Integrand size = 142, antiderivative size = 23 \[ \int \frac {e^{-\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (-1+e^{16} (-2+2 x)+e^{\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (2-6 x+6 x^2-2 x^3\right )\right )}{-2+6 x-6 x^2+2 x^3} \, dx=e^{-5+\left (e^{16}-\frac {1}{2 (-1+x)}\right )^2}-x \]
Time = 1.57 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (-1+e^{16} (-2+2 x)+e^{\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (2-6 x+6 x^2-2 x^3\right )\right )}{-2+6 x-6 x^2+2 x^3} \, dx=e^{-5+e^{32}+\frac {1}{4 (-1+x)^2}-\frac {e^{16}}{-1+x}}-x \]
Integrate[(-1 + E^16*(-2 + 2*x) + E^((19 - 40*x + 20*x^2 + E^16*(-4 + 4*x) + E^32*(-4 + 8*x - 4*x^2))/(4 - 8*x + 4*x^2))*(2 - 6*x + 6*x^2 - 2*x^3))/ (E^((19 - 40*x + 20*x^2 + E^16*(-4 + 4*x) + E^32*(-4 + 8*x - 4*x^2))/(4 - 8*x + 4*x^2))*(-2 + 6*x - 6*x^2 + 2*x^3)),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.
| \(\displaystyle \int \frac {\exp \left (-\frac {20 x^2+e^{32} \left (-4 x^2+8 x-4\right )-40 x+e^{16} (4 x-4)+19}{4 x^2-8 x+4}\right ) \left (\left (-2 x^3+6 x^2-6 x+2\right ) \exp \left (\frac {20 x^2+e^{32} \left (-4 x^2+8 x-4\right )-40 x+e^{16} (4 x-4)+19}{4 x^2-8 x+4}\right )+e^{16} (2 x-2)-1\right )}{2 x^3-6 x^2+6 x-2} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\exp \left (-\frac {20 x^2+e^{32} \left (-4 x^2+8 x-4\right )-40 x+e^{16} (4 x-4)+19}{4 x^2-8 x+4}\right ) \left (\left (-2 x^3+6 x^2-6 x+2\right ) \exp \left (\frac {20 x^2+e^{32} \left (-4 x^2+8 x-4\right )-40 x+e^{16} (4 x-4)+19}{4 x^2-8 x+4}\right )+e^{16} (2 x-2)-1\right )}{\left (\sqrt [3]{2} x-\sqrt [3]{2}\right )^3}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\exp \left (-\frac {4 \left (5-e^{32}\right ) x^2-4 \left (10-e^{16}-2 e^{32}\right ) x-4 e^{32}-4 e^{16}+19}{4 x^2-8 x+4}\right ) \left (-\left (-2 x^3+6 x^2-6 x+2\right ) \exp \left (\frac {20 x^2+e^{32} \left (-4 x^2+8 x-4\right )-40 x+e^{16} (4 x-4)+19}{4 x^2-8 x+4}\right )-e^{16} (2 x-2)+1\right )}{\left (\sqrt [3]{2}-\sqrt [3]{2} x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (2 e^{16} x-2 e^{16}-1\right ) \exp \left (-\frac {4 \left (5-e^{32}\right ) x^2-4 \left (10-e^{16}-2 e^{32}\right ) x-4 e^{32}-4 e^{16}+19}{4 x^2-8 x+4}\right )}{2 (x-1)^3}-\exp \left (\frac {5 x^2}{(x-1)^2}-\frac {4 \left (5-e^{32}\right ) x^2-4 \left (10-e^{16}-2 e^{32}\right ) x-4 e^{32}-4 e^{16}+19}{4 x^2-8 x+4}-\frac {10 x}{(x-1)^2}+\frac {e^{16}}{x-1}+\frac {19}{4 (x-1)^2}-e^{32}\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {\exp \left (\frac {4 \left (11+e^{32}\right ) x^2-4 \left (22+e^{16}+2 e^{32}\right ) x+4 e^{32}+4 e^{16}+45}{4 (x-1)^2}\right )}{(1-x)^2}dx-\frac {1}{2} \int \frac {\exp \left (-\frac {4 \left (5-e^{32}\right ) x^2-4 \left (10-e^{16}-2 e^{32}\right ) x-4 e^{32}-4 e^{16}+19}{4 x^2-8 x+4}\right )}{(x-1)^3}dx-x\) |
Int[(-1 + E^16*(-2 + 2*x) + E^((19 - 40*x + 20*x^2 + E^16*(-4 + 4*x) + E^3 2*(-4 + 8*x - 4*x^2))/(4 - 8*x + 4*x^2))*(2 - 6*x + 6*x^2 - 2*x^3))/(E^((1 9 - 40*x + 20*x^2 + E^16*(-4 + 4*x) + E^32*(-4 + 8*x - 4*x^2))/(4 - 8*x + 4*x^2))*(-2 + 6*x - 6*x^2 + 2*x^3)),x]
3.26.51.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(23)=46\).
