3.81.44 \(\int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} (8-6 x-8 x^3+12 x^4-3 x^5+e^x (-2 x^2+x^3+2 x^5-x^6))}{12 x-12 x^2+3 x^3} \, dx\)

Optimal. Leaf size=34 \[ \frac {\left (1+e^{e^x-\frac {2 (-2+x)}{x}}\right ) \left (-x+x^4\right )}{3 (2-x)} \]

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Rubi [F]  time = 2.99, 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 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{12 x-12 x^2+3 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x + 8*x^4 - 3*x^5 + E^((4 - 2*x + E^x*x)/x)*(8 - 6*x - 8*x^3 + 12*x^4 - 3*x^5 + E^x*(-2*x^2 + x^3 + 2*
x^5 - x^6)))/(12*x - 12*x^2 + 3*x^3),x]

[Out]

14/(3*(2 - x)) - (4*x)/3 - (2*x^2)/3 - x^3/3 + (4*Defer[Int][E^(-2 + E^x + 4/x), x])/3 - (7*Defer[Int][E^(-2 +
E^x + 4/x + x), x])/3 + (14*Defer[Int][E^(-2 + E^x + 4/x)/(-2 + x)^2, x])/3 + (14*Defer[Int][E^(-2 + E^x + 4/
x)/(-2 + x), x])/3 - (14*Defer[Int][E^(-2 + E^x + 4/x + x)/(-2 + x), x])/3 + (2*Defer[Int][E^(-2 + E^x + 4/x)/
x, x])/3 - (4*Defer[Int][E^(-2 + E^x + 4/x + x)*x, x])/3 - Defer[Int][E^(-2 + E^x + 4/x)*x^2, x] - (2*Defer[In
t][E^(-2 + E^x + 4/x + x)*x^2, x])/3 - Defer[Int][E^(-2 + E^x + 4/x + x)*x^3, x]/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{x \left (12-12 x+3 x^2\right )} \, dx\\ &=\int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{3 (-2+x)^2 x} \, dx\\ &=\frac {1}{3} \int \frac {-2 x+8 x^4-3 x^5+e^{\frac {4-2 x+e^x x}{x}} \left (8-6 x-8 x^3+12 x^4-3 x^5+e^x \left (-2 x^2+x^3+2 x^5-x^6\right )\right )}{(-2+x)^2 x} \, dx\\ &=\frac {1}{3} \int \left (-\frac {2}{(-2+x)^2}-\frac {6 e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2}+\frac {8 e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2 x}-\frac {8 e^{-2+e^x+\frac {4}{x}} x^2}{(-2+x)^2}+\frac {8 x^3}{(-2+x)^2}+\frac {12 e^{-2+e^x+\frac {4}{x}} x^3}{(-2+x)^2}-\frac {3 x^4}{(-2+x)^2}-\frac {3 e^{-2+e^x+\frac {4}{x}} x^4}{(-2+x)^2}-\frac {e^{-2+e^x+\frac {4}{x}+x} x \left (-1+x^3\right )}{-2+x}\right ) \, dx\\ &=-\frac {2}{3 (2-x)}-\frac {1}{3} \int \frac {e^{-2+e^x+\frac {4}{x}+x} x \left (-1+x^3\right )}{-2+x} \, dx-2 \int \frac {e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2} \, dx+\frac {8}{3} \int \frac {e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2 x} \, dx-\frac {8}{3} \int \frac {e^{-2+e^x+\frac {4}{x}} x^2}{(-2+x)^2} \, dx+\frac {8}{3} \int \frac {x^3}{(-2+x)^2} \, dx+4 \int \frac {e^{-2+e^x+\frac {4}{x}} x^3}{(-2+x)^2} \, dx-\int \frac {x^4}{(-2+x)^2} \, dx-\int \frac {e^{-2+e^x+\frac {4}{x}} x^4}{(-2+x)^2} \, dx\\ &=-\frac {2}{3 (2-x)}-\frac {1}{3} \int \left (7 e^{-2+e^x+\frac {4}{x}+x}+\frac {14 e^{-2+e^x+\frac {4}{x}+x}}{-2+x}+4 e^{-2+e^x+\frac {4}{x}+x} x+2 e^{-2+e^x+\frac {4}{x}+x} x^2+e^{-2+e^x+\frac {4}{x}+x} x^3\right ) \, dx-2 \int \frac {e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2} \, dx-\frac {8}{3} \int \left (e^{-2+e^x+\frac {4}{x}}+\frac {4 e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2}+\frac {4 e^{-2+e^x+\frac {4}{x}}}{-2+x}\right ) \, dx+\frac {8}{3} \int \left (\frac {e^{-2+e^x+\frac {4}{x}}}{2 (-2+x)^2}-\frac {e^{-2+e^x+\frac {4}{x}}}{4 (-2+x)}+\frac {e^{-2+e^x+\frac {4}{x}}}{4 x}\right ) \, dx+\frac {8}{3} \int \left (4+\frac {8}{(-2+x)^2}+\frac {12}{-2+x}+x\right ) \, dx+4 \int \left (4 e^{-2+e^x+\frac {4}{x}}+\frac {8 e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2}+\frac {12 e^{-2+e^x+\frac {4}{x}}}{-2+x}+e^{-2+e^x+\frac {4}{x}} x\right ) \, dx-\int \left (12+\frac {16}{(-2+x)^2}+\frac {32}{-2+x}+4 x+x^2\right ) \, dx-\int \left (12 e^{-2+e^x+\frac {4}{x}}+\frac {16 e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2}+\frac {32 e^{-2+e^x+\frac {4}{x}}}{-2+x}+4 e^{-2+e^x+\frac {4}{x}} x+e^{-2+e^x+\frac {4}{x}} x^2\right ) \, dx\\ &=\frac {14}{3 (2-x)}-\frac {4 x}{3}-\frac {2 x^2}{3}-\frac {x^3}{3}-\frac {1}{3} \int e^{-2+e^x+\frac {4}{x}+x} x^3 \, dx-\frac {2}{3} \int \frac {e^{-2+e^x+\frac {4}{x}}}{-2+x} \, dx+\frac {2}{3} \int \frac {e^{-2+e^x+\frac {4}{x}}}{x} \, dx-\frac {2}{3} \int e^{-2+e^x+\frac {4}{x}+x} x^2 \, dx+\frac {4}{3} \int \frac {e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2} \, dx-\frac {4}{3} \int e^{-2+e^x+\frac {4}{x}+x} x \, dx-2 \int \frac {e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2} \, dx-\frac {7}{3} \int e^{-2+e^x+\frac {4}{x}+x} \, dx-\frac {8}{3} \int e^{-2+e^x+\frac {4}{x}} \, dx-\frac {14}{3} \int \frac {e^{-2+e^x+\frac {4}{x}+x}}{-2+x} \, dx-\frac {32}{3} \int \frac {e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2} \, dx-\frac {32}{3} \int \frac {e^{-2+e^x+\frac {4}{x}}}{-2+x} \, dx-12 \int e^{-2+e^x+\frac {4}{x}} \, dx+16 \int e^{-2+e^x+\frac {4}{x}} \, dx-16 \int \frac {e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2} \, dx+32 \int \frac {e^{-2+e^x+\frac {4}{x}}}{(-2+x)^2} \, dx-32 \int \frac {e^{-2+e^x+\frac {4}{x}}}{-2+x} \, dx+48 \int \frac {e^{-2+e^x+\frac {4}{x}}}{-2+x} \, dx-\int e^{-2+e^x+\frac {4}{x}} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 5.15, size = 45, normalized size = 1.32 \begin {gather*} \frac {e^{e^x+\frac {4}{x}} \left (x-x^4\right )-e^2 \left (14-8 x+x^4\right )}{3 e^2 (-2+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x + 8*x^4 - 3*x^5 + E^((4 - 2*x + E^x*x)/x)*(8 - 6*x - 8*x^3 + 12*x^4 - 3*x^5 + E^x*(-2*x^2 + x^
3 + 2*x^5 - x^6)))/(12*x - 12*x^2 + 3*x^3),x]

