\(\int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} (48 x^3-32 x^4-768 x^5)+e^x (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6)}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx\) [8507]

Optimal result

Integrand size = 125, antiderivative size = 25 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\left (x+\frac {1-2 x}{16 x \left (-e^x+x\right )}\right )^2 \]

[Out]

(1/x/(16*x-16*exp(x))*(1-2*x)+x)^2

Rubi [F]

\[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx \]

[In]

Int[(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]

[Out]

x^2 + Defer[Int][(E^x - x)^(-3), x]/16 - (5*Defer[Int][(E^x - x)^(-2), x])/32 + (3*Defer[Int][(E^x - x)^(-1),
x])/8 - Defer[Int][1/((E^x - x)^2*x^3), x]/128 + Defer[Int][1/((E^x - x)^2*x^2), x]/128 + Defer[Int][1/((E^x -
x)^2*x), x]/32 - Defer[Int][x/(E^x - x)^3, x]/32 + (3*Defer[Int][x/(E^x - x)^2, x])/8 - Defer[Int][x/(E^x - x
), x]/4 - Defer[Int][x^2/(E^x - x)^2, x]/4 - Defer[Int][1/(x^2*(-E^x + x)^3), x]/128 + (5*Defer[Int][1/(x*(-E^
x + x)^3), x])/128

Rubi steps \begin{align*} \text {integral}& = \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 \left (e^x-x\right )^3 x^3} \, dx \\ & = \frac {1}{128} \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 )}{\left (e^x-x\right )^3 x^3} \, dx \\ & = \frac {1}{128} \int \left (256 x-\frac {16 (-3+2 x)}{e^x-x}+\frac {(-1+x) (-1+2 x)^2}{x^2 \left (-e^x+x\right )^3}-\frac {1-x-4 x^2+20 x^3-48 x^4+32 x^5}{\left (e^x-x\right )^2 x^3}\right ) \, dx \\ & = x^2+\frac {1}{128} \int \frac {(-1+x) (-1+2 x)^2}{x^2 \left (-e^x+x\right )^3} \, dx-\frac {1}{128} \int \frac {1-x-4 x^2+20 x^3-48 x^4+32 x^5}{\left (e^x-x\right )^2 x^3} \, dx-\frac {1}{8} \int \frac {-3+2 x}{e^x-x} \, dx \\ & = x^2-\frac {1}{128} \int \left (\frac {20}{\left (e^x-x\right )^2}+\frac {1}{\left (e^x-x\right )^2 x^3}-\frac {1}{\left (e^x-x\right )^2 x^2}-\frac {4}{\left (e^x-x\right )^2 x}-\frac {48 x}{\left (e^x-x\right )^2}+\frac {32 x^2}{\left (e^x-x\right )^2}\right ) \, dx+\frac {1}{128} \int \left (\frac {8}{\left (e^x-x\right )^3}-\frac {4 x}{\left (e^x-x\right )^3}-\frac {1}{x^2 \left (-e^x+x\right )^3}+\frac {5}{x \left (-e^x+x\right )^3}\right ) \, dx-\frac {1}{8} \int \left (-\frac {3}{e^x-x}+\frac {2 x}{e^x-x}\right ) \, dx \\ & = x^2-\frac {1}{128} \int \frac {1}{\left (e^x-x\right )^2 x^3} \, dx+\frac {1}{128} \int \frac {1}{\left (e^x-x\right )^2 x^2} \, dx-\frac {1}{128} \int \frac {1}{x^2 \left (-e^x+x\right )^3} \, dx+\frac {1}{32} \int \frac {1}{\left (e^x-x\right )^2 x} \, dx-\frac {1}{32} \int \frac {x}{\left (e^x-x\right )^3} \, dx+\frac {5}{128} \int \frac {1}{x \left (-e^x+x\right )^3} \, dx+\frac {1}{16} \int \frac {1}{\left (e^x-x\right )^3} \, dx-\frac {5}{32} \int \frac {1}{\left (e^x-x\right )^2} \, dx-\frac {1}{4} \int \frac {x}{e^x-x} \, dx-\frac {1}{4} \int \frac {x^2}{\left (e^x-x\right )^2} \, dx+\frac {3}{8} \int \frac {1}{e^x-x} \, dx+\frac {3}{8} \int \frac {x}{\left (e^x-x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 6.68 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {\left (1-2 x-16 e^x x^2+16 x^3\right )^2}{256 \left (e^x-x\right )^2 x^2} \]

[In]

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 - 12
8*x^6),x]

[Out]

(1 - 2*x - 16*E^x*x^2 + 16*x^3)^2/(256*(E^x - x)^2*x^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(51\) vs. \(2(23)=46\).

