Integrand size = 96, antiderivative size = 28 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-e^{x+\frac {3}{4+x^2}} x^2+\frac {4+x}{1+x} \] Output:
(4+x)/(1+x)-x^2*exp(x)*exp(3/(x^2+4))
Time = 0.54 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-e^{x+\frac {3}{4+x^2}} x^2+\frac {3}{1+x} \] Input:
Integrate[(-48 - 24*x^2 - 3*x^4 + E^(x + 3/(4 + x^2))*(-32*x - 80*x^2 - 74 *x^3 - 44*x^4 - 28*x^5 - 13*x^6 - 4*x^7 - x^8))/(16 + 32*x + 24*x^2 + 16*x ^3 + 9*x^4 + 2*x^5 + x^6),x]
Output:
-(E^(x + 3/(4 + x^2))*x^2) + 3/(1 + 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.
\(\displaystyle \int \frac {-3 x^4-24 x^2+e^{\frac {3}{x^2+4}+x} \left (-x^8-4 x^7-13 x^6-28 x^5-44 x^4-74 x^3-80 x^2-32 x\right )-48}{x^6+2 x^5+9 x^4+16 x^3+24 x^2+32 x+16} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {4 \left (-3 x^4-24 x^2+e^{\frac {3}{x^2+4}+x} \left (-x^8-4 x^7-13 x^6-28 x^5-44 x^4-74 x^3-80 x^2-32 x\right )-48\right )}{125 (x+1)}+\frac {(-4 x-1) \left (-3 x^4-24 x^2+e^{\frac {3}{x^2+4}+x} \left (-x^8-4 x^7-13 x^6-28 x^5-44 x^4-74 x^3-80 x^2-32 x\right )-48\right )}{125 \left (x^2+4\right )}+\frac {-3 x^4-24 x^2+e^{\frac {3}{x^2+4}+x} \left (-x^8-4 x^7-13 x^6-28 x^5-44 x^4-74 x^3-80 x^2-32 x\right )-48}{25 (x+1)^2}+\frac {(-2 x-3) \left (-3 x^4-24 x^2+e^{\frac {3}{x^2+4}+x} \left (-x^8-4 x^7-13 x^6-28 x^5-44 x^4-74 x^3-80 x^2-32 x\right )-48\right )}{25 \left (x^2+4\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \int \frac {e^{x+\frac {3}{x^2+4}}}{2 i-x}dx-2 \int e^{x+\frac {3}{x^2+4}} xdx-\int e^{x+\frac {3}{x^2+4}} x^2dx+3 \int \frac {e^{x+\frac {3}{x^2+4}}}{x+2 i}dx-24 \int \frac {e^{x+\frac {3}{x^2+4}} x}{\left (x^2+4\right )^2}dx-\frac {3 x^4}{125}-\frac {39 x^2}{125}+\frac {3}{125} \left (x^2+4\right )^2-\frac {9 x}{25}+\frac {3}{100} (2 x+3)^2+\frac {3}{x+1}\) |
Input:
Int[(-48 - 24*x^2 - 3*x^4 + E^(x + 3/(4 + x^2))*(-32*x - 80*x^2 - 74*x^3 - 44*x^4 - 28*x^5 - 13*x^6 - 4*x^7 - x^8))/(16 + 32*x + 24*x^2 + 16*x^3 + 9 *x^4 + 2*x^5 + x^6),x]
Output:
$Aborted
Time = 2.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11
method | result | size |
risch | \(\frac {3}{1+x}-{\mathrm e}^{\frac {x^{3}+4 x +3}{x^{2}+4}} x^{2}\) | \(31\) |
parallelrisch | \(-\frac {16 \,{\mathrm e}^{x} {\mathrm e}^{\frac {3}{x^{2}+4}} x^{3}+16 x^{2} {\mathrm e}^{x} {\mathrm e}^{\frac {3}{x^{2}+4}}-48}{16 \left (1+x \right )}\) | \(44\) |
parts | \(\frac {3}{1+x}+\frac {-4 x^{2} {\mathrm e}^{x} {\mathrm e}^{\frac {3}{x^{2}+4}}-{\mathrm e}^{x} {\mathrm e}^{\frac {3}{x^{2}+4}} x^{4}}{x^{2}+4}\) | \(52\) |