[Out]

(E^(E^x + 4/x)*(x - x^4) - E^2*(14 - 8*x + x^4))/(3*E^2*(-2 + x))

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fricas [A]  time = 0.86, size = 37, normalized size = 1.09 \begin {gather*} -\frac {x^{4} + {\left (x^{4} - x\right )} e^{\left (\frac {x e^{x} - 2 \, x + 4}{x}\right )} - 8 \, x + 14}{3 \, {\left (x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^6+2*x^5+x^3-2*x^2)*exp(x)-3*x^5+12*x^4-8*x^3-6*x+8)*exp((exp(x)*x+4-2*x)/x)-3*x^5+8*x^4-2*x)/(
3*x^3-12*x^2+12*x),x, algorithm="fricas")

[Out]

-1/3*(x^4 + (x^4 - x)*e^((x*e^x - 2*x + 4)/x) - 8*x + 14)/(x - 2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, x^{5} - 8 \, x^{4} + {\left (3 \, x^{5} - 12 \, x^{4} + 8 \, x^{3} + {\left (x^{6} - 2 \, x^{5} - x^{3} + 2 \, x^{2}\right )} e^{x} + 6 \, x - 8\right )} e^{\left (\frac {x e^{x} - 2 \, x + 4}{x}\right )} + 2 \, x}{3 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^6+2*x^5+x^3-2*x^2)*exp(x)-3*x^5+12*x^4-8*x^3-6*x+8)*exp((exp(x)*x+4-2*x)/x)-3*x^5+8*x^4-2*x)/(
3*x^3-12*x^2+12*x),x, algorithm="giac")

