3.38.69 \(\int \frac {e^{\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+(4 x^2+3 x^3+e^x (-4 x-x^2)) \log (x)}{2 e^x-2 x}} (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x (8 x+2 x^2)+(e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x (8 x+10 x^2-2 x^3)) \log (x))}{2 e^{2 x}-4 e^x x+2 x^2} \, dx\)

Optimal. Leaf size=38 \[ e^{4-e^x+x^2 \left (\frac {1}{2} \left (-1-\frac {4}{x}\right )+\frac {x}{e^x-x}\right ) \log (x)} \]

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Rubi [F]  time = 31.86, 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 {\exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 e^x-2 x}\right ) \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-2*E^(2*x) - 8*x + E^x*(8 + 2*x) + (4*x^2 + 3*x^3 + E^x*(-4*x - x^2))*Log[x])/(2*E^x - 2*x))*(-2*E^(3
*x) - 4*x^2 - 3*x^3 + E^(2*x)*(-4 + 3*x) + E^x*(8*x + 2*x^2) + (E^(2*x)*(-4 - 2*x) - 4*x^2 - 6*x^3 + E^x*(8*x
+ 10*x^2 - 2*x^3))*Log[x]))/(2*E^(2*x) - 4*E^x*x + 2*x^2),x]

[Out]

