3.34.2 \(\int \frac {2 e^{3 x} x-6 e^{2 x} x^2-2 x^4-8 e^4 x^4+e^x (6 x^3+e^4 (16 x^3-8 x^4))}{e^{4+3 x}-3 e^{4+2 x} x+3 e^{4+x} x^2-e^4 x^3} \, dx\)

Optimal. Leaf size=23 \[ -2+\frac {x^2}{e^4}+\frac {4 x^4}{\left (e^x-x\right )^2} \]

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Rubi [F]  time = 0.96, 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 {2 e^{3 x} x-6 e^{2 x} x^2-2 x^4-8 e^4 x^4+e^x \left (6 x^3+e^4 \left (16 x^3-8 x^4\right )\right )}{e^{4+3 x}-3 e^{4+2 x} x+3 e^{4+x} x^2-e^4 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*E^(3*x)*x - 6*E^(2*x)*x^2 - 2*x^4 - 8*E^4*x^4 + E^x*(6*x^3 + E^4*(16*x^3 - 8*x^4)))/(E^(4 + 3*x) - 3*E^
(4 + 2*x)*x + 3*E^(4 + x)*x^2 - E^4*x^3),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{3 x} x-6 e^{2 x} x^2+\left (-2-8 e^4\right ) x^4+e^x \left (6 x^3+e^4 \left (16 x^3-8 x^4\right )\right )}{e^{4+3 x}-3 e^{4+2 x} x+3 e^{4+x} x^2-e^4 x^3} \, dx\\ &=\int \frac {2 x \left (e^{3 x}-3 e^{2 x} x+3 e^x x^2-4 e^{4+x} (-2+x) x^2-\left (1+4 e^4\right ) x^3\right )}{e^4 \left (e^x-x\right )^3} \, dx\\ &=\frac {2 \int \frac {x \left (e^{3 x}-3 e^{2 x} x+3 e^x x^2-4 e^{4+x} (-2+x) x^2-\left (1+4 e^4\right ) x^3\right )}{\left (e^x-x\right )^3} \, dx}{e^4}\\ &=\frac {2 \int \left (x-\frac {4 e^4 (-2+x) x^3}{\left (e^x-x\right )^2}-\frac {4 e^4 (-1+x) x^4}{\left (e^x-x\right )^3}\right ) \, dx}{e^4}\\ &=\frac {x^2}{e^4}-8 \int \frac {(-2+x) x^3}{\left (e^x-x\right )^2} \, dx-8 \int \frac {(-1+x) x^4}{\left (e^x-x\right )^3} \, dx\\ &=\frac {x^2}{e^4}-8 \int \left (-\frac {2 x^3}{\left (e^x-x\right )^2}+\frac {x^4}{\left (e^x-x\right )^2}\right ) \, dx-8 \int \left (-\frac {x^4}{\left (e^x-x\right )^3}+\frac {x^5}{\left (e^x-x\right )^3}\right ) \, dx\\ &=\frac {x^2}{e^4}+8 \int \frac {x^4}{\left (e^x-x\right )^3} \, dx-8 \int \frac {x^4}{\left (e^x-x\right )^2} \, dx-8 \int \frac {x^5}{\left (e^x-x\right )^3} \, dx+16 \int \frac {x^3}{\left (e^x-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 30, normalized size = 1.30 \begin {gather*} -\frac {2 \left (-\frac {x^2}{2}-\frac {2 e^4 x^4}{\left (e^x-x\right )^2}\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*E^(3*x)*x - 6*E^(2*x)*x^2 - 2*x^4 - 8*E^4*x^4 + E^x*(6*x^3 + E^4*(16*x^3 - 8*x^4)))/(E^(4 + 3*x)
- 3*E^(4 + 2*x)*x + 3*E^(4 + x)*x^2 - E^4*x^3),x]

[Out]

(-2*(-1/2*x^2 - (2*E^4*x^4)/(E^x - x)^2))/E^4

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fricas [B]  time = 0.59, size = 56, normalized size = 2.43 \begin {gather*} \frac {4 \, x^{4} e^{12} + x^{4} e^{8} - 2 \, x^{3} e^{\left (x + 8\right )} + x^{2} e^{\left (2 \, x + 8\right )}}{x^{2} e^{12} - 2 \, x e^{\left (x + 12\right )} + e^{\left (2 \, x + 12\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)^3-6*exp(x)^2*x^2+((-8*x^4+16*x^3)*exp(4)+6*x^3)*exp(x)-8*x^4*exp(4)-2*x^4)/(exp(4)*exp(x
)^3-3*x*exp(4)*exp(x)^2+3*x^2*exp(4)*exp(x)-x^3*exp(4)),x, algorithm="fricas")

