3.8.93 \(\int \frac {x^2-512 x^3+96 x^4-4 x^5+e^{2 x} (1-512 x+96 x^2-4 x^3)+e^x (1-3 x+1024 x^2-192 x^3+8 x^4)}{e^{2 x}-2 e^x x+x^2} \, dx\)

Optimal. Leaf size=24 \[ 4+x+\frac {x}{e^x-x}-(-16+x)^2 x^2 \]

________________________________________________________________________________________

Rubi [F]  time = 0.67, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^2-512 x^3+96 x^4-4 x^5+e^{2 x} \left (1-512 x+96 x^2-4 x^3\right )+e^x \left (1-3 x+1024 x^2-192 x^3+8 x^4\right )}{e^{2 x}-2 e^x x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2 - 512*x^3 + 96*x^4 - 4*x^5 + E^(2*x)*(1 - 512*x + 96*x^2 - 4*x^3) + E^x*(1 - 3*x + 1024*x^2 - 192*x^3
+ 8*x^4))/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

x - 256*x^2 + 32*x^3 - x^4 + Defer[Int][(E^x - x)^(-1), x] + Defer[Int][x/(E^x - x)^2, x] - Defer[Int][x^2/(E^
x - x)^2, x] + Defer[Int][x/(-E^x + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2-512 x^3+96 x^4-4 x^5+e^{2 x} \left (1-512 x+96 x^2-4 x^3\right )+e^x \left (1-3 x+1024 x^2-192 x^3+8 x^4\right )}{\left (e^x-x\right )^2} \, dx\\ &=\int \left (1-512 x-\frac {(-1+x) x}{\left (e^x-x\right )^2}+96 x^2-4 x^3+\frac {-1+x}{-e^x+x}\right ) \, dx\\ &=x-256 x^2+32 x^3-x^4-\int \frac {(-1+x) x}{\left (e^x-x\right )^2} \, dx+\int \frac {-1+x}{-e^x+x} \, dx\\ &=x-256 x^2+32 x^3-x^4-\int \left (-\frac {x}{\left (e^x-x\right )^2}+\frac {x^2}{\left (e^x-x\right )^2}\right ) \, dx+\int \left (\frac {1}{e^x-x}+\frac {x}{-e^x+x}\right ) \, dx\\ &=x-256 x^2+32 x^3-x^4+\int \frac {1}{e^x-x} \, dx+\int \frac {x}{\left (e^x-x\right )^2} \, dx-\int \frac {x^2}{\left (e^x-x\right )^2} \, dx+\int \frac {x}{-e^x+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.19, size = 28, normalized size = 1.17 \begin {gather*} x+\frac {x}{e^x-x}-256 x^2+32 x^3-x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 - 512*x^3 + 96*x^4 - 4*x^5 + E^(2*x)*(1 - 512*x + 96*x^2 - 4*x^3) + E^x*(1 - 3*x + 1024*x^2 - 1
92*x^3 + 8*x^4))/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

x + x/(E^x - x) - 256*x^2 + 32*x^3 - x^4

________________________________________________________________________________________

fricas [B]  time = 0.80, size = 51, normalized size = 2.12 \begin {gather*} -\frac {x^{5} - 32 \, x^{4} + 256 \, x^{3} - x^{2} - {\left (x^{4} - 32 \, x^{3} + 256 \, x^{2} - x\right )} e^{x} + x}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+96*x^2-512*x+1)*exp(x)^2+(8*x^4-192*x^3+1024*x^2-3*x+1)*exp(x)-4*x^5+96*x^4-512*x^3+x^2)/(e
xp(x)^2-2*exp(x)*x+x^2),x, algorithm="fricas")

[Out]

-(x^5 - 32*x^4 + 256*x^3 - x^2 - (x^4 - 32*x^3 + 256*x^2 - x)*e^x + x)/(x - e^x)

________________________________________________________________________________________

giac [B]  time = 0.31, size = 55, normalized size = 2.29 \begin {gather*} -\frac {x^{5} - x^{4} e^{x} - 32 \, x^{4} + 32 \, x^{3} e^{x} + 256 \, x^{3} - 256 \, x^{2} e^{x} - x^{2} + x e^{x} + x}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+96*x^2-512*x+1)*exp(x)^2+(8*x^4-192*x^3+1024*x^2-3*x+1)*exp(x)-4*x^5+96*x^4-512*x^3+x^2)/(e
xp(x)^2-2*exp(x)*x+x^2),x, algorithm="giac")

[Out]

-(x^5 - x^4*e^x - 32*x^4 + 32*x^3*e^x + 256*x^3 - 256*x^2*e^x - x^2 + x*e^x + x)/(x - e^x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 29, normalized size = 1.21

method	result	size
risch	\(-x^{4}+32 x^{3}-256 x^{2}+x -\frac {x}{x -{\mathrm e}^{x}}\)	\(29\)
norman	\(\frac {x^{2}-x +{\mathrm e}^{x} x^{4}-256 x^{3}+32 x^{4}-x^{5}-{\mathrm e}^{x} x +256 \,{\mathrm e}^{x} x^{2}-32 \,{\mathrm e}^{x} x^{3}}{x -{\mathrm e}^{x}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3+96*x^2-512*x+1)*exp(x)^2+(8*x^4-192*x^3+1024*x^2-3*x+1)*exp(x)-4*x^5+96*x^4-512*x^3+x^2)/(exp(x)^
2-2*exp(x)*x+x^2),x,method=_RETURNVERBOSE)

[Out]

-x^4+32*x^3-256*x^2+x-x/(x-exp(x))

________________________________________________________________________________________

maxima [B]  time = 0.61, size = 51, normalized size = 2.12 \begin {gather*} -\frac {x^{5} - 32 \, x^{4} + 256 \, x^{3} - x^{2} - {\left (x^{4} - 32 \, x^{3} + 256 \, x^{2} - x\right )} e^{x} + x}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+96*x^2-512*x+1)*exp(x)^2+(8*x^4-192*x^3+1024*x^2-3*x+1)*exp(x)-4*x^5+96*x^4-512*x^3+x^2)/(e
xp(x)^2-2*exp(x)*x+x^2),x, algorithm="maxima")

[Out]

-(x^5 - 32*x^4 + 256*x^3 - x^2 - (x^4 - 32*x^3 + 256*x^2 - x)*e^x + x)/(x - e^x)

________________________________________________________________________________________

mupad [B]  time = 0.63, size = 28, normalized size = 1.17 \begin {gather*} x-\frac {x}{x-{\mathrm {e}}^x}-256\,x^2+32\,x^3-x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(1024*x^2 - 3*x - 192*x^3 + 8*x^4 + 1) - exp(2*x)*(512*x - 96*x^2 + 4*x^3 - 1) + x^2 - 512*x^3 + 9
6*x^4 - 4*x^5)/(exp(2*x) - 2*x*exp(x) + x^2),x)

[Out]

x - x/(x - exp(x)) - 256*x^2 + 32*x^3 - x^4

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 20, normalized size = 0.83 \begin {gather*} - x^{4} + 32 x^{3} - 256 x^{2} + x + \frac {x}{- x + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3+96*x**2-512*x+1)*exp(x)**2+(8*x**4-192*x**3+1024*x**2-3*x+1)*exp(x)-4*x**5+96*x**4-512*x**
3+x**2)/(exp(x)**2-2*exp(x)*x+x**2),x)

[Out]

-x**4 + 32*x**3 - 256*x**2 + x + x/(-x + exp(x))

________________________________________________________________________________________