Time = 0.54 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09
| method | result | size |
| risch | \(-x +{\mathrm e}^{-\frac {-4 x^{2} {\mathrm e}^{32}+8 x \,{\mathrm e}^{32}+4 x \,{\mathrm e}^{16}+20 x^{2}-4 \,{\mathrm e}^{32}-4 \,{\mathrm e}^{16}-40 x +19}{4 \left (-1+x \right )^{2}}}\) | \(48\) |
| parts | \(-x +\frac {{\mathrm e}^{{\mathrm e}^{32}} {\mathrm e}^{-5} \left (2 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{-1+x}+\frac {1}{4 \left (-1+x \right )^{2}}}-2 i {\mathrm e}^{16} \sqrt {\pi }\, {\mathrm e}^{-{\mathrm e}^{32}} \operatorname {erf}\left (\frac {i}{-2+2 x}-i {\mathrm e}^{16}\right )\right )}{2}+i {\mathrm e}^{{\mathrm e}^{32}} {\mathrm e}^{-5} {\mathrm e}^{16} \sqrt {\pi }\, {\mathrm e}^{-{\mathrm e}^{32}} \operatorname {erf}\left (\frac {i}{-2+2 x}-i {\mathrm e}^{16}\right )\) | \(106\) |
| derivativedivides | \(1-x +\frac {{\mathrm e}^{{\mathrm e}^{32}} {\mathrm e}^{-5} \left (2 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{-1+x}+\frac {1}{4 \left (-1+x \right )^{2}}}-2 i {\mathrm e}^{16} \sqrt {\pi }\, {\mathrm e}^{-{\mathrm e}^{32}} \operatorname {erf}\left (\frac {i}{-2+2 x}-i {\mathrm e}^{16}\right )\right )}{2}+i {\mathrm e}^{{\mathrm e}^{32}} {\mathrm e}^{-5} {\mathrm e}^{16} \sqrt {\pi }\, {\mathrm e}^{-{\mathrm e}^{32}} \operatorname {erf}\left (\frac {i}{-2+2 x}-i {\mathrm e}^{16}\right )\) | \(107\) |
| default | \(1-x +\frac {{\mathrm e}^{{\mathrm e}^{32}} {\mathrm e}^{-5} \left (2 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{-1+x}+\frac {1}{4 \left (-1+x \right )^{2}}}-2 i {\mathrm e}^{16} \sqrt {\pi }\, {\mathrm e}^{-{\mathrm e}^{32}} \operatorname {erf}\left (\frac {i}{-2+2 x}-i {\mathrm e}^{16}\right )\right )}{2}+i {\mathrm e}^{{\mathrm e}^{32}} {\mathrm e}^{-5} {\mathrm e}^{16} \sqrt {\pi }\, {\mathrm e}^{-{\mathrm e}^{32}} \operatorname {erf}\left (\frac {i}{-2+2 x}-i {\mathrm e}^{16}\right )\) | \(107\) |
| parallelrisch | \(-\frac {\left (-4+4 \,{\mathrm e}^{\frac {\left (-4 x^{2}+8 x -4\right ) {\mathrm e}^{32}+\left (-4+4 x \right ) {\mathrm e}^{16}+20 x^{2}-40 x +19}{4 x^{2}-8 x +4}} x^{3}+16 x^{2} {\mathrm e}^{\frac {\left (-4 x^{2}+8 x -4\right ) {\mathrm e}^{32}+\left (-4+4 x \right ) {\mathrm e}^{16}+20 x^{2}-40 x +19}{4 x^{2}-8 x +4}}-4 x^{2}-44 \,{\mathrm e}^{\frac {\left (-4 x^{2}+8 x -4\right ) {\mathrm e}^{32}+\left (-4+4 x \right ) {\mathrm e}^{16}+20 x^{2}-40 