Integrand size = 150, antiderivative size = 23 \[ \int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 \left (-8+8 e^{\frac {2}{x^2}}-2 x^3\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^2 \left (-12 x+10 x^3+e^{\frac {2}{x^2}} \left (8 x+2 x^3\right )\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx=2+\left (x+\frac {1}{5+e^{\frac {2}{x^2}}-\frac {2}{x^2}+x}\right )^2 \] Output:
2+(1/(exp(2/x^2)+x-2/x^2+5)+x)^2
Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.96 \[ \int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 \left (-8+8 e^{\frac {2}{x^2}}-2 x^3\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^2 \left (-12 x+10 x^3+e^{\frac {2}{x^2}} \left (8 x+2 x^3\right )\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx=\frac {x^2 \left (-2+x+\left (5+e^{\frac {2}{x^2}}\right ) x^2+x^3\right )^2}{\left (-2+\left (5+e^{\frac {2}{x^2}}\right ) x^2+x^3\right )^2} \] Input:
Integrate[(-4*x^2 + 10*x^4 + 2*E^(2/x^2)*x^4 + 2*x^5 + (x^4*(-8 + 8*E^(2/x ^2) - 2*x^3))/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3)^2 + (x^2*(-12*x + 10*x^3 + E^(2/x^2)*(8*x + 2*x^3)))/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3))/(-2*x + 5* x^3 + E^(2/x^2)*x^3 + x^4),x]
Output:
(x^2*(-2 + x + (5 + E^(2/x^2))*x^2 + x^3)^2)/(-2 + (5 + E^(2/x^2))*x^2 + x ^3)^2
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.
\(\displaystyle \int \frac {2 x^5+10 x^4-4 x^2+2 e^{\frac {2}{x^2}} x^4+\frac {\left (10 x^3+e^{\frac {2}{x^2}} \left (2 x^3+8 x\right )-12 x\right ) x^2}{x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2}+\frac {\left (-2 x^3+8 e^{\frac {2}{x^2}}-8\right ) x^4}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^2}}{x^4+5 x^3+e^{\frac {2}{x^2}} x^3-2 x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (x^2+4\right )}{x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2}-\frac {2 \left (x^5+4 x^3+24 x^2-8\right ) x}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^3}-\frac {2 \left (x^5+4 x^3+24 x^2-4 x-8\right )}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^2}+2 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 16 \int \frac {x}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^3}dx-48 \int \frac {x^3}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^3}dx+16 \int \frac {1}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^2}dx+8 \int \frac {x}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^2}dx-48 \int \frac {x^2}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^2}dx-8 \int \frac {x^3}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^2}dx+8 \int \frac {1}{x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2}dx+2 \int \frac {x^2}{x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2}dx-2 \int \frac {x^6}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^3}dx-2 \int \frac {x^5}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^2}dx-8 \int \frac {x^4}{\left (x^3+e^{\frac {2}{x^2}} x^2+5 x^2-2\right )^3}dx+x^2\) |
Input:
Int[(-4*x^2 + 10*x^4 + 2*E^(2/x^2)*x^4 + 2*x^5 + (x^4*(-8 + 8*E^(2/x^2) - 2*x^3))/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3)^2 + (x^2*(-12*x + 10*x^3 + E^(2 /x^2)*(8*x + 2*x^3)))/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3))/(-2*x + 5*x^3 + E^(2/x^2)*x^3 + x^4),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(22)=44\).
Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39
method | result | size |
risch | \(x^{2}+\frac {\left (2 x^{3}+2 x^{2} {\mathrm e}^{\frac {2}{x^{2}}}+10 x^{2}+x -4\right ) x^{3}}{\left (x^{2} {\mathrm e}^{\frac {2}{x^{2}}}+x^{3}+5 x^{2}-2\right )^{2}}\) | \(55\) |
parallelrisch | \(\frac {x^{8}+2 \,{\mathrm e}^{\frac {2}{x^{2}}} x^{7}+{\mathrm e}^{\frac {4}{x^{2}}} x^{6}+10 x^{7}+10 \,{\mathrm e}^{\frac {2}{x^{2}}} x^{6}+27 x^{6}+2 \,{\mathrm e}^{\frac {2}{x^{2}}} x^{5}+6 x^{5}-4 x^{4} {\mathrm e}^{\frac {2}{x^{2}}}-19 x^{4}-4 x^{3}+4 x^{2}}{\left (x^{2} {\mathrm e}^{\frac {2}{x^{2}}}+x^{3}+5 x^{2}-2\right )^{2}}\) | \(114\) |
Input:
int(((8*exp(2/x^2)-2*x^3-8)/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2*x^4+((2*x^3+8*x )*exp(2/x^2)+10*x^3-12*x)/(x^2*exp(2/x^2)+x^3+5*x^2-2)*x^2+2*x^4*exp(2/x^2 )+2*x^5+10*x^4-4*x^2)/(x^3*exp(2/x^2)+x^4+5*x^3-2*x),x,method=_RETURNVERBO SE)
Output:
x^2+(2*x^3+2*x^2*exp(2/x^2)+10*x^2+x-4)*x^3/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (22) = 44\).
Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.61 \[ \int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 \left (-8+8 e^{\frac {2}{x^2}}-2 x^3\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^2 \left (-12 x+10 x^3+e^{\frac {2}{x^2}} \left (8 x+2 x^3\right )\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx=\frac {x^{8} + 10 \, x^{7} + x^{6} e^{\left (\frac {4}{x^{2}}\right )} + 27 \, x^{6} + 6 \, x^{5} - 19 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} + 2 \, {\left (x^{7} + 5 \, x^{6} + x^{5} - 2 \, x^{4}\right )} e^{\left (\frac {2}{x^{2}}\right )}}{x^{6} + 10 \, x^{5} + x^{4} e^{\left (\frac {4}{x^{2}}\right )} + 25 \, x^{4} - 4 \, x^{3} - 20 \, x^{2} + 2 \, {\left (x^{5} + 5 \, x^{4} - 2 \, x^{2}\right )} e^{\left (\frac {2}{x^{2}}\right )} + 4} \] Input:
integrate(((8*exp(2/x^2)-2*x^3-8)/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2*x^4+((2*x ^3+8*x)*exp(2/x^2)+10*x^3-12*x)/(x^2*exp(2/x^2)+x^3+5*x^2-2)*x^2+2*x^4*exp (2/x^2)+2*x^5+10*x^4-4*x^2)/(x^3*exp(2/x^2)+x^4+5*x^3-2*x),x, algorithm="f ricas")
Output:
(x^8 + 10*x^7 + x^6*e^(4/x^2) + 27*x^6 + 6*x^5 - 19*x^4 - 4*x^3 + 4*x^2 + 2*(x^7 + 5*x^6 + x^5 - 2*x^4)*e^(2/x^2))/(x^6 + 10*x^5 + x^4*e^(4/x^2) + 2 5*x^4 - 4*x^3 - 20*x^2 + 2*(x^5 + 5*x^4 - 2*x^2)*e^(2/x^2) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (20) = 40\).
Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.91 \[ \int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 \left (-8+8 e^{\frac {2}{x^2}}-2 x^3\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^2 \left (-12 x+10 x^3+e^{\frac {2}{x^2}} \left (8 x+2 x^3\right )\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx=x^{2} + \frac {2 x^{6} + 2 x^{5} e^{\frac {2}{x^{2}}} + 10 x^{5} + x^{4} - 4 x^{3}}{x^{6} + 10 x^{5} + x^{4} e^{\frac {4}{x^{2}}} + 25 x^{4} - 4 x^{3} - 20 x^{2} + \left (2 x^{5} + 10 x^{4} - 4 x^{2}\right ) e^{\frac {2}{x^{2}}} + 4} \] Input:
integrate(((8*exp(2/x**2)-2*x**3-8)/(x**2*exp(2/x**2)+x**3+5*x**2-2)**2*x* *4+((2*x**3+8*x)*exp(2/x**2)+10*x**3-12*x)/(x**2*exp(2/x**2)+x**3+5*x**2-2 )*x**2+2*x**4*exp(2/x**2)+2*x**5+10*x**4-4*x**2)/(x**3*exp(2/x**2)+x**4+5* x**3-2*x),x)
Output:
x**2 + (2*x**6 + 2*x**5*exp(2/x**2) + 10*x**5 + x**4 - 4*x**3)/(x**6 + 10* x**5 + x**4*exp(4/x**2) + 25*x**4 - 4*x**3 - 20*x**2 + (2*x**5 + 10*x**4 - 4*x**2)*exp(2/x**2) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (22) = 44\).
Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.61 \[ \int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 \left (-8+8 e^{\frac {2}{x^2}}-2 x^3\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^2 \left (-12 x+10 x^3+e^{\frac {2}{x^2}} \left (8 x+2 x^3\right )\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx=\frac {x^{8} + 10 \, x^{7} + x^{6} e^{\left (\frac {4}{x^{2}}\right )} + 27 \, x^{6} + 6 \, x^{5} - 19 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} + 2 \, {\left (x^{7} + 5 \, x^{6} + x^{5} - 2 \, x^{4}\right )} e^{\left (\frac {2}{x^{2}}\right )}}{x^{6} + 10 \, x^{5} + x^{4} e^{\left (\frac {4}{x^{2}}\right )} + 25 \, x^{4} - 4 \, x^{3} - 20 \, x^{2} + 2 \, {\left (x^{5} + 5 \, x^{4} - 2 \, x^{2}\right )} e^{\left (\frac {2}{x^{2}}\right )} + 4} \] Input:
integrate(((8*exp(2/x^2)-2*x^3-8)/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2*x^4+((2*x ^3+8*x)*exp(2/x^2)+10*x^3-12*x)/(x^2*exp(2/x^2)+x^3+5*x^2-2)*x^2+2*x^4*exp (2/x^2)+2*x^5+10*x^4-4*x^2)/(x^3*exp(2/x^2)+x^4+5*x^3-2*x),x, algorithm="m axima")
Output:
(x^8 + 10*x^7 + x^6*e^(4/x^2) + 27*x^6 + 6*x^5 - 19*x^4 - 4*x^3 + 4*x^2 + 2*(x^7 + 5*x^6 + x^5 - 2*x^4)*e^(2/x^2))/(x^6 + 10*x^5 + x^4*e^(4/x^2) + 2 5*x^4 - 4*x^3 - 20*x^2 + 2*(x^5 + 5*x^4 - 2*x^2)*e^(2/x^2) + 4)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 159, normalized size of antiderivative = 6.91 \[ \int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 \left (-8+8 e^{\frac {2}{x^2}}-2 x^3\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^2 \left (-12 x+10 x^3+e^{\frac {2}{x^2}} \left (8 x+2 x^3\right )\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx=\frac {x^{8} + 2 \, x^{7} e^{\left (\frac {2}{x^{2}}\right )} + 10 \, x^{7} + x^{6} e^{\left (\frac {4}{x^{2}}\right )} + 10 \, x^{6} e^{\left (\frac {2}{x^{2}}\right )} + 27 \, x^{6} + 2 \, x^{5} e^{\left (\frac {2}{x^{2}}\right )} + 6 \, x^{5} - 4 \, x^{4} e^{\left (\frac {2}{x^{2}}\right )} - 19 \, x^{4} - 4 \, x^{3} + 4 \, x^{2}}{x^{6} + 2 \, x^{5} e^{\left (\frac {2}{x^{2}}\right )} + 10 \, x^{5} + x^{4} e^{\left (\frac {4}{x^{2}}\right )} + 10 \, x^{4} e^{\left (\frac {2}{x^{2}}\right )} + 25 \, x^{4} - 4 \, x^{3} - 4 \, x^{2} e^{\left (\frac {2}{x^{2}}\right )} - 20 \, x^{2} + 4} \] Input:
integrate(((8*exp(2/x^2)-2*x^3-8)/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2*x^4+((2*x ^3+8*x)*exp(2/x^2)+10*x^3-12*x)/(x^2*exp(2/x^2)+x^3+5*x^2-2)*x^2+2*x^4*exp (2/x^2)+2*x^5+10*x^4-4*x^2)/(x^3*exp(2/x^2)+x^4+5*x^3-2*x),x, algorithm="g iac")
