Integrand size = 145, antiderivative size = 24 \[ \int \frac {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} \left (-14-40 x+12 x^2\right )+e^{2 x} \left (8+58 x+36 x^2-8 x^3\right )+e^x \left (-30 x-60 x^2-8 x^3+2 x^4\right )}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x \left (48-24 x+3 x^2\right )} \, dx=\frac {\left (\left (e^x-x\right )^2+x\right )^2}{\left (4+e^x-x\right )^2} \] Output:
((exp(x)-x)^2+x)^2/(exp(x)-x+4)^2
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(24)=48\).
Time = 7.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} \left (-14-40 x+12 x^2\right )+e^{2 x} \left (8+58 x+36 x^2-8 x^3\right )+e^x \left (-30 x-60 x^2-8 x^3+2 x^4\right )}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x \left (48-24 x+3 x^2\right )} \, dx=e^{2 x}+10 x+x^2-2 e^x (4+x)-\frac {16 (16+x)}{4+e^x-x}+\frac {(16+x)^2}{\left (4+e^x-x\right )^2} \] Input:
Integrate[(2*E^(5*x) + E^(4*x)*(14 - 8*x) + 8*x + 24*x^2 + 14*x^3 - 2*x^4 + E^(3*x)*(-14 - 40*x + 12*x^2) + E^(2*x)*(8 + 58*x + 36*x^2 - 8*x^3) + E^ x*(-30*x - 60*x^2 - 8*x^3 + 2*x^4))/(64 + E^(3*x) + E^(2*x)*(12 - 3*x) - 4 8*x + 12*x^2 - x^3 + E^x*(48 - 24*x + 3*x^2)),x]
Output:
E^(2*x) + 10*x + x^2 - 2*E^x*(4 + x) - (16*(16 + x))/(4 + E^x - x) + (16 + x)^2/(4 + E^x - x)^2
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.
\(\displaystyle \int \frac {-2 x^4+14 x^3+24 x^2+e^{3 x} \left (12 x^2-40 x-14\right )+e^{2 x} \left (-8 x^3+36 x^2+58 x+8\right )+e^x \left (2 x^4-8 x^3-60 x^2-30 x\right )+8 x+2 e^{5 x}+e^{4 x} (14-8 x)}{-x^3+12 x^2+e^x \left (3 x^2-24 x+48\right )-48 x+e^{3 x}+e^{2 x} (12-3 x)+64} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-2 x^4+14 x^3+24 x^2+e^{3 x} \left (12 x^2-40 x-14\right )+e^{2 x} \left (-8 x^3+36 x^2+58 x+8\right )+e^x \left (2 x^4-8 x^3-60 x^2-30 x\right )+8 x+2 e^{5 x}+e^{4 x} (14-8 x)}{\left (-x+e^x+4\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (7 x^2+57 x-880\right )}{\left (-x+e^x+4\right )^2}-\frac {2 (x-5) (x+16)^2}{\left (-x+e^x+4\right )^3}+2 e^{2 x}-2 e^x (x+5)+2 (x+5)+\frac {16 (x+15)}{-x+e^x+4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {x^3}{\left (-x+e^x+4\right )^3}dx-54 \int \frac {x^2}{\left (-x+e^x+4\right )^3}dx+14 \int \frac {x^2}{\left (-x+e^x+4\right )^2}dx+2560 \int \frac {1}{\left (-x+e^x+4\right )^3}dx-1760 \int \frac {1}{\left (-x+e^x+4\right )^2}dx+240 \int \frac {1}{-x+e^x+4}dx-192 \int \frac {x}{\left (-x+e^x+4\right )^3}dx+114 \int \frac {x}{\left (-x+e^x+4\right )^2}dx+16 \int \frac {x}{-x+e^x+4}dx+(x+5)^2-2 e^x (x+5)+2 e^x+e^{2 x}\) |
Input:
Int[(2*E^(5*x) + E^(4*x)*(14 - 8*x) + 8*x + 24*x^2 + 14*x^3 - 2*x^4 + E^(3 *x)*(-14 - 40*x + 12*x^2) + E^(2*x)*(8 + 58*x + 36*x^2 - 8*x^3) + E^x*(-30 *x - 60*x^2 - 8*x^3 + 2*x^4))/(64 + E^(3*x) + E^(2*x)*(12 - 3*x) - 48*x + 12*x^2 - x^3 + E^x*(48 - 24*x + 3*x^2)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(22)=44\).