Time = 0.10 (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	\(\frac {256 \,{\mathrm e}^{2 x} x^{4}-512 x^{5} {\mathrm e}^{x}+256 x^{6}+64 \,{\mathrm e}^{x} x^{3}-64 x^{4}-32 \,{\mathrm e}^{x} x^{2}+32 x^{3}+4 x^{2}-4 x +1}{256 x^{2} \left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} x +x^{2}\right )}\)	\(76\)

[In]

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)

[Out]

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-exp(x))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.00 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {256 \, x^{6} + 256 \, x^{4} e^{\left (2 \, x\right )} - 64 \, x^{4} + 32 \, x^{3} + 4 \, x^{2} - 32 \, {\left (16 \, x^{5} - 2 \, x^{3} + x^{2}\right )} e^{x} - 4 \, x + 1}{256 \, {\left (x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )}} \]

[In]

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="frica
s")

[Out]

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))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (17) = 34\).

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=x^{2} + \frac {- 16 x^{4} + 8 x^{3} + x^{2} - x + \left (16 x^{3} - 8 x^{2}\right ) e^{x} + \frac {1}{4}}{64 x^{4} - 128 x^{3} e^{x} + 64 x^{2} e^{2 x}} \]

[In]

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)

[Out]

x**2 + (-16*x**4 + 8*x**3 + x**2 - x + (16*x**3 - 8*x**2)*exp(x) + 1/4)/(64*x**4 - 128*x**3*exp(x) + 64*x**2*e
xp(2*x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.00 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {256 \, x^{6} + 256 \, x^{4} e^{\left (2 \, x\right )} - 64 \, x^{4} + 32 \, x^{3} + 4 \, x^{2} - 32 \, {\left (16 \, x^{5} - 2 \, x^{3} + x^{2}\right )} e^{x} - 4 \, x + 1}{256 \, {\left (x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )}} \]

[In]

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="maxim
a")

[Out]

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))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (26) = 52\).

Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {256 \, x^{6} - 512 \, x^{5} e^{x} + 256 \, x^{4} e^{\left (2 \, x\right )} - 64 \, x^{4} + 64 \, x^{3} e^{x} + 32 \, x^{3} - 32 \, x^{2} e^{x} + 4 \, x^{2} - 4 \, x + 1}{256 \, {\left (x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}\right )}} \]

[In]

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"
)

[Out]

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))

Mupad [B] (verification not implemented)

Time = 12.68 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {2 x-6 x^2+4 x^3+16 x^4+256 e^{3 x} x^4-256 x^7+e^{2 x} \left (48 x^3-32 x^4-768 x^5\right )+e^x \left (-1+x+4 x^2-20 x^3-48 x^4+32 x^5+768 x^6\right )}{128 e^{3 x} x^3-384 e^{2 x} x^4+384 e^x x^5-128 x^6} \, dx=\frac {{\left (2\,x+16\,x^2\,{\mathrm {e}}^x-16\,x^3-1\right )}^2}{256\,x^2\,{\left (x-{\mathrm {e}}^x\right )}^2} \]

[In]

int((2*x + exp(x)*(x + 4*x^2 - 20*x^3 - 48*x^4 + 32*x^5 + 768*x^6 - 1) + 256*x^4*exp(3*x) - exp(2*x)*(32*x^4 -
48*x^3 + 768*x^5) - 6*x^2 + 4*x^3 + 16*x^4 - 256*x^7)/(384*x^5*exp(x) + 128*x^3*exp(3*x) - 384*x^4*exp(2*x) -
128*x^6),x)

[Out]

(2*x + 16*x^2*exp(x) - 16*x^3 - 1)^2/(256*x^2*(x - exp(x))^2)