Integrand size = 64, antiderivative size = 19 \[ \int \frac {e^{\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx=\frac {1}{16} e^{-3+\frac {9}{x^2}} \left (4+e^x\right ) x \] Output:
x*(exp(x)+4)*exp(9/x^2-4*ln(2)-3)
Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx=\frac {1}{16} e^{-3+\frac {9}{x^2}} \left (4+e^x\right ) x \] Input:
Integrate[(E^((9 - 3*x^2 - x^2*Log[16])/x^2)*(-72 + 4*x^2) + E^(x + (9 - 3 *x^2 - x^2*Log[16])/x^2)*(-18 + x^2 + x^3))/x^2,x]
Output:
(E^(-3 + 9/x^2)*(4 + E^x)*x)/16
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.
\(\displaystyle \int \frac {\left (4 x^2-72\right ) e^{\frac {-3 x^2+x^2 (-\log (16))+9}{x^2}}+\left (x^3+x^2-18\right ) e^{\frac {-3 x^2+x^2 (-\log (16))+9}{x^2}+x}}{x^2} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {1}{16} e^{\frac {9}{x^2}+x-3} x+\frac {1}{4} e^{\frac {9}{x^2}-3}+\frac {1}{16} e^{\frac {9}{x^2}+x-3}-\frac {9 e^{\frac {9}{x^2}-3}}{2 x^2}-\frac {9 e^{\frac {9}{x^2}+x-3}}{8 x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{16} \int e^{x-3+\frac {9}{x^2}}dx-\frac {9}{8} \int \frac {e^{x-3+\frac {9}{x^2}}}{x^2}dx+\frac {1}{16} \int e^{x-3+\frac {9}{x^2}} xdx+\frac {1}{4} e^{\frac {9}{x^2}-3} x\) |
Input:
Int[(E^((9 - 3*x^2 - x^2*Log[16])/x^2)*(-72 + 4*x^2) + E^(x + (9 - 3*x^2 - x^2*Log[16])/x^2)*(-18 + x^2 + x^3))/x^2,x]
Output:
$Aborted
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16
method | result | size |
risch | \(\frac {\left ({\mathrm e}^{x} x +4 x \right ) {\mathrm e}^{-\frac {3 \left (x^{2}-3\right )}{x^{2}}}}{16}\) | \(22\) |
parallelrisch | \(x \,{\mathrm e}^{x} {\mathrm e}^{-\frac {4 x^{2} \ln \left (2\right )+3 x^{2}-9}{x^{2}}}+4 x \,{\mathrm e}^{-\frac {4 x^{2} \ln \left (2\right )+3 x^{2}-9}{x^{2}}}\) | \(49\) |
norman | \(\frac {{\mathrm e}^{x} x^{2} {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}}+4 x^{2} {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}}}{x}\) | \(55\) |
parts | \({\mathrm e}^{x} x \,{\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}}+\frac {3 \,{\mathrm e}^{-3} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {3}{x}\right )}{4}-\frac {{\mathrm e}^{-3} \left (-{\mathrm e}^{\frac {9}{x^{2}}} x +3 \sqrt {\pi }\, \operatorname {erfi}\left (\frac {3}{x}\right )\right )}{4}\) | \(63\) |
orering | \(\frac {x \left (x^{3}+2 x^{2}-36\right ) \left (\left (x^{3}+x^{2}-18\right ) {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}} {\mathrm e}^{x}+\left (4 x^{2}-72\right ) {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}}\right )}{x^{5}+2 x^{4}-18 x^{3}-90 x^{2}+324}-\frac {x^{6} \left (\frac {\left (3 x^{2}+2 x \right ) {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}} {\mathrm e}^{x}+\left (x^{3}+x^{2}-18\right ) \left (\frac {-8 x \ln \left (2\right )-6 x}{x^{2}}-\frac {2 \left (-4 x^{2} \ln \left (2\right )-3 x^{2}+9\right )}{x^{3}}\right ) {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}} {\mathrm e}^{x}+\left (x^{3}+x^{2}-18\right ) {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}} {\mathrm e}^{x}+8 x \,{\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}}+\left (4 x^{2}-72\right ) \left (\frac {-8 x \ln \left (2\right )-6 x}{x^{2}}-\frac {2 \left (-4 x^{2} \ln \left (2\right )-3 x^{2}+9\right )}{x^{3}}\right ) {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}}}{x^{2}}-\frac {2 \left (\left (x^{3}+x^{2}-18\right ) {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}} {\mathrm e}^{x}+\left (4 x^{2}-72\right ) {\mathrm e}^{\frac {-4 x^{2} \ln \left (2\right )-3 x^{2}+9}{x^{2}}}\right )}{x^{3}}\right )}{x^{5}+2 x^{4}-18 x^{3}-90 x^{2}+324}\) | \(396\) |
