Integrand size = 97, antiderivative size = 30 \[ \int \frac {-189-24 e^{-8+2 x}-114 x+e^{-4+x} \left (135+47 x+35 x^2-4 x^3\right )}{27+324 x+1296 x^2+1728 x^3+e^{-4+x} \left (-18-216 x-864 x^2-1152 x^3\right )+e^{-8+2 x} \left (3+36 x+144 x^2+192 x^3\right )} \, dx=\frac {3-\frac {(-9+x) x}{3-e^{-4+x}}}{3 (1+4 x)^2} \] Output:
1/3*(3-x*(x-9)/(3-exp(-4+x)))/(1+4*x)^2
Time = 2.47 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {-189-24 e^{-8+2 x}-114 x+e^{-4+x} \left (135+47 x+35 x^2-4 x^3\right )}{27+324 x+1296 x^2+1728 x^3+e^{-4+x} \left (-18-216 x-864 x^2-1152 x^3\right )+e^{-8+2 x} \left (3+36 x+144 x^2+192 x^3\right )} \, dx=\frac {1}{3} \left (\frac {3}{(1+4 x)^2}+\frac {e^4 \left (-9 x+x^2\right )}{\left (-3 e^4+e^x\right ) (1+4 x)^2}\right ) \] Input:
Integrate[(-189 - 24*E^(-8 + 2*x) - 114*x + E^(-4 + x)*(135 + 47*x + 35*x^ 2 - 4*x^3))/(27 + 324*x + 1296*x^2 + 1728*x^3 + E^(-4 + x)*(-18 - 216*x - 864*x^2 - 1152*x^3) + E^(-8 + 2*x)*(3 + 36*x + 144*x^2 + 192*x^3)),x]
Output:
(3/(1 + 4*x)^2 + (E^4*(-9*x + x^2))/((-3*E^4 + E^x)*(1 + 4*x)^2))/3
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.
\(\displaystyle \int \frac {e^{x-4} \left (-4 x^3+35 x^2+47 x+135\right )-114 x-24 e^{2 x-8}-189}{1728 x^3+1296 x^2+e^{x-4} \left (-1152 x^3-864 x^2-216 x-18\right )+e^{2 x-8} \left (192 x^3+144 x^2+36 x+3\right )+324 x+27} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^8 \left (e^{x-4} \left (-4 x^3+35 x^2+47 x+135\right )-114 x-24 e^{2 x-8}-189\right )}{3 \left (3 e^4-e^x\right )^2 (4 x+1)^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} e^8 \int -\frac {114 x+24 e^{2 x-8}-e^{x-4} \left (-4 x^3+35 x^2+47 x+135\right )+189}{\left (3 e^4-e^x\right )^2 (4 x+1)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} e^8 \int \frac {114 x+24 e^{2 x-8}-e^{x-4} \left (-4 x^3+35 x^2+47 x+135\right )+189}{\left (3 e^4-e^x\right )^2 (4 x+1)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} e^8 \int \left (\frac {3 (x-9) x}{\left (-3 e^4+e^x\right )^2 (4 x+1)^2}-\frac {4 x^3-35 x^2-47 x+9}{e^4 \left (3 e^4-e^x\right ) (4 x+1)^3}+\frac {24}{e^8 (4 x+1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{3} e^8 \left (\frac {37 \int \frac {1}{\left (-3 e^4+e^x\right ) (4 x+1)^3}dx}{2 e^4}+\frac {111}{16} \int \frac {1}{\left (-3 e^4+e^x\right )^2 (4 x+1)^2}dx-\frac {115 \int \frac {1}{\left (-3 e^4+e^x\right ) (4 x+1)^2}dx}{16 e^4}-\frac {57}{8} \int \frac {1}{\left (-3 e^4+e^x\right )^2 (4 x+1)}dx-\frac {19 \int \frac {1}{\left (-3 e^4+e^x\right ) (4 x+1)}dx}{8 e^4}+\frac {1}{16 e^4 \left (3 e^4-e^x\right )}-\frac {3}{e^8 (4 x+1)^2}\right )\) |
Input:
Int[(-189 - 24*E^(-8 + 2*x) - 114*x + E^(-4 + x)*(135 + 47*x + 35*x^2 - 4* x^3))/(27 + 324*x + 1296*x^2 + 1728*x^3 + E^(-4 + x)*(-18 - 216*x - 864*x^ 2 - 1152*x^3) + E^(-8 + 2*x)*(3 + 36*x + 144*x^2 + 192*x^3)),x]
Output:
$Aborted
Time = 0.52 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {-3 x +{\mathrm e}^{x -4}+\frac {x^{2}}{3}-3}{\left ({\mathrm e}^{x -4}-3\right ) \left (1+4 x \right )^{2}}\) | \(31\) |
parallelrisch | \(\frac {-144+16 x^{2}-144 x +48 \,{\mathrm e}^{x -4}}{48 \left ({\mathrm e}^{x -4}-3\right ) \left (16 x^{2}+8 x +1\right )}\) | \(39\) |
risch | \(\frac {1}{16 x^{2}+8 x +1}+\frac {x \left (x -9\right )}{3 \left (16 x^{2}+8 x +1\right ) \left ({\mathrm e}^{x -4}-3\right )}\) | \(40\) |
Input:
int((-24*exp(x-4)^2+(-4*x^3+35*x^2+47*x+135)*exp(x-4)-114*x-189)/((192*x^3 +144*x^2+36*x+3)*exp(x-4)^2+(-1152*x^3-864*x^2-216*x-18)*exp(x-4)+1728*x^3 +1296*x^2+324*x+27),x,method=_RETURNVERBOSE)
Output:
(-3*x+exp(x-4)+1/3*x^2-3)/(exp(x-4)-3)/(1+4*x)^2
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {-189-24 e^{-8+2 x}-114 x+e^{-4+x} \left (135+47 x+35 x^2-4 x^3\right )}{27+324 x+1296 x^2+1728 x^3+e^{-4+x} \left (-18-216 x-864 x^2-1152 x^3\right )+e^{-8+2 x} \left (3+36 x+144 x^2+192 x^3\right )} \, dx=-\frac {x^{2} - 9 \, x + 3 \, e^{\left (x - 4\right )} - 9}{3 \, {\left (48 \, x^{2} - {\left (16 \, x^{2} + 8 \, x + 1\right )} e^{\left (x - 4\right )} + 24 \, x + 3\right )}} \] Input:
integrate((-24*exp(-4+x)^2+(-4*x^3+35*x^2+47*x+135)*exp(-4+x)-114*x-189)/( (192*x^3+144*x^2+36*x+3)*exp(-4+x)^2+(-1152*x^3-864*x^2-216*x-18)*exp(-4+x )+1728*x^3+1296*x^2+324*x+27),x, algorithm="fricas")
Output:
-1/3*(x^2 - 9*x + 3*e^(x - 4) - 9)/(48*x^2 - (16*x^2 + 8*x + 1)*e^(x - 4) + 24*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {-189-24 e^{-8+2 x}-114 x+e^{-4+x} \left (135+47 x+35 x^2-4 x^3\right )}{27+324 x+1296 x^2+1728 x^3+e^{-4+x} \left (-18-216 x-864 x^2-1152 x^3\right )+e^{-8+2 x} \left (3+36 x+144 x^2+192 x^3\right )} \, dx=\frac {16 x^{2} - 144 x}{- 2304 x^{2} - 1152 x + \left (768 x^{2} + 384 x + 48\right ) e^{x - 4} - 144} + \frac {8}{128 x^{2} + 64 x + 8} \] Input:
integrate((-24*exp(-4+x)**2+(-4*x**3+35*x**2+47*x+135)*exp(-4+x)-114*x-189 )/((192*x**3+144*x**2+36*x+3)*exp(-4+x)**2+(-1152*x**3-864*x**2-216*x-18)* exp(-4+x)+1728*x**3+1296*x**2+324*x+27),x)
Output:
(16*x**2 - 144*x)/(-2304*x**2 - 1152*x + (768*x**2 + 384*x + 48)*exp(x - 4 ) - 144) + 8/(128*x**2 + 64*x + 8)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83 \[ \int \frac {-189-24 e^{-8+2 x}-114 x+e^{-4+x} \left (135+47 x+35 x^2-4 x^3\right )}{27+324 x+1296 x^2+1728 x^3+e^{-4+x} \left (-18-216 x-864 x^2-1152 x^3\right )+e^{-8+2 x} \left (3+36 x+144 x^2+192 x^3\right )} \, dx=-\frac {x^{2} e^{4} - 9 \, x e^{4} - 9 \, e^{4} + 3 \, e^{x}}{3 \, {\left (48 \, x^{2} e^{4} + 24 \, x e^{4} - {\left (16 \, x^{2} + 8 \, x + 1\right )} e^{x} + 3 \, e^{4}\right )}} \] Input:
