Integrand size = 64, antiderivative size = 27 \[ \int \frac {1}{256} \left (-256+48 x^2-24 e x^2+3 e^2 x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx=-x+\frac {1}{256} (4-e)^2 x^2 \left (-e^{-5+x}+x\right ) \] Output:
1/256*(x-exp(-5+x))*x^2*(4-exp(1))^2-x
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(27)=54\).
Time = 0.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \frac {1}{256} \left (-256+48 x^2-24 e x^2+3 e^2 x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx=\frac {-256 e^5 x-16 e^x x^2+8 e^{1+x} x^2-e^{2+x} x^2+16 e^5 x^3-8 e^6 x^3+e^7 x^3}{256 e^5} \] Input:
Integrate[(-256 + 48*x^2 - 24*E*x^2 + 3*E^2*x^2 + E^(-5 + x)*(-32*x - 16*x ^2 + E^2*(-2*x - x^2) + E*(16*x + 8*x^2)))/256,x]
Output:
(-256*E^5*x - 16*E^x*x^2 + 8*E^(1 + x)*x^2 - E^(2 + x)*x^2 + 16*E^5*x^3 - 8*E^6*x^3 + E^7*x^3)/(256*E^5)
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6, 6, 27, 2009}
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.
\(\displaystyle \int \frac {1}{256} \left (3 e^2 x^2-24 e x^2+48 x^2+e^{x-5} \left (-16 x^2+e^2 \left (-x^2-2 x\right )+e \left (8 x^2+16 x\right )-32 x\right )-256\right ) \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {1}{256} \left (3 e^2 x^2+(48-24 e) x^2+e^{x-5} \left (-16 x^2+e^2 \left (-x^2-2 x\right )+e \left (8 x^2+16 x\right )-32 x\right )-256\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {1}{256} \left (\left (48-24 e+3 e^2\right ) x^2+e^{x-5} \left (-16 x^2+e^2 \left (-x^2-2 x\right )+e \left (8 x^2+16 x\right )-32 x\right )-256\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{256} \int \left (3 (4-e)^2 x^2-e^{x-5} \left (16 x^2+32 x+e^2 \left (x^2+2 x\right )-8 e \left (x^2+2 x\right )\right )-256\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{256} \left ((4-e)^2 x^3-16 e^{x-5} x^2+(8-e) e^{x-4} x^2-256 x\right )\) |
Input:
Int[(-256 + 48*x^2 - 24*E*x^2 + 3*E^2*x^2 + E^(-5 + x)*(-32*x - 16*x^2 + E ^2*(-2*x - x^2) + E*(16*x + 8*x^2)))/256,x]
Output:
(-256*x - 16*E^(-5 + x)*x^2 + (8 - E)*E^(-4 + x)*x^2 + (4 - E)^2*x^3)/256
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52
method | result | size |
norman | \(\left (\frac {{\mathrm e}^{2}}{256}-\frac {{\mathrm e}}{32}+\frac {1}{16}\right ) x^{3}+\left (-\frac {{\mathrm e}^{2}}{256}+\frac {{\mathrm e}}{32}-\frac {1}{16}\right ) x^{2} {\mathrm e}^{-5+x}-x\) | \(41\) |
risch | \(-\frac {\left ({\mathrm e}^{2}-8 \,{\mathrm e}+16\right ) x^{2} {\mathrm e}^{-5+x}}{256}+\frac {x^{3} {\mathrm e}^{2}}{256}-\frac {x^{3} {\mathrm e}}{32}+\frac {x^{3}}{16}-x\) | \(41\) |
parallelrisch | \(-\frac {{\mathrm e}^{-5+x} x^{2} {\mathrm e}^{2}}{256}+\frac {{\mathrm e} \,{\mathrm e}^{-5+x} x^{2}}{32}-\frac {{\mathrm e}^{-5+x} x^{2}}{16}+\frac {x^{3} {\mathrm e}^{2}}{256}-\frac {x^{3} {\mathrm e}}{32}+\frac {x^{3}}{16}-x\) | \(59\) |