x +19}{4 x^{2}-8 x +4}} x +8 x +24 \,{\mathrm e}^{\frac {\left (-4 x^{2}+8 x -4\right ) {\mathrm e}^{32}+\left (-4+4 x \right ) {\mathrm e}^{16}+20 x^{2}-40 x +19}{4 x^{2}-8 x +4}}\right ) {\mathrm e}^{-\frac {\left (-4 x^{2}+8 x -4\right ) {\mathrm e}^{32}+\left (-4+4 x \right ) {\mathrm e}^{16}+20 x^{2}-40 x +19}{4 \left (x^{2}-2 x +1\right )}}}{4 \left (-1+x \right )^{2}}\) | \(265\) |
int(((-2*x^3+6*x^2-6*x+2)*exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+2 0*x^2-40*x+19)/(4*x^2-8*x+4))+(-2+2*x)*exp(16)-1)/(2*x^3-6*x^2+6*x-2)/exp( ((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-8*x+4)), x,method=_RETURNVERBOSE)
-x+exp(-1/4*(-4*x^2*exp(32)+8*x*exp(32)+4*x*exp(16)+20*x^2-4*exp(32)-4*exp (16)-40*x+19)/(-1+x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {e^{-\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (-1+e^{16} (-2+2 x)+e^{\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (2-6 x+6 x^2-2 x^3\right )\right )}{-2+6 x-6 x^2+2 x^3} \, dx=-x + e^{\left (-\frac {20 \, x^{2} - 4 \, {\left (x^{2} - 2 \, x + 1\right )} e^{32} + 4 \, {\left (x - 1\right )} e^{16} - 40 \, x + 19}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )} \]
integrate(((-2*x^3+6*x^2-6*x+2)*exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp (16)+20*x^2-40*x+19)/(4*x^2-8*x+4))+(-2+2*x)*exp(16)-1)/(2*x^3-6*x^2+6*x-2 )/exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-8* x+4)),x, algorithm=\
-x + e^(-1/4*(20*x^2 - 4*(x^2 - 2*x + 1)*e^32 + 4*(x - 1)*e^16 - 40*x + 19 )/(x^2 - 2*x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (15) = 30\).
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {e^{-\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (-1+e^{16} (-2+2 x)+e^{\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (2-6 x+6 x^2-2 x^3\right )\right )}{-2+6 x-6 x^2+2 x^3} \, dx=- x + e^{- \frac {20 x^{2} - 40 x + \left (4 x - 4\right ) e^{16} + \left (- 4 x^{2} + 8 x - 4\right ) e^{32} + 19}{4 x^{2} - 8 x + 4}} \]
integrate(((-2*x**3+6*x**2-6*x+2)*exp(((-4*x**2+8*x-4)*exp(16)**2+(-4+4*x) *exp(16)+20*x**2-40*x+19)/(4*x**2-8*x+4))+(-2+2*x)*exp(16)-1)/(2*x**3-6*x* *2+6*x-2)/exp(((-4*x**2+8*x-4)*exp(16)**2+(-4+4*x)*exp(16)+20*x**2-40*x+19 )/(4*x**2-8*x+4)),x)
-x + exp(-(20*x**2 - 40*x + (4*x - 4)*exp(16) + (-4*x**2 + 8*x - 4)*exp(32 ) + 19)/(4*x**2 - 8*x + 4))
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (21) = 42\).
