Integrand size = 110, antiderivative size = 33 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{1-x-\frac {3 e^{16 x} x}{2+\frac {x^2}{3 (5-x)}}} \] Output:
exp(1-x-x*exp(16*x)/(2/3+1/9*x^2/(5-x)))
Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\frac {30-9 \left (4+5 e^{16 x}\right ) x+\left (7+9 e^{16 x}\right ) x^2-x^3}{30-6 x+x^2}} \] Input:
Integrate[(E^((30 - 36*x + 7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6 *x + x^2))*(-900 + 360*x - 96*x^2 + 12*x^3 - x^4 + E^(16*x)*(-1350 - 21060 *x + 8631*x^2 - 1584*x^3 + 144*x^4)))/(900 - 360*x + 96*x^2 - 12*x^3 + x^4 ),x]
Output:
E^((30 - 9*(4 + 5*E^(16*x))*x + (7 + 9*E^(16*x))*x^2 - x^3)/(30 - 6*x + x^ 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.
| \(\displaystyle \int \frac {\left (-x^4+12 x^3-96 x^2+e^{16 x} \left (144 x^4-1584 x^3+8631 x^2-21060 x-1350\right )+360 x-900\right ) \exp \left (\frac {-x^3+7 x^2+e^{16 x} \left (9 x^2-45 x\right )-36 x+30}{x^2-6 x+30}\right )}{x^4-12 x^3+96 x^2-360 x+900} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {i \left (-x^4+12 x^3-96 x^2+e^{16 x} \left (144 x^4-1584 x^3+8631 x^2-21060 x-1350\right )+360 x-900\right ) \exp \left (\frac {-x^3+7 x^2+e^{16 x} \left (9 x^2-45 x\right )-36 x+30}{x^2-6 x+30}\right )}{42 \sqrt {21} \left (-2 x+2 i \sqrt {21}+6\right )}+\frac {i \left (-x^4+12 x^3-96 x^2+e^{16 x} \left (144 x^4-1584 x^3+8631 x^2-21060 x-1350\right )+360 x-900\right ) \exp \left (\frac {-x^3+7 x^2+e^{16 x} \left (9 x^2-45 x\right )-36 x+30}{x^2-6 x+30}\right )}{42 \sqrt {21} \left (2 x+2 i \sqrt {21}-6\right )}-\frac {\left (-x^4+12 x^3-96 x^2+e^{16 x} \left (144 x^4-1584 x^3+8631 x^2-21060 x-1350\right )+360 x-900\right ) \exp \left (\frac {-x^3+7 x^2+e^{16 x} \left (9 x^2-45 x\right )-36 x+30}{x^2-6 x+30}\right )}{21 \left (-2 x+2 i \sqrt {21}+6\right )^2}-\frac {\left (-x^4+12 x^3-96 x^2+e^{16 x} \left (144 x^4-1584 x^3+8631 x^2-21060 x-1350\right )+360 x-900\right ) \exp \left (\frac {-x^3+7 x^2+e^{16 x} \left (9 x^2-45 x\right )-36 x+30}{x^2-6 x+30}\right )}{21 \left (2 x+2 i \sqrt {21}-6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (9 e^{16 x} \left (16 x^4-176 x^3+959 x^2-2340 x-150\right )-\left (x^2-6 x+30\right )^2\right ) \exp \left (\frac {-x^3+\left (9 e^{16 x}+7\right ) x^2-9 \left (5 e^{16 x}+4\right ) x+30}{x^2-6 x+30}\right )}{\left (x^2-6 x+30\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {9 \left (16 x^4-176 x^3+959 x^2-2340 x-150\right ) \exp \left (\frac {-x^3+\left (9 e^{16 x}+7\right ) x^2-9 \left (5 e^{16 x}+4\right ) x+30}{x^2-6 x+30}+16 x\right )}{\left (x^2-6 x+30\right )^2}-\exp \left (\frac {-x^3+\left (9 e^{16 x}+7\right ) x^2-9 \left (5 e^{16 x}+4\right ) x+30}{x^2-6 x+30}\right )\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \left (\frac {9 \left (16 x^4-176 x^3+959 x^2-2340 x-150\right ) \exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{\left (x^2-6 x+30\right )^2}-\exp \left (\frac {-x^3+\left (9 e^{16 x}+7\right ) x^2-9 \left (5 e^{16 x}+4\right ) x+30}{x^2-6 x+30}\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\int \exp \left (\frac {-x^3+\left (7+9 e^{16 x}\right ) x^2-9 \left (4+5 e^{16 x}\right ) x+30}{x^2-6 x+30}\right )dx+144 \int \exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )dx-\frac {162}{7} \left (3+i \sqrt {21}\right ) \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{\left (-2 x+2 i \sqrt {21}+6\right )^2}dx+\frac {360}{7} \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{\left (-2 x+2 i \sqrt {21}+6\right )^2}dx+3 i \sqrt {\frac {3}{7}} \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{-2 x+2 i \sqrt {21}+6}dx+\frac {3}{7} \left (336+433 i \sqrt {21}\right ) \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{2 x-2 i \sqrt {21}-6}dx-\frac {162}{7} \left (3-i \sqrt {21}\right ) \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{\left (2 x+2 i \sqrt {21}-6\right )^2}dx+\frac {360}{7} \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{\left (2 x+2 i \sqrt {21}-6\right )^2}dx+\frac {3}{7} \left (336-433 i \sqrt {21}\right ) \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{2 x+2 i \sqrt {21}-6}dx+3 i \sqrt {\frac {3}{7}} \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{2 x+2 i \sqrt {21}-6}dx\) |
Input:
Int[(E^((30 - 36*x + 7*x^2 - x^3 + E^(16*x)*(-45*x + 9*x^2))/(30 - 6*x + x ^2))*(-900 + 360*x - 96*x^2 + 12*x^3 - x^4 + E^(16*x)*(-1350 - 21060*x + 8 631*x^2 - 1584*x^3 + 144*x^4)))/(900 - 360*x + 96*x^2 - 12*x^3 + x^4),x]
Output:
$Aborted
Time = 0.93 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}\) | \(42\) |
| risch | \({\mathrm e}^{-\frac {-9 \,{\mathrm e}^{16 x} x^{2}+x^{3}+45 \,{\mathrm e}^{16 x} x -7 x^{2}+36 x -30}{x^{2}-6 x +30}}\) | \(43\) |
| norman | \(\frac {x^{2} {\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}-6 x \,{\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}+30 \,{\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}}{x^{2}-6 x +30}\) | \(145\) |
Input:
int(((144*x^4-1584*x^3+8631*x^2-21060*x-1350)*exp(16*x)-x^4+12*x^3-96*x^2+ 360*x-900)*exp(((9*x^2-45*x)*exp(16*x)-x^3+7*x^2-36*x+30)/(x^2-6*x+30))/(x ^4-12*x^3+96*x^2-360*x+900),x,method=_RETURNVERBOSE)
Output:
exp(((9*x^2-45*x)*exp(16*x)-x^3+7*x^2-36*x+30)/(x^2-6*x+30))
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\left (-\frac {x^{3} - 7 \, x^{2} - 9 \, {\left (x^{2} - 5 \, x\right )} e^{\left (16 \, x\right )} + 36 \, x - 30}{x^{2} - 6 \, x + 30}\right )} \] Input:
integrate(((144*x^4-1584*x^3+8631*x^2-21060*x-1350)*exp(16*x)-x^4+12*x^3-9 6*x^2+360*x-900)*exp(((9*x^2-45*x)*exp(16*x)-x^3+7*x^2-36*x+30)/(x^2-6*x+3 0))/(x^4-12*x^3+96*x^2-360*x+900),x, algorithm="fricas")
Output:
e^(-(x^3 - 7*x^2 - 9*(x^2 - 5*x)*e^(16*x) + 36*x - 30)/(x^2 - 6*x + 30))
Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\frac {- x^{3} + 7 x^{2} - 36 x + \left (9 x^{2} - 45 x\right ) e^{16 x} + 30}{x^{2} - 6 x + 30}} \] Input:
integrate(((144*x**4-1584*x**3+8631*x**2-21060*x-1350)*exp(16*x)-x**4+12*x **3-96*x**2+360*x-900)*exp(((9*x**2-45*x)*exp(16*x)-x**3+7*x**2-36*x+30)/( x**2-6*x+30))/(x**4-12*x**3+96*x**2-360*x+900),x)
Output:
exp((-x**3 + 7*x**2 - 36*x + (9*x**2 - 45*x)*exp(16*x) + 30)/(x**2 - 6*x + 30))
Time = 0.36 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\left (-x + \frac {9 \, x e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} - \frac {270 \, e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} + 9 \, e^{\left (16 \, x\right )} + 1\right )} \] Input:
integrate(((144*x^4-1584*x^3+8631*x^2-21060*x-1350)*exp(16*x)-x^4+12*x^3-9 6*x^2+360*x-900)*exp(((9*x^2-45*x)*exp(16*x)-x^3+7*x^2-36*x+30)/(x^2-6*x+3 0))/(x^4-12*x^3+96*x^2-360*x+900),x, algorithm="maxima")
Output:
e^(-x + 9*x*e^(16*x)/(x^2 - 6*x + 30) - 270*e^(16*x)/(x^2 - 6*x + 30) + 9* e^(16*x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (26) = 52\).