default | \(-x -\frac {5 \,{\mathrm e}^{-5+x} \left (-5+x \right )}{8}-\frac {25 \,{\mathrm e}^{-5+x}}{16}-\frac {{\mathrm e}^{-5+x} \left (-5+x \right )^{2}}{16}+\frac {35 \,{\mathrm e} \,{\mathrm e}^{-5+x}}{32}-\frac {35 \,{\mathrm e}^{-5+x} {\mathrm e}^{2}}{256}+\frac {{\mathrm e} \left ({\mathrm e}^{-5+x} \left (-5+x \right )^{2}-2 \,{\mathrm e}^{-5+x} \left (-5+x \right )+2 \,{\mathrm e}^{-5+x}\right )}{32}+\frac {3 \,{\mathrm e} \left ({\mathrm e}^{-5+x} \left (-5+x \right )-{\mathrm e}^{-5+x}\right )}{8}-\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}^{-5+x} \left (-5+x \right )-{\mathrm e}^{-5+x}\right )}{64}-\frac {{\mathrm e}^{2} \left ({\mathrm e}^{-5+x} \left (-5+x \right )^{2}-2 \,{\mathrm e}^{-5+x} \left (-5+x \right )+2 \,{\mathrm e}^{-5+x}\right )}{256}+\frac {x^{3}}{16}-\frac {x^{3} {\mathrm e}}{32}+\frac {x^{3} {\mathrm e}^{2}}{256}\) | \(172\) |
parts | \(-x -\frac {5 \,{\mathrm e}^{-5+x} \left (-5+x \right )}{8}-\frac {25 \,{\mathrm e}^{-5+x}}{16}-\frac {{\mathrm e}^{-5+x} \left (-5+x \right )^{2}}{16}+\frac {35 \,{\mathrm e} \,{\mathrm e}^{-5+x}}{32}-\frac {35 \,{\mathrm e}^{-5+x} {\mathrm e}^{2}}{256}+\frac {{\mathrm e} \left ({\mathrm e}^{-5+x} \left (-5+x \right )^{2}-2 \,{\mathrm e}^{-5+x} \left (-5+x \right )+2 \,{\mathrm e}^{-5+x}\right )}{32}+\frac {3 \,{\mathrm e} \left ({\mathrm e}^{-5+x} \left (-5+x \right )-{\mathrm e}^{-5+x}\right )}{8}-\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}^{-5+x} \left (-5+x \right )-{\mathrm e}^{-5+x}\right )}{64}-\frac {{\mathrm e}^{2} \left ({\mathrm e}^{-5+x} \left (-5+x \right )^{2}-2 \,{\mathrm e}^{-5+x} \left (-5+x \right )+2 \,{\mathrm e}^{-5+x}\right )}{256}+\frac {x^{3}}{16}-\frac {x^{3} {\mathrm e}}{32}+\frac {x^{3} {\mathrm e}^{2}}{256}\) | \(172\) |
derivativedivides | \(5-x +\frac {x^{3}}{16}-\frac {x^{3} {\mathrm e}}{32}+\frac {x^{3} {\mathrm e}^{2}}{256}-\frac {5 \,{\mathrm e}^{-5+x} \left (-5+x \right )}{8}-\frac {25 \,{\mathrm e}^{-5+x}}{16}-\frac {{\mathrm e}^{-5+x} \left (-5+x \right )^{2}}{16}+\frac {35 \,{\mathrm e} \,{\mathrm e}^{-5+x}}{32}-\frac {35 \,{\mathrm e}^{-5+x} {\mathrm e}^{2}}{256}+\frac {3 \,{\mathrm e} \left ({\mathrm e}^{-5+x} \left (-5+x \right )-{\mathrm e}^{-5+x}\right )}{8}+\frac {{\mathrm e} \left ({\mathrm e}^{-5+x} \left (-5+x \right )^{2}-2 \,{\mathrm e}^{-5+x} \left (-5+x \right )+2 \,{\mathrm e}^{-5+x}\right )}{32}-\frac {3 \,{\mathrm e}^{2} \left ({\mathrm e}^{-5+x} \left (-5+x \right )-{\mathrm e}^{-5+x}\right )}{64}-\frac {{\mathrm e}^{2} \left ({\mathrm e}^{-5+x} \left (-5+x \right )^{2}-2 \,{\mathrm e}^{-5+x} \left (-5+x \right )+2 \,{\mathrm e}^{-5+x}\right )}{256}\) | \(173\) |
Input:
int(1/256*((-x^2-2*x)*exp(1)^2+(8*x^2+16*x)*exp(1)-16*x^2-32*x)*exp(-5+x)+ 3/256*x^2*exp(1)^2-3/32*x^2*exp(1)+3/16*x^2-1,x,method=_RETURNVERBOSE)
Output:
(1/256*exp(1)^2-1/32*exp(1)+1/16)*x^3+(-1/256*exp(1)^2+1/32*exp(1)-1/16)*x ^2*exp(-5+x)-x
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {1}{256} \left (-256+48 x^2-24 e x^2+3 e^2 x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx=\frac {1}{256} \, x^{3} e^{2} - \frac {1}{32} \, x^{3} e + \frac {1}{16} \, x^{3} - \frac {1}{256} \, {\left (x^{2} e^{2} - 8 \, x^{2} e + 16 \, x^{2}\right )} e^{\left (x - 5\right )} - x \] Input:
integrate(1/256*((-x^2-2*x)*exp(1)^2+(8*x^2+16*x)*exp(1)-16*x^2-32*x)*exp( -5+x)+3/256*x^2*exp(1)^2-3/32*x^2*exp(1)+3/16*x^2-1,x, algorithm="fricas")
