Integrand size = 125, antiderivative size = 25 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\left (x+\frac {1-2 x}{16 x \left (-e^x+x\right )}\right )^2 \] Output:
(1/x/(16*x-16*exp(x))*(1-2*x)+x)^2
Time = 5.94 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {\left (1-2 x-16 e^x x^2+16 x^3\right )^2}{256 \left (e^x-x\right )^2 x^2} \] Input:
Integrate[(2*x - 6*x^2 + 4*x^3 + 16*x^4 + 256*E^(3*x)*x^4 - 256*x^7 + E^(2 *x)*(48*x^3 - 32*x^4 - 768*x^5) + E^x*(-1 + x + 4*x^2 - 20*x^3 - 48*x^4 + 32*x^5 + 768*x^6))/(128*E^(3*x)*x^3 - 384*E^(2*x)*x^4 + 384*E^x*x^5 - 128* x^6),x]
Output:
(1 - 2*x - 16*E^x*x^2 + 16*x^3)^2/(256*(E^x - x)^2*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.
\(\displaystyle \int \frac {-256 x^7+256 e^{3 x} x^4+16 x^4+4 x^3-6 x^2+e^{2 x} \left (-768 x^5-32 x^4+48 x^3\right )+e^x \left (768 x^6+32 x^5-48 x^4-20 x^3+4 x^2+x-1\right )+2 x}{-128 x^6+384 e^x x^5-384 e^{2 x} x^4+128 e^{3 x} x^3} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-256 x^7+256 e^{3 x} x^4+16 x^4+4 x^3-6 x^2+e^{2 x} \left (-768 x^5-32 x^4+48 x^3\right )+e^x \left (768 x^6+32 x^5-48 x^4-20 x^3+4 x^2+x-1\right )+2 x}{128 \left (e^x-x\right )^3 x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{128} \int \frac {-256 x^7+256 e^{3 x} x^4+16 x^4+4 x^3-6 x^2+2 x+16 e^{2 x} \left (-48 x^5-2 x^4+3 x^3\right )-e^x \left (-768 x^6-32 x^5+48 x^4+20 x^3-4 x^2-x+1\right )}{\left (e^x-x\right )^3 x^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{128} \int \left (\frac {(x-1) (2 x-1)^2}{x^2 \left (x-e^x\right )^3}+256 x-\frac {16 (2 x-3)}{e^x-x}-\frac {32 x^5-48 x^4+20 x^3-4 x^2-x+1}{\left (e^x-x\right )^2 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{128} \left (-\int \frac {1}{\left (e^x-x\right )^2 x^3}dx+\int \frac {1}{\left (e^x-x\right )^2 x^2}dx-32 \int \frac {x^2}{\left (e^x-x\right )^2}dx-\int \frac {1}{x^2 \left (x-e^x\right )^3}dx+8 \int \frac {1}{\left (e^x-x\right )^3}dx-20 \int \frac {1}{\left (e^x-x\right )^2}dx+48 \int \frac {1}{e^x-x}dx+4 \int \frac {1}{\left (e^x-x\right )^2 x}dx-4 \int \frac {x}{\left (e^x-x\right )^3}dx+48 \int \frac {x}{\left (e^x-x\right )^2}dx-32 \int \frac {x}{e^x-x}dx+5 \int \frac {1}{x \left (x-e^x\right )^3}dx+128 x^2\right )\) |
Input:
Int[(2*x - 6*x^2 + 4*x^3 + 16*x^4 + 256*E^(3*x)*x^4 - 256*x^7 + E^(2*x)*(4 8*x^3 - 32*x^4 - 768*x^5) + E^x*(-1 + x + 4*x^2 - 20*x^3 - 48*x^4 + 32*x^5 + 768*x^6))/(128*E^(3*x)*x^3 - 384*E^(2*x)*x^4 + 384*E^x*x^5 - 128*x^6),x ]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(23)=46\).