Time = 0.84 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04
method | result | size |
risch | \(x^{2}+{\mathrm e}^{2 x}+10 x +\left (-2 x -8\right ) {\mathrm e}^{x}+\frac {17 x^{2}-16 \,{\mathrm e}^{x} x +224 x -256 \,{\mathrm e}^{x}-768}{\left (x -{\mathrm e}^{x}-4\right )^{2}}\) | \(49\) |
parallelrisch | \(\frac {x^{4}-4 \,{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{2 x} x^{2}-4 x \,{\mathrm e}^{3 x}+{\mathrm e}^{4 x}+2 x^{3}-4 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+x^{2}}{x^{2}-2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}-8 x +8 \,{\mathrm e}^{x}+16}\) | \(78\) |
norman | \(\frac {x^{4}+{\mathrm e}^{4 x}-8 \,{\mathrm e}^{x}-{\mathrm e}^{2 x}+8 x +2 \,{\mathrm e}^{x} x +2 x^{3}+2 x \,{\mathrm e}^{2 x}-4 x \,{\mathrm e}^{3 x}-4 \,{\mathrm e}^{x} x^{2}-4 \,{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{2 x} x^{2}-16}{\left (x -{\mathrm e}^{x}-4\right )^{2}}\) | \(80\) |
Input:
int((2*exp(x)^5+(-8*x+14)*exp(x)^4+(12*x^2-40*x-14)*exp(x)^3+(-8*x^3+36*x^ 2+58*x+8)*exp(x)^2+(2*x^4-8*x^3-60*x^2-30*x)*exp(x)-2*x^4+14*x^3+24*x^2+8* x)/(exp(x)^3+(-3*x+12)*exp(x)^2+(3*x^2-24*x+48)*exp(x)-x^3+12*x^2-48*x+64) ,x,method=_RETURNVERBOSE)
Output:
x^2+exp(2*x)+10*x+(-2*x-8)*exp(x)+(17*x^2-16*exp(x)*x+224*x-256*exp(x)-768 )/(x-exp(x)-4)^2
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33 \[ \int \frac {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} \left (-14-40 x+12 x^2\right )+e^{2 x} \left (8+58 x+36 x^2-8 x^3\right )+e^x \left (-30 x-60 x^2-8 x^3+2 x^4\right )}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x \left (48-24 x+3 x^2\right )} \, dx=\frac {x^{4} + 2 \, x^{3} - 47 \, x^{2} - 4 \, x e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{2} + x - 24\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{3} + x^{2} - 24 \, x + 96\right )} e^{x} + 384 \, x + e^{\left (4 \, x\right )} - 768}{x^{2} - 2 \, {\left (x - 4\right )} e^{x} - 8 \, x + e^{\left (2 \, x\right )} + 16} \] Input:
integrate((2*exp(x)^5+(-8*x+14)*exp(x)^4+(12*x^2-40*x-14)*exp(x)^3+(-8*x^3 +36*x^2+58*x+8)*exp(x)^2+(2*x^4-8*x^3-60*x^2-30*x)*exp(x)-2*x^4+14*x^3+24* x^2+8*x)/(exp(x)^3+(-3*x+12)*exp(x)^2+(3*x^2-24*x+48)*exp(x)-x^3+12*x^2-48 *x+64),x, algorithm="fricas")
Output:
(x^4 + 2*x^3 - 47*x^2 - 4*x*e^(3*x) + 2*(3*x^2 + x - 24)*e^(2*x) - 4*(x^3 + x^2 - 24*x + 96)*e^x + 384*x + e^(4*x) - 768)/(x^2 - 2*(x - 4)*e^x - 8*x + e^(2*x) + 16)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (17) = 34\).