orering | \(-\frac {\left (x^{12}+6 x^{11}+25 x^{10}+92 x^{9}+212 x^{8}+584 x^{7}+846 x^{6}+1800 x^{5}+1780 x^{4}+2432 x^{3}+2336 x^{2}+2048 x +512\right ) \left (\left (-x^{8}-4 x^{7}-13 x^{6}-28 x^{5}-44 x^{4}-74 x^{3}-80 x^{2}-32 x \right ) {\mathrm e}^{\frac {3}{x^{2}+4}} {\mathrm e}^{x}-3 x^{4}-24 x^{2}-48\right )}{\left (x^{11}+7 x^{10}+26 x^{9}+102 x^{8}+226 x^{7}+602 x^{6}+916 x^{5}+1684 x^{4}+1952 x^{3}+1824 x^{2}+2560 x +512\right ) \left (x^{6}+2 x^{5}+9 x^{4}+16 x^{3}+24 x^{2}+32 x +16\right )}+\frac {x \left (x^{6}+4 x^{5}+10 x^{4}+26 x^{3}+26 x^{2}+64 x +32\right ) \left (1+x \right ) \left (x^{2}+4\right )^{2} \left (\frac {\left (-8 x^{7}-28 x^{6}-78 x^{5}-140 x^{4}-176 x^{3}-222 x^{2}-160 x -32\right ) {\mathrm e}^{\frac {3}{x^{2}+4}} {\mathrm e}^{x}-\frac {6 \left (-x^{8}-4 x^{7}-13 x^{6}-28 x^{5}-44 x^{4}-74 x^{3}-80 x^{2}-32 x \right ) x \,{\mathrm e}^{\frac {3}{x^{2}+4}} {\mathrm e}^{x}}{\left (x^{2}+4\right )^{2}}+\left (-x^{8}-4 x^{7}-13 x^{6}-28 x^{5}-44 x^{4}-74 x^{3}-80 x^{2}-32 x \right ) {\mathrm e}^{\frac {3}{x^{2}+4}} {\mathrm e}^{x}-12 x^{3}-48 x}{x^{6}+2 x^{5}+9 x^{4}+16 x^{3}+24 x^{2}+32 x +16}-\frac {\left (\left (-x^{8}-4 x^{7}-13 x^{6}-28 x^{5}-44 x^{4}-74 x^{3}-80 x^{2}-32 x \right ) {\mathrm e}^{\frac {3}{x^{2}+4}} {\mathrm e}^{x}-3 x^{4}-24 x^{2}-48\right ) \left (6 x^{5}+10 x^{4}+36 x^{3}+48 x^{2}+48 x +32\right )}{\left (x^{6}+2 x^{5}+9 x^{4}+16 x^{3}+24 x^{2}+32 x +16\right )^{2}}\right )}{x^{11}+7 x^{10}+26 x^{9}+102 x^{8}+226 x^{7}+602 x^{6}+916 x^{5}+1684 x^{4}+1952 x^{3}+1824 x^{2}+2560 x +512}\) | \(629\) |
Input:
int(((-x^8-4*x^7-13*x^6-28*x^5-44*x^4-74*x^3-80*x^2-32*x)*exp(3/(x^2+4))*e xp(x)-3*x^4-24*x^2-48)/(x^6+2*x^5+9*x^4+16*x^3+24*x^2+32*x+16),x,method=_R ETURNVERBOSE)
Output:
3/(1+x)-exp((x^3+4*x+3)/(x^2+4))*x^2
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-\frac {{\left (x^{3} + x^{2}\right )} e^{\left (\frac {x^{3} + 4 \, x + 3}{x^{2} + 4}\right )} - 3}{x + 1} \] Input:
integrate(((-x^8-4*x^7-13*x^6-28*x^5-44*x^4-74*x^3-80*x^2-32*x)*exp(3/(x^2 +4))*exp(x)-3*x^4-24*x^2-48)/(x^6+2*x^5+9*x^4+16*x^3+24*x^2+32*x+16),x, al gorithm="fricas")
Output:
-((x^3 + x^2)*e^((x^3 + 4*x + 3)/(x^2 + 4)) - 3)/(x + 1)
Time = 12.89 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=- x^{2} e^{x} e^{\frac {3}{x^{2} + 4}} + \frac {3}{x + 1} \] Input:
integrate(((-x**8-4*x**7-13*x**6-28*x**5-44*x**4-74*x**3-80*x**2-32*x)*exp (3/(x**2+4))*exp(x)-3*x**4-24*x**2-48)/(x**6+2*x**5+9*x**4+16*x**3+24*x**2 +32*x+16),x)
Output:
-x**2*exp(x)*exp(3/(x**2 + 4)) + 3/(x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (27) = 54\).
Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.25 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-x^{2} e^{\left (x + \frac {3}{x^{2} + 4}\right )} + \frac {6 \, {\left (11 \, x^{2} - 5 \, x + 24\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} + \frac {3 \, {\left (7 \, x^{2} - 10 \, x - 12\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} - \frac {12 \, {\left (x^{2} - 5 \, x - 16\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} \] Input:
integrate(((-x^8-4*x^7-13*x^6-28*x^5-44*x^4-74*x^3-80*x^2-32*x)*exp(3/(x^2 +4))*exp(x)-3*x^4-24*x^2-48)/(x^6+2*x^5+9*x^4+16*x^3+24*x^2+32*x+16),x, al gorithm="maxima")
Output:
-x^2*e^(x + 3/(x^2 + 4)) + 6/25*(11*x^2 - 5*x + 24)/(x^3 + x^2 + 4*x + 4) + 3/25*(7*x^2 - 10*x - 12)/(x^3 + x^2 + 4*x + 4) - 12/25*(x^2 - 5*x - 16)/ (x^3 + x^2 + 4*x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (27) = 54\).
Time = 0.14 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-\frac {x^{3} e^{\left (\frac {4 \, x^{3} - 3 \, x^{2} + 16 \, x}{4 \, {\left (x^{2} + 4\right )}} + \frac {3}{4}\right )} + x^{2} e^{\left (\frac {4 \, x^{3} - 3 \, x^{2} + 16 \, x}{4 \, {\left (x^{2} + 4\right )}} + \frac {3}{4}\right )} - 3}{x + 1} \] Input:
integrate(((-x^8-4*x^7-13*x^6-28*x^5-44*x^4-74*x^3-80*x^2-32*x)*exp(3/(x^2 +4))*exp(x)-3*x^4-24*x^2-48)/(x^6+2*x^5+9*x^4+16*x^3+24*x^2+32*x+16),x, al gorithm="giac")
Output:
-(x^3*e^(1/4*(4*x^3 - 3*x^2 + 16*x)/(x^2 + 4) + 3/4) + x^2*e^(1/4*(4*x^3 - 3*x^2 + 16*x)/(x^2 + 4) + 3/4) - 3)/(x + 1)
Time = 3.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=\frac {3}{x+1}-x^2\,{\mathrm {e}}^{\frac {3}{x^2+4}}\,{\mathrm {e}}^x \] Input:
int(-(24*x^2 + 3*x^4 + exp(3/(x^2 + 4))*exp(x)*(32*x + 80*x^2 + 74*x^3 + 4 4*x^4 + 28*x^5 + 13*x^6 + 4*x^7 + x^8) + 48)/(32*x + 24*x^2 + 16*x^3 + 9*x ^4 + 2*x^5 + x^6 + 16),x)
Output:
3/(x + 1) - x^2*exp(3/(x^2 + 4))*exp(x)
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=\frac {x \left (-e^{\frac {x^{3}+4 x +3}{x^{2}+4}} x^{2}-e^{\frac {x^{3}+4 x +3}{x^{2}+4}} x -3\right )}{x +1} \] Input:
int(((-x^8-4*x^7-13*x^6-28*x^5-44*x^4-74*x^3-80*x^2-32*x)*exp(3/(x^2+4))*e xp(x)-3*x^4-24*x^2-48)/(x^6+2*x^5+9*x^4+16*x^3+24*x^2+32*x+16),x)
Output:
(x*( - e**((x**3 + 4*x + 3)/(x**2 + 4))*x**2 - e**((x**3 + 4*x + 3)/(x**2 + 4))*x - 3))/(x + 1)