Time = 0.55 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08
method | result | size |
risch | \(x^{2}-\frac {64 x^{4}-64 \,{\mathrm e}^{x} x^{3}-32 x^{3}+32 \,{\mathrm e}^{x} x^{2}-4 x^{2}+4 x -1}{256 x^{2} \left (x -{\mathrm e}^{x}\right )^{2}}\) | \(52\) |
norman | \(\frac {\frac {1}{256}+x^{6}+{\mathrm e}^{2 x} x^{4}-\frac {{\mathrm e}^{x} x^{3}}{4}+\frac {{\mathrm e}^{2 x} x^{2}}{4}-\frac {x}{64}+\frac {x^{2}}{64}+\frac {x^{3}}{8}-2 x^{5} {\mathrm e}^{x}-\frac {{\mathrm e}^{x} x^{2}}{8}}{x^{2} \left (x -{\mathrm e}^{x}\right )^{2}}\) | \(69\) |
parallelrisch | \(\frac {256 \,{\mathrm e}^{2 x} x^{4}-512 x^{5} {\mathrm e}^{x}+256 x^{6}+64 \,{\mathrm e}^{x} x^{3}-64 x^{4}-32 \,{\mathrm e}^{x} x^{2}+32 x^{3}+4 x^{2}-4 x +1}{256 x^{2} \left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x +x^{2}\right )}\) | \(76\) |
Input:
int((256*x^4*exp(x)^3+(-768*x^5-32*x^4+48*x^3)*exp(x)^2+(768*x^6+32*x^5-48 *x^4-20*x^3+4*x^2+x-1)*exp(x)-256*x^7+16*x^4+4*x^3-6*x^2+2*x)/(128*x^3*exp (x)^3-384*exp(x)^2*x^4+384*x^5*exp(x)-128*x^6),x,method=_RETURNVERBOSE)
Output:
x^2-1/256/x^2*(64*x^4-64*exp(x)*x^3-32*x^3+32*exp(x)*x^2-4*x^2+4*x-1)/(x-e xp(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).
Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.00 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {256 \, x^{6} + 256 \, x^{4} e^{\left (2 \, x\right )} - 64 \, x^{4} + 32 \, x^{3} + 4 \, x^{2} - 32 \, {\left (16 \, x^{5} - 2 \, x^{3} + x^{2}\right )} e^{x} - 4 \, x + 1}{256 \, {\left (x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )}} \] Input:
integrate((256*x^4*exp(x)^3+(-768*x^5-32*x^4+48*x^3)*exp(x)^2+(768*x^6+32* x^5-48*x^4-20*x^3+4*x^2+x-1)*exp(x)-256*x^7+16*x^4+4*x^3-6*x^2+2*x)/(128*x ^3*exp(x)^3-384*exp(x)^2*x^4+384*x^5*exp(x)-128*x^6),x, algorithm="fricas" )
Output:
1/256*(256*x^6 + 256*x^4*e^(2*x) - 64*x^4 + 32*x^3 + 4*x^2 - 32*(16*x^5 - 2*x^3 + x^2)*e^x - 4*x + 1)/(x^4 - 2*x^3*e^x + x^2*e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=x^{2} + \frac {- 16 x^{4} + 8 x^{3} + x^{2} - x + \left (16 x^{3} - 8 x^{2}\right ) e^{x} + \frac {1}{4}}{64 x^{4} - 128 x^{3} e^{x} + 64 x^{2} e^{2 x}} \] Input:
integrate((256*x**4*exp(x)**3+(-768*x**5-32*x**4+48*x**3)*exp(x)**2+(768*x **6+32*x**5-48*x**4-20*x**3+4*x**2+x-1)*exp(x)-256*x**7+16*x**4+4*x**3-6*x **2+2*x)/(128*x**3*exp(x)**3-384*exp(x)**2*x**4+384*x**5*exp(x)-128*x**6), x)
Output:
x**2 + (-16*x**4 + 8*x**3 + x**2 - x + (16*x**3 - 8*x**2)*exp(x) + 1/4)/(6 4*x**4 - 128*x**3*exp(x) + 64*x**2*exp(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.00 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {256 \, x^{6} + 256 \, x^{4} e^{\left (2 \, x\right )} - 64 \, x^{4} + 32 \, x^{3} + 4 \, x^{2} - 32 \, {\left (16 \, x^{5} - 2 \, x^{3} + x^{2}\right )} e^{x} - 4 \, x + 1}{256 \, {\left (x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )}} \] Input:
integrate((256*x^4*exp(x)^3+(-768*x^5-32*x^4+48*x^3)*exp(x)^2+(768*x^6+32* x^5-48*x^4-20*x^3+4*x^2+x-1)*exp(x)-256*x^7+16*x^4+4*x^3-6*x^2+2*x)/(128*x ^3*exp(x)^3-384*exp(x)^2*x^4+384*x^5*exp(x)-128*x^6),x, algorithm="maxima" )
Output:
1/256*(256*x^6 + 256*x^4*e^(2*x) - 64*x^4 + 32*x^3 + 4*x^2 - 32*(16*x^5 - 2*x^3 + x^2)*e^x - 4*x + 1)/(x^4 - 2*x^3*e^x + x^2*e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (26) = 52\).