[Out]

integrate(-1/3*(3*x^5 - 8*x^4 + (3*x^5 - 12*x^4 + 8*x^3 + (x^6 - 2*x^5 - x^3 + 2*x^2)*e^x + 6*x - 8)*e^((x*e^x
- 2*x + 4)/x) + 2*x)/(x^3 - 4*x^2 + 4*x), x)

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maple [A]  time = 0.20, size = 49, normalized size = 1.44

method	result	size
risch	\(-\frac {x^{3}}{3}-\frac {2 x^{2}}{3}-\frac {4 x}{3}-\frac {14}{3 \left (x -2\right )}-\frac {\left (x^{3}-1\right ) x \,{\mathrm e}^{\frac {{\mathrm e}^{x} x +4-2 x}{x}}}{3 \left (x -2\right )}\)	\(49\)
norman	\(\frac {-\frac {x^{4}}{3}+\frac {{\mathrm e}^{\frac {{\mathrm e}^{x} x +4-2 x}{x}} x}{3}-\frac {{\mathrm e}^{\frac {{\mathrm e}^{x} x +4-2 x}{x}} x^{4}}{3}+\frac {2}{3}}{x -2}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^6+2*x^5+x^3-2*x^2)*exp(x)-3*x^5+12*x^4-8*x^3-6*x+8)*exp((exp(x)*x+4-2*x)/x)-3*x^5+8*x^4-2*x)/(3*x^3-
12*x^2+12*x),x,method=_RETURNVERBOSE)

[Out]

-1/3*x^3-2/3*x^2-4/3*x-14/3/(x-2)-1/3*(x^3-1)*x/(x-2)*exp((exp(x)*x+4-2*x)/x)

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maxima [A]  time = 0.43, size = 50, normalized size = 1.47 \begin {gather*} -\frac {1}{3} \, x^{3} - \frac {2}{3} \, x^{2} - \frac {4}{3} \, x - \frac {{\left (x^{4} - x\right )} e^{\left (\frac {4}{x} + e^{x}\right )}}{3 \, {\left (x e^{2} - 2 \, e^{2}\right )}} - \frac {14}{3 \, {\left (x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^6+2*x^5+x^3-2*x^2)*exp(x)-3*x^5+12*x^4-8*x^3-6*x+8)*exp((exp(x)*x+4-2*x)/x)-3*x^5+8*x^4-2*x)/(
3*x^3-12*x^2+12*x),x, algorithm="maxima")

[Out]

-1/3*x^3 - 2/3*x^2 - 4/3*x - 1/3*(x^4 - x)*e^(4/x + e^x)/(x*e^2 - 2*e^2) - 14/3/(x - 2)

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mupad [B]  time = 6.32, size = 64, normalized size = 1.88 \begin {gather*} \frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x+\frac {4}{x}-2}}{3\,\left (x-2\right )}-\frac {14}{3\,\left (x-2\right )}-\frac {2\,x^2}{3}-\frac {x^3}{3}-\frac {x^4\,{\mathrm {e}}^{{\mathrm {e}}^x+\frac {4}{x}-2}}{3\,\left (x-2\right )}-\frac {4\,x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 8*x^4 + 3*x^5 + exp((x*exp(x) - 2*x + 4)/x)*(6*x + exp(x)*(2*x^2 - x^3 - 2*x^5 + x^6) + 8*x^3 - 12
*x^4 + 3*x^5 - 8))/(12*x - 12*x^2 + 3*x^3),x)

[Out]

(x*exp(exp(x) + 4/x - 2))/(3*(x - 2)) - 14/(3*(x - 2)) - (2*x^2)/3 - x^3/3 - (x^4*exp(exp(x) + 4/x - 2))/(3*(x
- 2)) - (4*x)/3

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sympy [A]  time = 0.27, size = 46, normalized size = 1.35 \begin {gather*} - \frac {x^{3}}{3} - \frac {2 x^{2}}{3} - \frac {4 x}{3} + \frac {\left (- x^{4} + x\right ) e^{\frac {x e^{x} - 2 x + 4}{x}}}{3 x - 6} - \frac {14}{3 x - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**6+2*x**5+x**3-2*x**2)*exp(x)-3*x**5+12*x**4-8*x**3-6*x+8)*exp((exp(x)*x+4-2*x)/x)-3*x**5+8*x*
*4-2*x)/(3*x**3-12*x**2+12*x),x)

[Out]

-x**3/3 - 2*x**2/3 - 4*x/3 + (-x**4 + x)*exp((x*exp(x) - 2*x + 4)/x)/(3*x - 6) - 14/(3*x - 6)

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