Integrand size = 104, antiderivative size = 31 \[ \int \frac {1}{9} e^{-24-x} \left (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x \left (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7\right )\right ) \, dx=-e^{-x} \left (-e^{4 x}+x\right )+\frac {4}{9} \left (x-\frac {x}{e^3}\right )^8 \] Output:
4/9*(x-x/exp(3))^8-(x-exp(4*x))/exp(x)
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1}{9} e^{-24-x} \left (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x \left (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7\right )\right ) \, dx=e^{3 x}-e^{-x} x+\frac {4 \left (-1+e^3\right )^8 x^8}{9 e^{24}} \] Input:
Integrate[(E^(-24 - x)*(27*E^(24 + 4*x) + E^24*(-9 + 9*x) + E^x*(32*x^7 - 256*E^3*x^7 + 896*E^6*x^7 - 1792*E^9*x^7 + 2240*E^12*x^7 - 1792*E^15*x^7 + 896*E^18*x^7 - 256*E^21*x^7 + 32*E^24*x^7)))/9,x]
Output:
E^(3*x) - x/E^x + (4*(-1 + E^3)^8*x^8)/(9*E^24)
Time = 0.44 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.55, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {27, 7239, 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.
\(\displaystyle \int \frac {1}{9} e^{-x-24} \left (e^x \left (32 e^{24} x^7-256 e^{21} x^7+896 e^{18} x^7-1792 e^{15} x^7+2240 e^{12} x^7-1792 e^9 x^7+896 e^6 x^7-256 e^3 x^7+32 x^7\right )+e^{24} (9 x-9)+27 e^{4 x+24}\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int e^{-x-24} \left (-9 e^{24} (1-x)+27 e^{4 x+24}+32 e^x \left (e^{24} x^7-8 e^{21} x^7+28 e^{18} x^7-56 e^{15} x^7+70 e^{12} x^7-56 e^9 x^7+28 e^6 x^7-8 e^3 x^7+x^7\right )\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \frac {1}{9} \int \left (\frac {32 \left (-1+e^3\right )^8 x^7}{e^{24}}+27 e^{3 x}+9 e^{-x} (x-1)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{9} \left (\frac {4 \left (1-e^3\right )^8 x^8}{e^{24}}-9 e^{-x}+9 e^{3 x}+9 e^{-x} (1-x)\right )\) |
Input:
Int[(E^(-24 - x)*(27*E^(24 + 4*x) + E^24*(-9 + 9*x) + E^x*(32*x^7 - 256*E^ 3*x^7 + 896*E^6*x^7 - 1792*E^9*x^7 + 2240*E^12*x^7 - 1792*E^15*x^7 + 896*E ^18*x^7 - 256*E^21*x^7 + 32*E^24*x^7)))/9,x]
Output:
(-9/E^x + 9*E^(3*x) + (9*(1 - x))/E^x + (4*(1 - E^3)^8*x^8)/E^24)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 0.71 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19
method | result | size |
parts | \(\frac {4 \,{\mathrm e}^{-24} x^{8} \left ({\mathrm e}^{24}-8 \,{\mathrm e}^{21}+28 \,{\mathrm e}^{18}-56 \,{\mathrm e}^{15}+70 \,{\mathrm e}^{12}-56 \,{\mathrm e}^{9}+28 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}+1\right )}{9}+{\mathrm e}^{3 x}-{\mathrm e}^{-x} x\) | \(68\) |
risch | \(\frac {4 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{24}}{9}-\frac {32 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{21}}{9}+\frac {112 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{18}}{9}-\frac {224 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{15}}{9}+\frac {280 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{12}}{9}-\frac {224 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{9}}{9}+\frac {112 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{6}}{9}-\frac {32 \,{\mathrm e}^{-24} x^{8} {\mathrm e}^{3}}{9}+\frac {4 \,{\mathrm e}^{-24} x^{8}}{9}+{\mathrm e}^{3 x}-{\mathrm e}^{-x} x\) | \(92\) |