Output:
1/256*x^3*e^2 - 1/32*x^3*e + 1/16*x^3 - 1/256*(x^2*e^2 - 8*x^2*e + 16*x^2) *e^(x - 5) - x
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {1}{256} \left (-256+48 x^2-24 e x^2+3 e^2 x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx=x^{3} \left (- \frac {e}{32} + \frac {e^{2}}{256} + \frac {1}{16}\right ) - x + \frac {\left (- 16 x^{2} - x^{2} e^{2} + 8 e x^{2}\right ) e^{x - 5}}{256} \] Input:
integrate(1/256*((-x**2-2*x)*exp(1)**2+(8*x**2+16*x)*exp(1)-16*x**2-32*x)* exp(-5+x)+3/256*x**2*exp(1)**2-3/32*x**2*exp(1)+3/16*x**2-1,x)
Output:
x**3*(-E/32 + exp(2)/256 + 1/16) - x + (-16*x**2 - x**2*exp(2) + 8*E*x**2) *exp(x - 5)/256
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {1}{256} \left (-256+48 x^2-24 e x^2+3 e^2 x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx=\frac {1}{256} \, x^{3} e^{2} - \frac {1}{32} \, x^{3} e - \frac {1}{256} \, x^{2} {\left (e^{2} - 8 \, e + 16\right )} e^{\left (x - 5\right )} + \frac {1}{16} \, x^{3} - x \] Input:
integrate(1/256*((-x^2-2*x)*exp(1)^2+(8*x^2+16*x)*exp(1)-16*x^2-32*x)*exp( -5+x)+3/256*x^2*exp(1)^2-3/32*x^2*exp(1)+3/16*x^2-1,x, algorithm="maxima")
Output:
1/256*x^3*e^2 - 1/32*x^3*e - 1/256*x^2*(e^2 - 8*e + 16)*e^(x - 5) + 1/16*x ^3 - x
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (23) = 46\).
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int \frac {1}{256} \left (-256+48 x^2-24 e x^2+3 e^2 x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx=\frac {1}{256} \, x^{3} e^{2} - \frac {1}{32} \, x^{3} e + \frac {1}{16} \, x^{3} - \frac {1}{256} \, x^{2} e^{\left (x - 3\right )} + \frac {1}{32} \, x^{2} e^{\left (x - 4\right )} - \frac {1}{16} \, x^{2} e^{\left (x - 5\right )} - x \] Input:
integrate(1/256*((-x^2-2*x)*exp(1)^2+(8*x^2+16*x)*exp(1)-16*x^2-32*x)*exp( -5+x)+3/256*x^2*exp(1)^2-3/32*x^2*exp(1)+3/16*x^2-1,x, algorithm="giac")
Output:
1/256*x^3*e^2 - 1/32*x^3*e + 1/16*x^3 - 1/256*x^2*e^(x - 3) + 1/32*x^2*e^( x - 4) - 1/16*x^2*e^(x - 5) - x
Time = 3.89 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {1}{256} \left (-256+48 x^2-24 e x^2+3 e^2 x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx=\frac {x^3\,{\left (\mathrm {e}-4\right )}^2}{256}-x-\frac {x^2\,{\mathrm {e}}^{x-5}\,{\left (\mathrm {e}-4\right )}^2}{256} \] Input:
int((3*x^2*exp(2))/256 - (3*x^2*exp(1))/32 - (exp(x - 5)*(32*x - exp(1)*(1 6*x + 8*x^2) + exp(2)*(2*x + x^2) + 16*x^2))/256 + (3*x^2)/16 - 1,x)
Output:
(x^3*(exp(1) - 4)^2)/256 - x - (x^2*exp(x - 5)*(exp(1) - 4)^2)/256
Time = 0.18 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.11 \[ \int \frac {1}{256} \left (-256+48 x^2-24 e x^2+3 e^2 x^2+e^{-5+x} \left (-32 x-16 x^2+e^2 \left (-2 x-x^2\right )+e \left (16 x+8 x^2\right )\right )\right ) \, dx=\frac {x \left (-e^{x} e^{2} x +8 e^{x} e x -16 e^{x} x +e^{7} x^{2}-8 e^{6} x^{2}+16 e^{5} x^{2}-256 e^{5}\right )}{256 e^{5}} \] Input:
int(1/256*((-x^2-2*x)*exp(1)^2+(8*x^2+16*x)*exp(1)-16*x^2-32*x)*exp(-5+x)+ 3/256*x^2*exp(1)^2-3/32*x^2*exp(1)+3/16*x^2-1,x)
Output:
(x*( - e**x*e**2*x + 8*e**x*e*x - 16*e**x*x + e**7*x**2 - 8*e**6*x**2 + 16 *e**5*x**2 - 256*e**5))/(256*e**5)