[Out]

(4*x^4*e^12 + x^4*e^8 - 2*x^3*e^(x + 8) + x^2*e^(2*x + 8))/(x^2*e^12 - 2*x*e^(x + 12) + e^(2*x + 12))

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giac [B]  time = 0.19, size = 49, normalized size = 2.13 \begin {gather*} \frac {4 \, x^{4} e^{4} + x^{4} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}}{x^{2} e^{4} - 2 \, x e^{\left (x + 4\right )} + e^{\left (2 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)^3-6*exp(x)^2*x^2+((-8*x^4+16*x^3)*exp(4)+6*x^3)*exp(x)-8*x^4*exp(4)-2*x^4)/(exp(4)*exp(x
)^3-3*x*exp(4)*exp(x)^2+3*x^2*exp(4)*exp(x)-x^3*exp(4)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.15, size = 21, normalized size = 0.91

method	result	size
risch	\(x^{2} {\mathrm e}^{-4}+\frac {4 x^{4}}{\left (x -{\mathrm e}^{x}\right )^{2}}\)	\(21\)
norman	\(\frac {x^{2} {\mathrm e}^{-4} {\mathrm e}^{2 x}+\left (4 \,{\mathrm e}^{4}+1\right ) {\mathrm e}^{-4} x^{4}-2 \,{\mathrm e}^{-4} x^{3} {\mathrm e}^{x}}{\left (x -{\mathrm e}^{x}\right )^{2}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(x)^3-6*exp(x)^2*x^2+((-8*x^4+16*x^3)*exp(4)+6*x^3)*exp(x)-8*x^4*exp(4)-2*x^4)/(exp(4)*exp(x)^3-3*
x*exp(4)*exp(x)^2+3*x^2*exp(4)*exp(x)-x^3*exp(4)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.87, size = 49, normalized size = 2.13 \begin {gather*} \frac {x^{4} {\left (4 \, e^{4} + 1\right )} - 2 \, x^{3} e^{x} + x^{2} e^{\left (2 \, x\right )}}{x^{2} e^{4} - 2 \, x e^{\left (x + 4\right )} + e^{\left (2 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)^3-6*exp(x)^2*x^2+((-8*x^4+16*x^3)*exp(4)+6*x^3)*exp(x)-8*x^4*exp(4)-2*x^4)/(exp(4)*exp(x
)^3-3*x*exp(4)*exp(x)^2+3*x^2*exp(4)*exp(x)-x^3*exp(4)),x, algorithm="maxima")

[Out]

(x^4*(4*e^4 + 1) - 2*x^3*e^x + x^2*e^(2*x))/(x^2*e^4 - 2*x*e^(x + 4) + e^(2*x + 4))

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mupad [B]  time = 0.19, size = 49, normalized size = 2.13 \begin {gather*} \frac {x^4\,\left (4\,{\mathrm {e}}^4+1\right )-2\,x^3\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}}{{\mathrm {e}}^{2\,x+4}-2\,x\,{\mathrm {e}}^{x+4}+x^2\,{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x^2*exp(2*x) - exp(x)*(exp(4)*(16*x^3 - 8*x^4) + 6*x^3) - 2*x*exp(3*x) + 8*x^4*exp(4) + 2*x^4)/(exp(3*
x)*exp(4) - x^3*exp(4) - 3*x*exp(2*x)*exp(4) + 3*x^2*exp(4)*exp(x)),x)

[Out]

(x^4*(4*exp(4) + 1) - 2*x^3*exp(x) + x^2*exp(2*x))/(exp(2*x + 4) - 2*x*exp(x + 4) + x^2*exp(4))

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sympy [A]  time = 0.15, size = 26, normalized size = 1.13 \begin {gather*} \frac {4 x^{4}}{x^{2} - 2 x e^{x} + e^{2 x}} + \frac {x^{2}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(x)**3-6*exp(x)**2*x**2+((-8*x**4+16*x**3)*exp(4)+6*x**3)*exp(x)-8*x**4*exp(4)-2*x**4)/(exp(
4)*exp(x)**3-3*x*exp(4)*exp(x)**2+3*x**2*exp(4)*exp(x)-x**3*exp(4)),x)

[Out]

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

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