\) |
parallelrisch | \(\frac {x -2+{\mathrm e}^{x^{2}}}{5+{\mathrm e}^{{\mathrm e}^{36 x^{4}}+5}}\) | \(22\) |
Input:
int((((-144*x^3*exp(x^2)-144*x^4+288*x^3)*exp(36*x^4)+2*exp(x^2)*x+1)*exp( exp(36*x^4)+5)+10*exp(x^2)*x+5)/(exp(exp(36*x^4)+5)^2+10*exp(exp(36*x^4)+5 )+25),x,method=_RETURNVERBOSE)
Output:
(x-2+exp(x^2))/(5+exp(exp(36*x^4)+5))
Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x+e^{36 x^4} \left (288 x^3-144 e^{x^2} x^3-144 x^4\right )\right )}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx=\frac {x + e^{\left (x^{2}\right )} - 2}{e^{\left (e^{\left (36 \, x^{4}\right )} + 5\right )} + 5} \] Input:
integrate((((-144*x^3*exp(x^2)-144*x^4+288*x^3)*exp(36*x^4)+2*exp(x^2)*x+1 )*exp(exp(36*x^4)+5)+10*exp(x^2)*x+5)/(exp(exp(36*x^4)+5)^2+10*exp(exp(36* x^4)+5)+25),x, algorithm="fricas")
Output:
(x + e^(x^2) - 2)/(e^(e^(36*x^4) + 5) + 5)
Time = 0.13 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x+e^{36 x^4} \left (288 x^3-144 e^{x^2} x^3-144 x^4\right )\right )}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx=\frac {x + e^{x^{2}} - 2}{e^{e^{36 x^{4}} + 5} + 5} \] Input:
integrate((((-144*exp(x**2)*x**3-144*x**4+288*x**3)*exp(36*x**4)+2*exp(x** 2)*x+1)*exp(exp(36*x**4)+5)+10*exp(x**2)*x+5)/(exp(exp(36*x**4)+5)**2+10*e xp(exp(36*x**4)+5)+25),x)
Output:
(x + exp(x**2) - 2)/(exp(exp(36*x**4) + 5) + 5)
Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x+e^{36 x^4} \left (288 x^3-144 e^{x^2} x^3-144 x^4\right )\right )}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx=\frac {x + e^{\left (x^{2}\right )} - 2}{e^{\left (e^{\left (36 \, x^{4}\right )} + 5\right )} + 5} \] Input:
integrate((((-144*x^3*exp(x^2)-144*x^4+288*x^3)*exp(36*x^4)+2*exp(x^2)*x+1 )*exp(exp(36*x^4)+5)+10*exp(x^2)*x+5)/(exp(exp(36*x^4)+5)^2+10*exp(exp(36* x^4)+5)+25),x, algorithm="maxima")
Output:
(x + e^(x^2) - 2)/(e^(e^(36*x^4) + 5) + 5)
Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x+e^{36 x^4} \left (288 x^3-144 e^{x^2} x^3-144 x^4\right )\right )}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx=\frac {x + e^{\left (x^{2}\right )} - 2}{e^{\left (e^{\left (36 \, x^{4}\right )} + 5\right )} + 5} \] Input:
integrate((((-144*x^3*exp(x^2)-144*x^4+288*x^3)*exp(36*x^4)+2*exp(x^2)*x+1 )*exp(exp(36*x^4)+5)+10*exp(x^2)*x+5)/(exp(exp(36*x^4)+5)^2+10*exp(exp(36* x^4)+5)+25),x, algorithm="giac")
Output:
(x + e^(x^2) - 2)/(e^(e^(36*x^4) + 5) + 5)
Time = 1.44 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46 \[ \int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x+e^{36 x^4} \left (288 x^3-144 e^{x^2} x^3-144 x^4\right )\right )}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx=\frac {{\mathrm {e}}^{-36\,x^4}\,\left (x^3\,{\mathrm {e}}^{36\,x^4+x^2}-2\,x^3\,{\mathrm {e}}^{36\,x^4}+x^4\,{\mathrm {e}}^{36\,x^4}\right )}{x^3\,\left ({\mathrm {e}}^{{\mathrm {e}}^{36\,x^4}+5}+5\right )} \] Input:
int((10*x*exp(x^2) + exp(exp(36*x^4) + 5)*(2*x*exp(x^2) - exp(36*x^4)*(144 *x^3*exp(x^2) - 288*x^3 + 144*x^4) + 1) + 5)/(10*exp(exp(36*x^4) + 5) + ex p(2*exp(36*x^4) + 10) + 25),x)
Output:
(exp(-36*x^4)*(x^3*exp(x^2 + 36*x^4) - 2*x^3*exp(36*x^4) + x^4*exp(36*x^4) ))/(x^3*(exp(exp(36*x^4) + 5) + 5))
Time = 0.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {5+10 e^{x^2} x+e^{5+e^{36 x^4}} \left (1+2 e^{x^2} x+e^{36 x^4} \left (288 x^3-144 e^{x^2} x^3-144 x^4\right )\right )}{25+10 e^{5+e^{36 x^4}}+e^{10+2 e^{36 x^4}}} \, dx=\frac {2 e^{e^{36 x^{4}}} e^{5}+5 e^{x^{2}}+5 x}{5 e^{e^{36 x^{4}}} e^{5}+25} \] Input:
int((((-144*exp(x^2)*x^3-144*x^4+288*x^3)*exp(36*x^4)+2*exp(x^2)*x+1)*exp( exp(36*x^4)+5)+10*exp(x^2)*x+5)/(exp(exp(36*x^4)+5)^2+10*exp(exp(36*x^4)+5 )+25),x)
Output:
(2*e**(e**(36*x**4))*e**5 + 5*e**(x**2) + 5*x)/(5*(e**(e**(36*x**4))*e**5 + 5))