Output:
(x^8 + 2*x^7*e^(2/x^2) + 10*x^7 + x^6*e^(4/x^2) + 10*x^6*e^(2/x^2) + 27*x^ 6 + 2*x^5*e^(2/x^2) + 6*x^5 - 4*x^4*e^(2/x^2) - 19*x^4 - 4*x^3 + 4*x^2)/(x ^6 + 2*x^5*e^(2/x^2) + 10*x^5 + x^4*e^(4/x^2) + 10*x^4*e^(2/x^2) + 25*x^4 - 4*x^3 - 4*x^2*e^(2/x^2) - 20*x^2 + 4)
Time = 1.78 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 \left (-8+8 e^{\frac {2}{x^2}}-2 x^3\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^2 \left (-12 x+10 x^3+e^{\frac {2}{x^2}} \left (8 x+2 x^3\right )\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx=\frac {x^2\,{\left (x+x^2\,{\mathrm {e}}^{\frac {2}{x^2}}+5\,x^2+x^3-2\right )}^2}{{\left (x^2\,{\mathrm {e}}^{\frac {2}{x^2}}+5\,x^2+x^3-2\right )}^2} \] Input:
int((2*x^4*exp(2/x^2) - 4*x^2 + 10*x^4 + 2*x^5 - (x^4*(2*x^3 - 8*exp(2/x^2 ) + 8))/(x^2*exp(2/x^2) + 5*x^2 + x^3 - 2)^2 + (x^2*(exp(2/x^2)*(8*x + 2*x ^3) - 12*x + 10*x^3))/(x^2*exp(2/x^2) + 5*x^2 + x^3 - 2))/(x^3*exp(2/x^2) - 2*x + 5*x^3 + x^4),x)
Output:
(x^2*(x + x^2*exp(2/x^2) + 5*x^2 + x^3 - 2)^2)/(x^2*exp(2/x^2) + 5*x^2 + x ^3 - 2)^2
Time = 0.20 (sec) , antiderivative size = 193, normalized size of antiderivative = 8.39 \[ \int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 \left (-8+8 e^{\frac {2}{x^2}}-2 x^3\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^2 \left (-12 x+10 x^3+e^{\frac {2}{x^2}} \left (8 x+2 x^3\right )\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx=\frac {e^{\frac {4}{x^{2}}} x^{6}+48 e^{\frac {4}{x^{2}}} x^{4}+2 e^{\frac {2}{x^{2}}} x^{7}+10 e^{\frac {2}{x^{2}}} x^{6}+98 e^{\frac {2}{x^{2}}} x^{5}+476 e^{\frac {2}{x^{2}}} x^{4}-192 e^{\frac {2}{x^{2}}} x^{2}+x^{8}+10 x^{7}+75 x^{6}+486 x^{5}+1181 x^{4}-196 x^{3}-956 x^{2}+192}{e^{\frac {4}{x^{2}}} x^{4}+2 e^{\frac {2}{x^{2}}} x^{5}+10 e^{\frac {2}{x^{2}}} x^{4}-4 e^{\frac {2}{x^{2}}} x^{2}+x^{6}+10 x^{5}+25 x^{4}-4 x^{3}-20 x^{2}+4} \] Input:
int(((8*exp(2/x^2)-2*x^3-8)/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2*x^4+((2*x^3+8*x )*exp(2/x^2)+10*x^3-12*x)/(x^2*exp(2/x^2)+x^3+5*x^2-2)*x^2+2*x^4*exp(2/x^2 )+2*x^5+10*x^4-4*x^2)/(x^3*exp(2/x^2)+x^4+5*x^3-2*x),x)
Output:
(e**(4/x**2)*x**6 + 48*e**(4/x**2)*x**4 + 2*e**(2/x**2)*x**7 + 10*e**(2/x* *2)*x**6 + 98*e**(2/x**2)*x**5 + 476*e**(2/x**2)*x**4 - 192*e**(2/x**2)*x* *2 + x**8 + 10*x**7 + 75*x**6 + 486*x**5 + 1181*x**4 - 196*x**3 - 956*x**2 + 192)/(e**(4/x**2)*x**4 + 2*e**(2/x**2)*x**5 + 10*e**(2/x**2)*x**4 - 4*e **(2/x**2)*x**2 + x**6 + 10*x**5 + 25*x**4 - 4*x**3 - 20*x**2 + 4)