3.99.18 \(\int \frac {e^{2 x} (1-6 x)-3 x^2-6 x^3+e^x (6 x+8 x^2)}{e^{2 x}-2 e^x x+x^2} \, dx\)

Optimal. Leaf size=30 \[ \frac {4}{3}-x+(1+x)^2+4 x \left (-x+\frac {x}{e^x-x}\right ) \]

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Rubi [F]  time = 0.63, 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 {e^{2 x} (1-6 x)-3 x^2-6 x^3+e^x \left (6 x+8 x^2\right )}{e^{2 x}-2 e^x x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(1 - 6*x) - 3*x^2 - 6*x^3 + E^x*(6*x + 8*x^2))/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

x - 3*x^2 + 8*Defer[Int][x/(E^x - x), x] + 4*Defer[Int][x^2/(E^x - x)^2, x] - 4*Defer[Int][x^2/(E^x - x), x] -
4*Defer[Int][x^3/(E^x - x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} (1-6 x)-3 x^2-6 x^3+e^x \left (6 x+8 x^2\right )}{\left (e^x-x\right )^2} \, dx\\ &=\int \left (1-6 x-\frac {4 (-2+x) x}{e^x-x}-\frac {4 (-1+x) x^2}{\left (e^x-x\right )^2}\right ) \, dx\\ &=x-3 x^2-4 \int \frac {(-2+x) x}{e^x-x} \, dx-4 \int \frac {(-1+x) x^2}{\left (e^x-x\right )^2} \, dx\\ &=x-3 x^2-4 \int \left (-\frac {2 x}{e^x-x}+\frac {x^2}{e^x-x}\right ) \, dx-4 \int \left (-\frac {x^2}{\left (e^x-x\right )^2}+\frac {x^3}{\left (e^x-x\right )^2}\right ) \, dx\\ &=x-3 x^2+4 \int \frac {x^2}{\left (e^x-x\right )^2} \, dx-4 \int \frac {x^2}{e^x-x} \, dx-4 \int \frac {x^3}{\left (e^x-x\right )^2} \, dx+8 \int \frac {x}{e^x-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 21, normalized size = 0.70 \begin {gather*} x-3 x^2+\frac {4 x^2}{e^x-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(1 - 6*x) - 3*x^2 - 6*x^3 + E^x*(6*x + 8*x^2))/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

x - 3*x^2 + (4*x^2)/(E^x - x)

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fricas [A]  time = 0.76, size = 34, normalized size = 1.13 \begin {gather*} -\frac {3 \, x^{3} + 3 \, x^{2} - {\left (3 \, x^{2} - x\right )} e^{x}}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1-6*x)*exp(x)^2+(8*x^2+6*x)*exp(x)-6*x^3-3*x^2)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="fricas")

[Out]

-(3*x^3 + 3*x^2 - (3*x^2 - x)*e^x)/(x - e^x)

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giac [A]  time = 0.18, size = 32, normalized size = 1.07 \begin {gather*} -\frac {3 \, x^{3} - 3 \, x^{2} e^{x} + 3 \, x^{2} + x e^{x}}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1-6*x)*exp(x)^2+(8*x^2+6*x)*exp(x)-6*x^3-3*x^2)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="giac")

[Out]

-(3*x^3 - 3*x^2*e^x + 3*x^2 + x*e^x)/(x - e^x)

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maple [A]  time = 0.04, size = 21, normalized size = 0.70

method	result	size
risch	\(-3 x^{2}+x -\frac {4 x^{2}}{x -{\mathrm e}^{x}}\)	\(21\)
norman	\(\frac {-3 x^{2}-3 x^{3}-{\mathrm e}^{x} x +3 \,{\mathrm e}^{x} x^{2}}{x -{\mathrm e}^{x}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1-6*x)*exp(x)^2+(8*x^2+6*x)*exp(x)-6*x^3-3*x^2)/(exp(x)^2-2*exp(x)*x+x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.42, size = 34, normalized size = 1.13 \begin {gather*} -\frac {3 \, x^{3} + 3 \, x^{2} - {\left (3 \, x^{2} - x\right )} e^{x}}{x - e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1-6*x)*exp(x)^2+(8*x^2+6*x)*exp(x)-6*x^3-3*x^2)/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="maxima")

[Out]

-(3*x^3 + 3*x^2 - (3*x^2 - x)*e^x)/(x - e^x)

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mupad [B]  time = 5.64, size = 20, normalized size = 0.67 \begin {gather*} x-\frac {4\,x^2}{x-{\mathrm {e}}^x}-3\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(6*x - 1) - exp(x)*(6*x + 8*x^2) + 3*x^2 + 6*x^3)/(exp(2*x) - 2*x*exp(x) + x^2),x)

[Out]

x - (4*x^2)/(x - exp(x)) - 3*x^2

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sympy [A]  time = 0.10, size = 15, normalized size = 0.50 \begin {gather*} - 3 x^{2} + \frac {4 x^{2}}{- x + e^{x}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1-6*x)*exp(x)**2+(8*x**2+6*x)*exp(x)-6*x**3-3*x**2)/(exp(x)**2-2*exp(x)*x+x**2),x)

[Out]

-3*x**2 + 4*x**2/(-x + exp(x)) + x

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