-2*Defer[Int][E^(4 - E^x)*x^((x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))), x] - 2*Log[x]*Defer[Int][E^(4
- E^x)*x^((x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))), x] - Defer[Int][E^(4 - E^x + x)*x^((x*(-(E^x*(4 +
x)) + x*(4 + 3*x)))/(2*(E^x - x))), x] - Defer[Int][E^(4 - E^x)*x^(1 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*
(E^x - x))), x]/2 - Log[x]*Defer[Int][E^(4 - E^x)*x^(1 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))), x]
+ Defer[Int][(E^(4 - E^x)*x^(2 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))))/(E^x - x), x] + 3*Log[x]*
Defer[Int][(E^(4 - E^x)*x^(2 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))))/(E^x - x), x] + Log[x]*Defer
[Int][(E^(4 - E^x)*x^(3 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))))/(E^x - x)^2, x] - Log[x]*Defer[In
t][(E^(4 - E^x)*x^(3 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))))/(E^x - x), x] - Log[x]*Defer[Int][(E
^(4 - E^x)*x^(4 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))))/(E^x - x)^2, x] + 2*Defer[Int][Defer[Int]
[E^(4 - E^x)*x^((x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))), x]/x, x] + Defer[Int][Defer[Int][E^(4 - E^x
)*x^(1 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))), x]/x, x] - 3*Defer[Int][Defer[Int][(E^(4 - E^x)*x^
(2 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))))/(E^x - x), x]/x, x] - Defer[Int][Defer[Int][(E^(4 - E^
x)*x^(3 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))))/(E^x - x)^2, x]/x, x] + Defer[Int][Defer[Int][(E^
(4 - E^x)*x^(3 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))))/(E^x - x), x]/x, x] + Defer[Int][Defer[Int
][(E^(4 - E^x)*x^(4 + (x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^x - x))))/(E^x - x)^2, x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 \left (e^x-x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {\exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{\left (e^x-x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-4 \exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right )-2 \exp \left (x+\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right )-\exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) x-4 \exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) \log (x)-2 \exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) x \log (x)-\frac {2 \exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) (-1+x) x^3 \log (x)}{\left (e^x-x\right )^2}-\frac {2 \exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) x^2 (-1-3 \log (x)+x \log (x))}{e^x-x}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) x \, dx\right )-2 \int \exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) \, dx-2 \int \exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) \log (x) \, dx-\int \exp \left (x+\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) \, dx-\int \exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) x \log (x) \, dx-\int \frac {\exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) (-1+x) x^3 \log (x)}{\left (e^x-x\right )^2} \, dx-\int \frac {\exp \left (\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 \left (e^x-x\right )}\right ) x^2 (-1-3 \log (x)+x \log (x))}{e^x-x} \, dx\\ &=-\left (\frac {1}{2} \int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx\right )-2 \int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+2 \int \frac {\int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx}{x} \, dx-\log (x) \int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+\log (x) \int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx-\log (x) \int \frac {e^{4-e^x} x^{4+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx-(2 \log (x)) \int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx-\int e^{4-e^x+x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx-\int \frac {e^{4-e^x} x^{2+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} (-1+(-3+x) \log (x))}{e^x-x} \, dx+\int \frac {\int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx}{x} \, dx+\int \frac {-\int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx+\int \frac {e^{4-e^x} x^{4+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx}{x} \, dx\\ &=-\left (\frac {1}{2} \int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx\right )-2 \int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+2 \int \frac {\int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx}{x} \, dx-\log (x) \int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+\log (x) \int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx-\log (x) \int \frac {e^{4-e^x} x^{4+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx-(2 \log (x)) \int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx-\int e^{4-e^x+x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx-\int \left (-\frac {e^{4-e^x} x^{2+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{e^x-x}-\frac {3 e^{4-e^x} x^{2+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \log (x)}{e^x-x}+\frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \log (x)}{e^x-x}\right ) \, dx+\int \frac {\int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx}{x} \, dx+\int \left (-\frac {\int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx}{x}+\frac {\int \frac {e^{4-e^x} x^{4+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx}{x}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx\right )-2 \int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+2 \int \frac {\int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx}{x} \, dx+3 \int \frac {e^{4-e^x} x^{2+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \log (x)}{e^x-x} \, dx-\log (x) \int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+\log (x) \int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx-\log (x) \int \frac {e^{4-e^x} x^{4+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx-(2 \log (x)) \int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx-\int e^{4-e^x+x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+\int \frac {e^{4-e^x} x^{2+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{e^x-x} \, dx-\int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \log (x)}{e^x-x} \, dx+\int \frac {\int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx}{x} \, dx-\int \frac {\int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx}{x} \, dx+\int \frac {\int \frac {e^{4-e^x} x^{4+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx}{x} \, dx\\ &=-\left (\frac {1}{2} \int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx\right )-2 \int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+2 \int \frac {\int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx}{x} \, dx-3 \int \frac {\int \frac {e^{4-e^x} x^{2+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{e^x-x} \, dx}{x} \, dx-\log (x) \int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+\log (x) \int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx-\log (x) \int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{e^x-x} \, dx-\log (x) \int \frac {e^{4-e^x} x^{4+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx-(2 \log (x)) \int e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+(3 \log (x)) \int \frac {e^{4-e^x} x^{2+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{e^x-x} \, dx-\int e^{4-e^x+x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx+\int \frac {e^{4-e^x} x^{2+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{e^x-x} \, dx+\int \frac {\int e^{4-e^x} x^{1+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \, dx}{x} \, dx-\int \frac {\int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx}{x} \, dx+\int \frac {\int \frac {e^{4-e^x} x^{3+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{e^x-x} \, dx}{x} \, dx+\int \frac {\int \frac {e^{4-e^x} x^{4+\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}}}{\left (e^x-x\right )^2} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 42, normalized size = 1.11 \begin {gather*} e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-2*E^(2*x) - 8*x + E^x*(8 + 2*x) + (4*x^2 + 3*x^3 + E^x*(-4*x - x^2))*Log[x])/(2*E^x - 2*x))*(-
2*E^(3*x) - 4*x^2 - 3*x^3 + E^(2*x)*(-4 + 3*x) + E^x*(8*x + 2*x^2) + (E^(2*x)*(-4 - 2*x) - 4*x^2 - 6*x^3 + E^x
*(8*x + 10*x^2 - 2*x^3))*Log[x]))/(2*E^(2*x) - 4*E^x*x + 2*x^2),x]

[Out]