Time = 0.82 (sec) , antiderivative size = 93, normalized size of antiderivative = 4.04 \[ \int \frac {e^{-\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (-1+e^{16} (-2+2 x)+e^{\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (2-6 x+6 x^2-2 x^3\right )\right )}{-2+6 x-6 x^2+2 x^3} \, dx=-x + \frac {6 \, x - 5}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {3 \, {\left (4 \, x - 3\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {3 \, {\left (2 \, x - 1\right )}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {1}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} + e^{\left (-\frac {e^{16}}{x - 1} + \frac {1}{4 \, {\left (x^{2} - 2 \, x + 1\right )}} + e^{32} - 5\right )} \]
integrate(((-2*x^3+6*x^2-6*x+2)*exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp (16)+20*x^2-40*x+19)/(4*x^2-8*x+4))+(-2+2*x)*exp(16)-1)/(2*x^3-6*x^2+6*x-2 )/exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-8* x+4)),x, algorithm=\
-x + 1/2*(6*x - 5)/(x^2 - 2*x + 1) - 3/2*(4*x - 3)/(x^2 - 2*x + 1) + 3/2*( 2*x - 1)/(x^2 - 2*x + 1) - 1/2/(x^2 - 2*x + 1) + e^(-e^16/(x - 1) + 1/4/(x ^2 - 2*x + 1) + e^32 - 5)
\[ \int \frac {e^{-\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (-1+e^{16} (-2+2 x)+e^{\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (2-6 x+6 x^2-2 x^3\right )\right )}{-2+6 x-6 x^2+2 x^3} \, dx=\int { \frac {{\left (2 \, {\left (x - 1\right )} e^{16} - 2 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} e^{\left (\frac {20 \, x^{2} - 4 \, {\left (x^{2} - 2 \, x + 1\right )} e^{32} + 4 \, {\left (x - 1\right )} e^{16} - 40 \, x + 19}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )} - 1\right )} e^{\left (-\frac {20 \, x^{2} - 4 \, {\left (x^{2} - 2 \, x + 1\right )} e^{32} + 4 \, {\left (x - 1\right )} e^{16} - 40 \, x + 19}{4 \, {\left (x^{2} - 2 \, x + 1\right )}}\right )}}{2 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \,d x } \]
integrate(((-2*x^3+6*x^2-6*x+2)*exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp (16)+20*x^2-40*x+19)/(4*x^2-8*x+4))+(-2+2*x)*exp(16)-1)/(2*x^3-6*x^2+6*x-2 )/exp(((-4*x^2+8*x-4)*exp(16)^2+(-4+4*x)*exp(16)+20*x^2-40*x+19)/(4*x^2-8* x+4)),x, algorithm=\
integrate(1/2*(2*(x - 1)*e^16 - 2*(x^3 - 3*x^2 + 3*x - 1)*e^(1/4*(20*x^2 - 4*(x^2 - 2*x + 1)*e^32 + 4*(x - 1)*e^16 - 40*x + 19)/(x^2 - 2*x + 1)) - 1 )*e^(-1/4*(20*x^2 - 4*(x^2 - 2*x + 1)*e^32 + 4*(x - 1)*e^16 - 40*x + 19)/( x^2 - 2*x + 1))/(x^3 - 3*x^2 + 3*x - 1), x)
Time = 12.76 (sec) , antiderivative size = 144, normalized size of antiderivative = 6.26 \[ \int \frac {e^{-\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (-1+e^{16} (-2+2 x)+e^{\frac {19-40 x+20 x^2+e^{16} (-4+4 x)+e^{32} \left (-4+8 x-4 x^2\right )}{4-8 x+4 x^2}} \left (2-6 x+6 x^2-2 x^3\right )\right )}{-2+6 x-6 x^2+2 x^3} \, dx={\mathrm {e}}^{\frac {40\,x}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{-\frac {20\,x^2}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{-\frac {4\,x\,{\mathrm {e}}^{16}}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{-\frac {8\,x\,{\mathrm {e}}^{32}}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{-\frac {19}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^{32}}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{16}}{4\,x^2-8\,x+4}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{32}}{4\,x^2-8\,x+4}}-x \]
int(-(exp(-(20*x^2 - exp(32)*(4*x^2 - 8*x + 4) - 40*x + exp(16)*(4*x - 4) + 19)/(4*x^2 - 8*x + 4))*(exp((20*x^2 - exp(32)*(4*x^2 - 8*x + 4) - 40*x + exp(16)*(4*x - 4) + 19)/(4*x^2 - 8*x + 4))*(6*x - 6*x^2 + 2*x^3 - 2) - ex p(16)*(2*x - 2) + 1))/(6*x - 6*x^2 + 2*x^3 - 2),x)