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {256 \, x^{6} - 512 \, x^{5} e^{x} + 256 \, x^{4} e^{\left (2 \, x\right )} - 64 \, x^{4} + 64 \, x^{3} e^{x} + 32 \, x^{3} - 32 \, x^{2} e^{x} + 4 \, x^{2} - 4 \, x + 1}{256 \, {\left (x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )}} \] Input:
integrate((256*x^4*exp(x)^3+(-768*x^5-32*x^4+48*x^3)*exp(x)^2+(768*x^6+32* x^5-48*x^4-20*x^3+4*x^2+x-1)*exp(x)-256*x^7+16*x^4+4*x^3-6*x^2+2*x)/(128*x ^3*exp(x)^3-384*exp(x)^2*x^4+384*x^5*exp(x)-128*x^6),x, algorithm="giac")
Output:
1/256*(256*x^6 - 512*x^5*e^x + 256*x^4*e^(2*x) - 64*x^4 + 64*x^3*e^x + 32* x^3 - 32*x^2*e^x + 4*x^2 - 4*x + 1)/(x^4 - 2*x^3*e^x + x^2*e^(2*x))
Time = 3.52 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {{\left (2\,x+16\,x^2\,{\mathrm {e}}^x-16\,x^3-1\right )}^2}{256\,x^2\,{\left (x-{\mathrm {e}}^x\right )}^2} \] Input:
int((2*x + exp(x)*(x + 4*x^2 - 20*x^3 - 48*x^4 + 32*x^5 + 768*x^6 - 1) + 2 56*x^4*exp(3*x) - exp(2*x)*(32*x^4 - 48*x^3 + 768*x^5) - 6*x^2 + 4*x^3 + 1 6*x^4 - 256*x^7)/(384*x^5*exp(x) + 128*x^3*exp(3*x) - 384*x^4*exp(2*x) - 1 28*x^6),x)
Output:
(2*x + 16*x^2*exp(x) - 16*x^3 - 1)^2/(256*x^2*(x - exp(x))^2)
Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {256 e^{2 x} x^{4}+32 e^{2 x} x^{2}-512 e^{x} x^{5}-32 e^{x} x^{2}+256 x^{6}-32 x^{4}+32 x^{3}+4 x^{2}-4 x +1}{256 x^{2} \left (e^{2 x}-2 e^{x} x +x^{2}\right )} \] Input:
int((256*x^4*exp(x)^3+(-768*x^5-32*x^4+48*x^3)*exp(x)^2+(768*x^6+32*x^5-48 *x^4-20*x^3+4*x^2+x-1)*exp(x)-256*x^7+16*x^4+4*x^3-6*x^2+2*x)/(128*x^3*exp (x)^3-384*exp(x)^2*x^4+384*x^5*exp(x)-128*x^6),x)
Output:
(256*e**(2*x)*x**4 + 32*e**(2*x)*x**2 - 512*e**x*x**5 - 32*e**x*x**2 + 256 *x**6 - 32*x**4 + 32*x**3 + 4*x**2 - 4*x + 1)/(256*x**2*(e**(2*x) - 2*e**x *x + x**2))