Time = 0.29 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.82 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\left (-\frac {x^{3}}{x^{2} - 6 \, x + 30} + \frac {9 \, x^{2} e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} + \frac {7 \, x^{2}}{x^{2} - 6 \, x + 30} - \frac {45 \, x e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} - \frac {36 \, x}{x^{2} - 6 \, x + 30} + \frac {30}{x^{2} - 6 \, x + 30}\right )} \] Input:
integrate(((144*x^4-1584*x^3+8631*x^2-21060*x-1350)*exp(16*x)-x^4+12*x^3-9 6*x^2+360*x-900)*exp(((9*x^2-45*x)*exp(16*x)-x^3+7*x^2-36*x+30)/(x^2-6*x+3 0))/(x^4-12*x^3+96*x^2-360*x+900),x, algorithm="giac")
Output:
e^(-x^3/(x^2 - 6*x + 30) + 9*x^2*e^(16*x)/(x^2 - 6*x + 30) + 7*x^2/(x^2 - 6*x + 30) - 45*x*e^(16*x)/(x^2 - 6*x + 30) - 36*x/(x^2 - 6*x + 30) + 30/(x ^2 - 6*x + 30))
Time = 1.61 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.97 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx={\mathrm {e}}^{-\frac {x^3}{x^2-6\,x+30}}\,{\mathrm {e}}^{\frac {7\,x^2}{x^2-6\,x+30}}\,{\mathrm {e}}^{\frac {30}{x^2-6\,x+30}}\,{\mathrm {e}}^{\frac {9\,x^2\,{\mathrm {e}}^{16\,x}}{x^2-6\,x+30}}\,{\mathrm {e}}^{-\frac {36\,x}{x^2-6\,x+30}}\,{\mathrm {e}}^{-\frac {45\,x\,{\mathrm {e}}^{16\,x}}{x^2-6\,x+30}} \] Input:
int(-(exp(-(36*x + exp(16*x)*(45*x - 9*x^2) - 7*x^2 + x^3 - 30)/(x^2 - 6*x + 30))*(exp(16*x)*(21060*x - 8631*x^2 + 1584*x^3 - 144*x^4 + 1350) - 360* x + 96*x^2 - 12*x^3 + x^4 + 900))/(96*x^2 - 360*x - 12*x^3 + x^4 + 900),x)
Output:
exp(-x^3/(x^2 - 6*x + 30))*exp((7*x^2)/(x^2 - 6*x + 30))*exp(30/(x^2 - 6*x + 30))*exp((9*x^2*exp(16*x))/(x^2 - 6*x + 30))*exp(-(36*x)/(x^2 - 6*x + 3 0))*exp(-(45*x*exp(16*x))/(x^2 - 6*x + 30))
Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.27 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=\frac {e^{\frac {9 e^{16 x} x^{2}-45 e^{16 x} x +270 e^{16 x}}{x^{2}-6 x +30}} e}{e^{\frac {270 e^{16 x}+x^{3}-6 x^{2}+30 x}{x^{2}-6 x +30}}} \] Input:
int(((144*x^4-1584*x^3+8631*x^2-21060*x-1350)*exp(16*x)-x^4+12*x^3-96*x^2+ 360*x-900)*exp(((9*x^2-45*x)*exp(16*x)-x^3+7*x^2-36*x+30)/(x^2-6*x+30))/(x ^4-12*x^3+96*x^2-360*x+900),x)
Output:
(e**((9*e**(16*x)*x**2 - 45*e**(16*x)*x + 270*e**(16*x))/(x**2 - 6*x + 30) )*e)/e**((270*e**(16*x) + x**3 - 6*x**2 + 30*x)/(x**2 - 6*x + 30))