parallelrisch | \(\frac {{\mathrm e}^{-24} \left (4 \,{\mathrm e}^{24} x^{8} {\mathrm e}^{x}-32 \,{\mathrm e}^{21} x^{8} {\mathrm e}^{x}+112 \,{\mathrm e}^{18} x^{8} {\mathrm e}^{x}-224 \,{\mathrm e}^{15} x^{8} {\mathrm e}^{x}+280 \,{\mathrm e}^{12} x^{8} {\mathrm e}^{x}-224 \,{\mathrm e}^{9} x^{8} {\mathrm e}^{x}+112 \,{\mathrm e}^{6} x^{8} {\mathrm e}^{x}-32 \,{\mathrm e}^{3} x^{8} {\mathrm e}^{x}-9 x \,{\mathrm e}^{24}+9 \,{\mathrm e}^{24} {\mathrm e}^{4 x}+4 x^{8} {\mathrm e}^{x}\right ) {\mathrm e}^{-x}}{9}\) | \(122\) |
default | \(\frac {{\mathrm e}^{-24} \left (4 x^{8}-32 \,{\mathrm e}^{3} x^{8}+112 \,{\mathrm e}^{6} x^{8}-224 \,{\mathrm e}^{9} x^{8}+280 \,{\mathrm e}^{12} x^{8}-224 \,{\mathrm e}^{15} x^{8}+112 \,{\mathrm e}^{18} x^{8}-32 \,{\mathrm e}^{21} x^{8}+4 \,{\mathrm e}^{24} x^{8}+9 \,{\mathrm e}^{-x} {\mathrm e}^{24}+9 \,{\mathrm e}^{3 x} {\mathrm e}^{24}+9 \,{\mathrm e}^{24} \left (-{\mathrm e}^{-x} x -{\mathrm e}^{-x}\right )\right )}{9}\) | \(123\) |
orering | \(\text {Expression too large to display}\) | \(1074\) |
Input:
int(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*ex p(3)^6-1792*x^7*exp(3)^5+2240*x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3 )^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x),x,meth od=_RETURNVERBOSE)
Output:
4/9/exp(3)^8*x^8*(exp(3)^8-8*exp(3)^7+28*exp(3)^6-56*exp(3)^5+70*exp(3)^4- 56*exp(3)^3+28*exp(3)^2-8*exp(3)+1)+exp(x)^3-x/exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.74 \[ \int \frac {1}{9} e^{-24-x} \left (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x \left (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7\right )\right ) \, dx=-\frac {1}{9} \, {\left (9 \, x e^{24} - 4 \, {\left (x^{8} e^{24} - 8 \, x^{8} e^{21} + 28 \, x^{8} e^{18} - 56 \, x^{8} e^{15} + 70 \, x^{8} e^{12} - 56 \, x^{8} e^{9} + 28 \, x^{8} e^{6} - 8 \, x^{8} e^{3} + x^{8}\right )} e^{x} - 9 \, e^{\left (4 \, x + 24\right )}\right )} e^{\left (-x - 24\right )} \] Input:
integrate(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896* x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7 *exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x), x, algorithm="fricas")
Output:
-1/9*(9*x*e^24 - 4*(x^8*e^24 - 8*x^8*e^21 + 28*x^8*e^18 - 56*x^8*e^15 + 70 *x^8*e^12 - 56*x^8*e^9 + 28*x^8*e^6 - 8*x^8*e^3 + x^8)*e^x - 9*e^(4*x + 24 ))*e^(-x - 24)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.94 \[ \int \frac {1}{9} e^{-24-x} \left (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x \left (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7\right )\right ) \, dx=\frac {x^{8} \left (- 32 e^{21} - 224 e^{15} - 224 e^{9} - 32 e^{3} + 4 + 112 e^{6} + 280 e^{12} + 112 e^{18} + 4 e^{24}\right )}{9 e^{24}} - x e^{- x} + e^{3 x} \] Input:
integrate(1/9*(27*exp(3)**8*exp(4*x)+(32*x**7*exp(3)**8-256*x**7*exp(3)**7 +896*x**7*exp(3)**6-1792*x**7*exp(3)**5+2240*x**7*exp(3)**4-1792*x**7*exp( 3)**3+896*x**7*exp(3)**2-256*x**7*exp(3)+32*x**7)*exp(x)+(9*x-9)*exp(3)**8 )/exp(3)**8/exp(x),x)
Output:
x**8*(-32*exp(21) - 224*exp(15) - 224*exp(9) - 32*exp(3) + 4 + 112*exp(6) + 280*exp(12) + 112*exp(18) + 4*exp(24))*exp(-24)/9 - x*exp(-x) + exp(3*x)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (27) = 54\).
Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int \frac {1}{9} e^{-24-x} \left (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x \left (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7\right )\right ) \, dx=-\frac {32}{9} \, x^{8} e^{\left (-3\right )} + \frac {112}{9} \, x^{8} e^{\left (-6\right )} - \frac {224}{9} \, x^{8} e^{\left (-9\right )} + \frac {280}{9} \, x^{8} e^{\left (-12\right )} - \frac {224}{9} \, x^{8} e^{\left (-15\right )} + \frac {112}{9} \, x^{8} e^{\left (-18\right )} - \frac {32}{9} \, x^{8} e^{\left (-21\right )} + \frac {4}{9} \, x^{8} e^{\left (-24\right )} + \frac {4}{9} \, x^{8} - {\left (x + 1\right )} e^{\left (-x\right )} + e^{\left (3 \, x\right )} + e^{\left (-x\right )} \] Input:
integrate(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896* x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7 *exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x), x, algorithm="maxima")
Output:
-32/9*x^8*e^(-3) + 112/9*x^8*e^(-6) - 224/9*x^8*e^(-9) + 280/9*x^8*e^(-12) - 224/9*x^8*e^(-15) + 112/9*x^8*e^(-18) - 32/9*x^8*e^(-21) + 4/9*x^8*e^(- 24) + 4/9*x^8 - (x + 1)*e^(-x) + e^(3*x) + e^(-x)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.68 \[ \int \frac {1}{9} e^{-24-x} \left (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x \left (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7\right )\right ) \, dx=\frac {1}{9} \, {\left (4 \, x^{8} e^{24} - 32 \, x^{8} e^{21} + 112 \, x^{8} e^{18} - 224 \, x^{8} e^{15} + 280 \, x^{8} e^{12} - 224 \, x^{8} e^{9} + 112 \, x^{8} e^{6} - 32 \, x^{8} e^{3} + 4 \, x^{8} - 9 \, x e^{\left (-x + 24\right )} + 9 \, e^{\left (3 \, x + 24\right )}\right )} e^{\left (-24\right )} \] Input:
integrate(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896* x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7 *exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x), x, algorithm="giac")
Output:
1/9*(4*x^8*e^24 - 32*x^8*e^21 + 112*x^8*e^18 - 224*x^8*e^15 + 280*x^8*e^12 - 224*x^8*e^9 + 112*x^8*e^6 - 32*x^8*e^3 + 4*x^8 - 9*x*e^(-x + 24) + 9*e^ (3*x + 24))*e^(-24)
Time = 3.46 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {1}{9} e^{-24-x} \left (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x \left (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7\right )\right ) \, dx={\mathrm {e}}^{3\,x}-x\,{\mathrm {e}}^{-x}+x^8\,\left (\frac {112\,{\mathrm {e}}^{-6}}{9}-\frac {32\,{\mathrm {e}}^{-3}}{9}-\frac {224\,{\mathrm {e}}^{-9}}{9}+\frac {280\,{\mathrm {e}}^{-12}}{9}-\frac {224\,{\mathrm {e}}^{-15}}{9}+\frac {112\,{\mathrm {e}}^{-18}}{9}-\frac {32\,{\mathrm {e}}^{-21}}{9}+\frac {4\,{\mathrm {e}}^{-24}}{9}+\frac {4}{9}\right ) \] Input:
int(exp(-x)*exp(-24)*(3*exp(4*x)*exp(24) + (exp(x)*(896*x^7*exp(6) - 256*x ^7*exp(3) - 1792*x^7*exp(9) + 2240*x^7*exp(12) - 1792*x^7*exp(15) + 896*x^ 7*exp(18) - 256*x^7*exp(21) + 32*x^7*exp(24) + 32*x^7))/9 + (exp(24)*(9*x - 9))/9),x)
Output:
exp(3*x) - x*exp(-x) + x^8*((112*exp(-6))/9 - (32*exp(-3))/9 - (224*exp(-9 ))/9 + (280*exp(-12))/9 - (224*exp(-15))/9 + (112*exp(-18))/9 - (32*exp(-2 1))/9 + (4*exp(-24))/9 + 4/9)
Time = 0.51 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.97 \[ \int \frac {1}{9} e^{-24-x} \left (27 e^{24+4 x}+e^{24} (-9+9 x)+e^x \left (32 x^7-256 e^3 x^7+896 e^6 x^7-1792 e^9 x^7+2240 e^{12} x^7-1792 e^{15} x^7+896 e^{18} x^7-256 e^{21} x^7+32 e^{24} x^7\right )\right ) \, dx=\frac {9 e^{4 x} e^{24}+4 e^{x} e^{24} x^{8}-32 e^{x} e^{21} x^{8}+112 e^{x} e^{18} x^{8}-224 e^{x} e^{15} x^{8}+280 e^{x} e^{12} x^{8}-224 e^{x} e^{9} x^{8}+112 e^{x} e^{6} x^{8}-32 e^{x} e^{3} x^{8}+4 e^{x} x^{8}-9 e^{24} x}{9 e^{x} e^{24}} \] Input:
int(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*ex p(3)^6-1792*x^7*exp(3)^5+2240*x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3 )^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x),x)
Output:
(9*e**(4*x)*e**24 + 4*e**x*e**24*x**8 - 32*e**x*e**21*x**8 + 112*e**x*e**1 8*x**8 - 224*e**x*e**15*x**8 + 280*e**x*e**12*x**8 - 224*e**x*e**9*x**8 + 112*e**x*e**6*x**8 - 32*e**x*e**3*x**8 + 4*e**x*x**8 - 9*e**24*x)/(9*e**x* e**24)