Input:
int(((x^3+x^2-18)*exp((-4*x^2*ln(2)-3*x^2+9)/x^2)*exp(x)+(4*x^2-72)*exp((- 4*x^2*ln(2)-3*x^2+9)/x^2))/x^2,x,method=_RETURNVERBOSE)
Output:
1/16*(exp(x)*x+4*x)*exp(-3*(x^2-3)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.53 \[ \int \frac {e^{\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx=x e^{\left (\frac {x^{3} - 4 \, x^{2} \log \left (2\right ) - 3 \, x^{2} + 9}{x^{2}}\right )} + 4 \, x e^{\left (-\frac {4 \, x^{2} \log \left (2\right ) + 3 \, x^{2} - 9}{x^{2}}\right )} \] Input:
integrate(((x^3+x^2-18)*exp((-4*x^2*log(2)-3*x^2+9)/x^2)*exp(x)+(4*x^2-72) *exp((-4*x^2*log(2)-3*x^2+9)/x^2))/x^2,x, algorithm="fricas")
Output:
x*e^((x^3 - 4*x^2*log(2) - 3*x^2 + 9)/x^2) + 4*x*e^(-(4*x^2*log(2) + 3*x^2 - 9)/x^2)
Time = 0.67 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx=\left (x e^{x} + 4 x\right ) e^{\frac {- 3 x^{2} - 4 x^{2} \log {\left (2 \right )} + 9}{x^{2}}} \] Input:
integrate(((x**3+x**2-18)*exp((-4*x**2*ln(2)-3*x**2+9)/x**2)*exp(x)+(4*x** 2-72)*exp((-4*x**2*ln(2)-3*x**2+9)/x**2))/x**2,x)
Output:
(x*exp(x) + 4*x)*exp((-3*x**2 - 4*x**2*log(2) + 9)/x**2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.21 \[ \int \frac {e^{\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx=\frac {3}{8} \, x \sqrt {-\frac {1}{x^{2}}} e^{\left (-3\right )} \Gamma \left (-\frac {1}{2}, -\frac {9}{x^{2}}\right ) + \frac {1}{16} \, x e^{\left (x + \frac {9}{x^{2}} - 3\right )} + \frac {3 \, \sqrt {\pi } {\left (\operatorname {erf}\left (3 \, \sqrt {-\frac {1}{x^{2}}}\right ) - 1\right )} e^{\left (-3\right )}}{4 \, x \sqrt {-\frac {1}{x^{2}}}} \] Input:
integrate(((x^3+x^2-18)*exp((-4*x^2*log(2)-3*x^2+9)/x^2)*exp(x)+(4*x^2-72) *exp((-4*x^2*log(2)-3*x^2+9)/x^2))/x^2,x, algorithm="maxima")
Output:
3/8*x*sqrt(-1/x^2)*e^(-3)*gamma(-1/2, -9/x^2) + 1/16*x*e^(x + 9/x^2 - 3) + 3/4*sqrt(pi)*(erf(3*sqrt(-1/x^2)) - 1)*e^(-3)/(x*sqrt(-1/x^2))
Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {e^{\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx=\frac {1}{16} \, x e^{\left (\frac {x^{3} - 3 \, x^{2} + 9}{x^{2}}\right )} + \frac {1}{4} \, x e^{\left (-\frac {3 \, {\left (x^{2} - 3\right )}}{x^{2}}\right )} \] Input:
integrate(((x^3+x^2-18)*exp((-4*x^2*log(2)-3*x^2+9)/x^2)*exp(x)+(4*x^2-72) *exp((-4*x^2*log(2)-3*x^2+9)/x^2))/x^2,x, algorithm="giac")
Output:
1/16*x*e^((x^3 - 3*x^2 + 9)/x^2) + 1/4*x*e^(-3*(x^2 - 3)/x^2)
Time = 0.77 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx=\frac {x\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^{\frac {9}{x^2}}\,\left ({\mathrm {e}}^x+4\right )}{16} \] Input:
int((exp(-(4*x^2*log(2) + 3*x^2 - 9)/x^2)*(4*x^2 - 72) + exp(-(4*x^2*log(2 ) + 3*x^2 - 9)/x^2)*exp(x)*(x^2 + x^3 - 18))/x^2,x)
Output:
(x*exp(-3)*exp(9/x^2)*(exp(x) + 4))/16
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-72+4 x^2\right )+e^{x+\frac {9-3 x^2-x^2 \log (16)}{x^2}} \left (-18+x^2+x^3\right )}{x^2} \, dx=\frac {e^{\frac {9}{x^{2}}} x \left (e^{x}+4\right )}{16 e^{3}} \] Input:
int(((x^3+x^2-18)*exp((-4*x^2*log(2)-3*x^2+9)/x^2)*exp(x)+(4*x^2-72)*exp(( -4*x^2*log(2)-3*x^2+9)/x^2))/x^2,x)
Output:
(e**(9/x**2)*x*(e**x + 4))/(16*e**3)