Integrand size = 110, antiderivative size = 33 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{1-x-\frac {3 e^{16 x} x}{2+\frac {x^2}{3 (5-x)}}} \]
Time = 0.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\frac {30-9 \left (4+5 e^{16 x}\right ) x+\left (7+9 e^{16 x}\right ) x^2-x^3}{30-6 x+x^2}} \]
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]
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 |
\(\displaystyle \int \left (\frac {9 \left (16 x^4-176 x^3+959 x^2-2340 x-150\right ) \exp \left (\frac {-x^3+\left (9 e^{16 x}+7\right ) x^2-9 \left (5 e^{16 x}+4\right ) x+30}{x^2-6 x+30}+16 x\right )}{\left (x^2-6 x+30\right )^2}-\exp \left (\frac {-x^3+\left (9 e^{16 x}+7\right ) x^2-9 \left (5 e^{16 x}+4\right ) x+30}{x^2-6 x+30}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \exp \left (\frac {-x^3+\left (7+9 e^{16 x}\right ) x^2-9 \left (4+5 e^{16 x}\right ) x+30}{x^2-6 x+30}\right )dx+144 \int \exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )dx-\frac {162}{7} \left (3+i \sqrt {21}\right ) \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{\left (-2 x+2 i \sqrt {21}+6\right )^2}dx+\frac {360}{7} \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{\left (-2 x+2 i \sqrt {21}+6\right )^2}dx+3 i \sqrt {\frac {3}{7}} \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{-2 x+2 i \sqrt {21}+6}dx+\frac {3}{7} \left (336+433 i \sqrt {21}\right ) \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{2 x-2 i \sqrt {21}-6}dx-\frac {162}{7} \left (3-i \sqrt {21}\right ) \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{\left (2 x+2 i \sqrt {21}-6\right )^2}dx+\frac {360}{7} \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{\left (2 x+2 i \sqrt {21}-6\right )^2}dx+\frac {3}{7} \left (336-433 i \sqrt {21}\right ) \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{2 x+2 i \sqrt {21}-6}dx+3 i \sqrt {\frac {3}{7}} \int \frac {\exp \left (\frac {15 x^3+9 e^{16 x} x^2-89 x^2-45 e^{16 x} x+444 x+30}{x^2-6 x+30}\right )}{2 x+2 i \sqrt {21}-6}dx\) |
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]
3.15.32.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.59 (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 | \(\frac {x^{2} {\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}-6 x \,{\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}+30 \,{\mathrm e}^{\frac {\left (9 x^{2}-45 x \right ) {\mathrm e}^{16 x}-x^{3}+7 x^{2}-36 x +30}{x^{2}-6 x +30}}}{x^{2}-6 x +30}\) | \(145\) |
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)
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\left (-\frac {x^{3} - 7 \, x^{2} - 9 \, {\left (x^{2} - 5 \, x\right )} e^{\left (16 \, x\right )} + 36 \, x - 30}{x^{2} - 6 \, x + 30}\right )} \]
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=\
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\frac {- x^{3} + 7 x^{2} - 36 x + \left (9 x^{2} - 45 x\right ) e^{16 x} + 30}{x^{2} - 6 x + 30}} \]
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)
Time = 0.47 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\left (-x + \frac {9 \, x e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} - \frac {270 \, e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} + 9 \, e^{\left (16 \, x\right )} + 1\right )} \]
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=\
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (26) = 52\).
Time = 0.45 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.82 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx=e^{\left (-\frac {x^{3}}{x^{2} - 6 \, x + 30} + \frac {9 \, x^{2} e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} + \frac {7 \, x^{2}}{x^{2} - 6 \, x + 30} - \frac {45 \, x e^{\left (16 \, x\right )}}{x^{2} - 6 \, x + 30} - \frac {36 \, x}{x^{2} - 6 \, x + 30} + \frac {30}{x^{2} - 6 \, x + 30}\right )} \]
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=\
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 = 0.31 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.97 \[ \int \frac {e^{\frac {30-36 x+7 x^2-x^3+e^{16 x} \left (-45 x+9 x^2\right )}{30-6 x+x^2}} \left (-900+360 x-96 x^2+12 x^3-x^4+e^{16 x} \left (-1350-21060 x+8631 x^2-1584 x^3+144 x^4\right )\right )}{900-360 x+96 x^2-12 x^3+x^4} \, dx={\mathrm {e}}^{-\frac {x^3}{x^2-6\,x+30}}\,{\mathrm {e}}^{\frac {7\,x^2}{x^2-6\,x+30}}\,{\mathrm {e}}^{\frac {30}{x^2-6\,x+30}}\,{\mathrm {e}}^{\frac {9\,x^2\,{\mathrm {e}}^{16\,x}}{x^2-6\,x+30}}\,{\mathrm {e}}^{-\frac {36\,x}{x^2-6\,x+30}}\,{\mathrm {e}}^{-\frac {45\,x\,{\mathrm {e}}^{16\,x}}{x^2-6\,x+30}} \]
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)