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} \left (-14-40 x+12 x^2\right )+e^{2 x} \left (8+58 x+36 x^2-8 x^3\right )+e^x \left (-30 x-60 x^2-8 x^3+2 x^4\right )}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x \left (48-24 x+3 x^2\right )} \, dx=x^{2} + 10 x + \left (- 2 x - 8\right ) e^{x} + \frac {17 x^{2} + 224 x + \left (- 16 x - 256\right ) e^{x} - 768}{x^{2} - 8 x + \left (8 - 2 x\right ) e^{x} + e^{2 x} + 16} + e^{2 x} \] Input:
integrate((2*exp(x)**5+(-8*x+14)*exp(x)**4+(12*x**2-40*x-14)*exp(x)**3+(-8 *x**3+36*x**2+58*x+8)*exp(x)**2+(2*x**4-8*x**3-60*x**2-30*x)*exp(x)-2*x**4 +14*x**3+24*x**2+8*x)/(exp(x)**3+(-3*x+12)*exp(x)**2+(3*x**2-24*x+48)*exp( x)-x**3+12*x**2-48*x+64),x)
Output:
x**2 + 10*x + (-2*x - 8)*exp(x) + (17*x**2 + 224*x + (-16*x - 256)*exp(x) - 768)/(x**2 - 8*x + (8 - 2*x)*exp(x) + exp(2*x) + 16) + exp(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33 \[ \int \frac {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} \left (-14-40 x+12 x^2\right )+e^{2 x} \left (8+58 x+36 x^2-8 x^3\right )+e^x \left (-30 x-60 x^2-8 x^3+2 x^4\right )}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x \left (48-24 x+3 x^2\right )} \, dx=\frac {x^{4} + 2 \, x^{3} - 47 \, x^{2} - 4 \, x e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{2} + x - 24\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{3} + x^{2} - 24 \, x + 96\right )} e^{x} + 384 \, x + e^{\left (4 \, x\right )} - 768}{x^{2} - 2 \, {\left (x - 4\right )} e^{x} - 8 \, x + e^{\left (2 \, x\right )} + 16} \] Input:
integrate((2*exp(x)^5+(-8*x+14)*exp(x)^4+(12*x^2-40*x-14)*exp(x)^3+(-8*x^3 +36*x^2+58*x+8)*exp(x)^2+(2*x^4-8*x^3-60*x^2-30*x)*exp(x)-2*x^4+14*x^3+24* x^2+8*x)/(exp(x)^3+(-3*x+12)*exp(x)^2+(3*x^2-24*x+48)*exp(x)-x^3+12*x^2-48 *x+64),x, algorithm="maxima")
Output:
(x^4 + 2*x^3 - 47*x^2 - 4*x*e^(3*x) + 2*(3*x^2 + x - 24)*e^(2*x) - 4*(x^3 + x^2 - 24*x + 96)*e^x + 384*x + e^(4*x) - 768)/(x^2 - 2*(x - 4)*e^x - 8*x + e^(2*x) + 16)
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.08 \[ \int \frac {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} \left (-14-40 x+12 x^2\right )+e^{2 x} \left (8+58 x+36 x^2-8 x^3\right )+e^x \left (-30 x-60 x^2-8 x^3+2 x^4\right )}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x \left (48-24 x+3 x^2\right )} \, dx=\frac {x^{4} - 4 \, x^{3} e^{x} + 2 \, x^{3} + 6 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{x} - 47 \, x^{2} - 4 \, x e^{\left (3 \, x\right )} + 2 \, x e^{\left (2 \, x\right )} + 96 \, x e^{x} + 384 \, x + e^{\left (4 \, x\right )} - 48 \, e^{\left (2 \, x\right )} - 384 \, e^{x} - 768}{x^{2} - 2 \, x e^{x} - 8 \, x + e^{\left (2 \, x\right )} + 8 \, e^{x} + 16} \] Input:
integrate((2*exp(x)^5+(-8*x+14)*exp(x)^4+(12*x^2-40*x-14)*exp(x)^3+(-8*x^3 +36*x^2+58*x+8)*exp(x)^2+(2*x^4-8*x^3-60*x^2-30*x)*exp(x)-2*x^4+14*x^3+24* x^2+8*x)/(exp(x)^3+(-3*x+12)*exp(x)^2+(3*x^2-24*x+48)*exp(x)-x^3+12*x^2-48 *x+64),x, algorithm="giac")
Output:
(x^4 - 4*x^3*e^x + 2*x^3 + 6*x^2*e^(2*x) - 4*x^2*e^x - 47*x^2 - 4*x*e^(3*x ) + 2*x*e^(2*x) + 96*x*e^x + 384*x + e^(4*x) - 48*e^(2*x) - 384*e^x - 768) /(x^2 - 2*x*e^x - 8*x + e^(2*x) + 8*e^x + 16)
Time = 2.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} \left (-14-40 x+12 x^2\right )+e^{2 x} \left (8+58 x+36 x^2-8 x^3\right )+e^x \left (-30 x-60 x^2-8 x^3+2 x^4\right )}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x \left (48-24 x+3 x^2\right )} \, dx=\frac {{\left (x+{\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^x+x^2\right )}^2}{{\left ({\mathrm {e}}^x-x+4\right )}^2} \] Input:
int((8*x + 2*exp(5*x) - exp(3*x)*(40*x - 12*x^2 + 14) - exp(x)*(30*x + 60* x^2 + 8*x^3 - 2*x^4) + exp(2*x)*(58*x + 36*x^2 - 8*x^3 + 8) - exp(4*x)*(8* x - 14) + 24*x^2 + 14*x^3 - 2*x^4)/(exp(3*x) - 48*x + exp(x)*(3*x^2 - 24*x + 48) - exp(2*x)*(3*x - 12) + 12*x^2 - x^3 + 64),x)
Output:
(x + exp(2*x) - 2*x*exp(x) + x^2)^2/(exp(x) - x + 4)^2
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58 \[ \int \frac {2 e^{5 x}+e^{4 x} (14-8 x)+8 x+24 x^2+14 x^3-2 x^4+e^{3 x} \left (-14-40 x+12 x^2\right )+e^{2 x} \left (8+58 x+36 x^2-8 x^3\right )+e^x \left (-30 x-60 x^2-8 x^3+2 x^4\right )}{64+e^{3 x}+e^{2 x} (12-3 x)-48 x+12 x^2-x^3+e^x \left (48-24 x+3 x^2\right )} \, dx=\frac {e^{4 x}-4 e^{3 x} x +6 e^{2 x} x^{2}+2 e^{2 x} x -4 e^{x} x^{3}-4 e^{x} x^{2}+x^{4}+2 x^{3}+x^{2}}{e^{2 x}-2 e^{x} x +8 e^{x}+x^{2}-8 x +16} \] Input:
int((2*exp(x)^5+(-8*x+14)*exp(x)^4+(12*x^2-40*x-14)*exp(x)^3+(-8*x^3+36*x^ 2+58*x+8)*exp(x)^2+(2*x^4-8*x^3-60*x^2-30*x)*exp(x)-2*x^4+14*x^3+24*x^2+8* x)/(exp(x)^3+(-3*x+12)*exp(x)^2+(3*x^2-24*x+48)*exp(x)-x^3+12*x^2-48*x+64) ,x)
Output:
(e**(4*x) - 4*e**(3*x)*x + 6*e**(2*x)*x**2 + 2*e**(2*x)*x - 4*e**x*x**3 - 4*e**x*x**2 + x**4 + 2*x**3 + x**2)/(e**(2*x) - 2*e**x*x + 8*e**x + x**2 - 8*x + 16)