integrate((-24*exp(-4+x)^2+(-4*x^3+35*x^2+47*x+135)*exp(-4+x)-114*x-189)/( (192*x^3+144*x^2+36*x+3)*exp(-4+x)^2+(-1152*x^3-864*x^2-216*x-18)*exp(-4+x )+1728*x^3+1296*x^2+324*x+27),x, algorithm="maxima")
Output:
-1/3*(x^2*e^4 - 9*x*e^4 - 9*e^4 + 3*e^x)/(48*x^2*e^4 + 24*x*e^4 - (16*x^2 + 8*x + 1)*e^x + 3*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.90 \[ \int \frac {-189-24 e^{-8+2 x}-114 x+e^{-4+x} \left (135+47 x+35 x^2-4 x^3\right )}{27+324 x+1296 x^2+1728 x^3+e^{-4+x} \left (-18-216 x-864 x^2-1152 x^3\right )+e^{-8+2 x} \left (3+36 x+144 x^2+192 x^3\right )} \, dx=-\frac {x^{2} e^{4} - 9 \, x e^{4} - 9 \, e^{4} + 3 \, e^{x}}{3 \, {\left (48 \, x^{2} e^{4} - 16 \, x^{2} e^{x} + 24 \, x e^{4} - 8 \, x e^{x} + 3 \, e^{4} - e^{x}\right )}} \] Input:
integrate((-24*exp(-4+x)^2+(-4*x^3+35*x^2+47*x+135)*exp(-4+x)-114*x-189)/( (192*x^3+144*x^2+36*x+3)*exp(-4+x)^2+(-1152*x^3-864*x^2-216*x-18)*exp(-4+x )+1728*x^3+1296*x^2+324*x+27),x, algorithm="giac")
Output:
-1/3*(x^2*e^4 - 9*x*e^4 - 9*e^4 + 3*e^x)/(48*x^2*e^4 - 16*x^2*e^x + 24*x*e ^4 - 8*x*e^x + 3*e^4 - e^x)
Time = 0.57 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-189-24 e^{-8+2 x}-114 x+e^{-4+x} \left (135+47 x+35 x^2-4 x^3\right )}{27+324 x+1296 x^2+1728 x^3+e^{-4+x} \left (-18-216 x-864 x^2-1152 x^3\right )+e^{-8+2 x} \left (3+36 x+144 x^2+192 x^3\right )} \, dx=\frac {1}{{\left (4\,x+1\right )}^2}-\frac {3\,x-\frac {x^2}{3}}{{\left (4\,x+1\right )}^2\,\left ({\mathrm {e}}^{x-4}-3\right )} \] Input:
int(-(114*x + 24*exp(2*x - 8) - exp(x - 4)*(47*x + 35*x^2 - 4*x^3 + 135) + 189)/(324*x - exp(x - 4)*(216*x + 864*x^2 + 1152*x^3 + 18) + exp(2*x - 8) *(36*x + 144*x^2 + 192*x^3 + 3) + 1296*x^2 + 1728*x^3 + 27),x)
Output:
1/(4*x + 1)^2 - (3*x - x^2/3)/((4*x + 1)^2*(exp(x - 4) - 3))
Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {-189-24 e^{-8+2 x}-114 x+e^{-4+x} \left (135+47 x+35 x^2-4 x^3\right )}{27+324 x+1296 x^2+1728 x^3+e^{-4+x} \left (-18-216 x-864 x^2-1152 x^3\right )+e^{-8+2 x} \left (3+36 x+144 x^2+192 x^3\right )} \, dx=\frac {3 e^{x}+e^{4} x^{2}-9 e^{4} x -9 e^{4}}{48 e^{x} x^{2}+24 e^{x} x +3 e^{x}-144 e^{4} x^{2}-72 e^{4} x -9 e^{4}} \] Input:
int((-24*exp(-4+x)^2+(-4*x^3+35*x^2+47*x+135)*exp(-4+x)-114*x-189)/((192*x ^3+144*x^2+36*x+3)*exp(-4+x)^2+(-1152*x^3-864*x^2-216*x-18)*exp(-4+x)+1728 *x^3+1296*x^2+324*x+27),x)
Output:
(3*e**x + e**4*x**2 - 9*e**4*x - 9*e**4)/(3*(16*e**x*x**2 + 8*e**x*x + e** x - 48*e**4*x**2 - 24*e**4*x - 3*e**4))