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

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fricas [A]  time = 0.82, size = 53, normalized size = 1.39 \begin {gather*} e^{\left (-\frac {2 \, {\left (x + 4\right )} e^{x} + {\left (3 \, x^{3} + 4 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x}\right )} \log \relax (x) - 8 \, x - 2 \, e^{\left (2 \, x\right )}}{2 \, {\left (x - e^{x}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-4)*exp(x)^2+(-2*x^3+10*x^2+8*x)*exp(x)-6*x^3-4*x^2)*log(x)-2*exp(x)^3+(3*x-4)*exp(x)^2+(2*x^
2+8*x)*exp(x)-3*x^3-4*x^2)*exp((((-x^2-4*x)*exp(x)+3*x^3+4*x^2)*log(x)-2*exp(x)^2+(2*x+8)*exp(x)-8*x)/(2*exp(x
)-2*x))/(2*exp(x)^2-4*exp(x)*x+2*x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.68, size = 60, normalized size = 1.58 \begin {gather*} e^{\left (-\frac {3 \, x^{3} \log \relax (x) - x^{2} e^{x} \log \relax (x) + 4 \, x^{2} \log \relax (x) - 4 \, x e^{x} \log \relax (x) + 2 \, x e^{x} - 8 \, x - 2 \, e^{\left (2 \, x\right )} + 8 \, e^{x}}{2 \, {\left (x - e^{x}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-4)*exp(x)^2+(-2*x^3+10*x^2+8*x)*exp(x)-6*x^3-4*x^2)*log(x)-2*exp(x)^3+(3*x-4)*exp(x)^2+(2*x^
2+8*x)*exp(x)-3*x^3-4*x^2)*exp((((-x^2-4*x)*exp(x)+3*x^3+4*x^2)*log(x)-2*exp(x)^2+(2*x+8)*exp(x)-8*x)/(2*exp(x
)-2*x))/(2*exp(x)^2-4*exp(x)*x+2*x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 60, normalized size = 1.58

method	result	size
risch	\({\mathrm e}^{-\frac {x^{2} {\mathrm e}^{x} \ln \relax (x )-3 x^{3} \ln \relax (x )+4 x \,{\mathrm e}^{x} \ln \relax (x )-4 x^{2} \ln \relax (x )-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{2 x}-8 \,{\mathrm e}^{x}+8 x}{2 \left ({\mathrm e}^{x}-x \right )}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x-4)*exp(x)^2+(-2*x^3+10*x^2+8*x)*exp(x)-6*x^3-4*x^2)*ln(x)-2*exp(x)^3+(3*x-4)*exp(x)^2+(2*x^2+8*x)*
exp(x)-3*x^3-4*x^2)*exp((((-x^2-4*x)*exp(x)+3*x^3+4*x^2)*ln(x)-2*exp(x)^2+(2*x+8)*exp(x)-8*x)/(2*exp(x)-2*x))/
(2*exp(x)^2-4*exp(x)*x+2*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(-1/2*(x^2*exp(x)*ln(x)-3*x^3*ln(x)+4*x*exp(x)*ln(x)-4*x^2*ln(x)-2*exp(x)*x+2*exp(2*x)-8*exp(x)+8*x)/(exp(x
)-x))

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maxima [A]  time = 0.74, size = 50, normalized size = 1.32 \begin {gather*} e^{\left (-\frac {3}{2} \, x^{2} \log \relax (x) - x e^{x} \log \relax (x) - 2 \, x \log \relax (x) - e^{\left (2 \, x\right )} \log \relax (x) - \frac {e^{\left (3 \, x\right )} \log \relax (x)}{x - e^{x}} - e^{x} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-4)*exp(x)^2+(-2*x^3+10*x^2+8*x)*exp(x)-6*x^3-4*x^2)*log(x)-2*exp(x)^3+(3*x-4)*exp(x)^2+(2*x^
2+8*x)*exp(x)-3*x^3-4*x^2)*exp((((-x^2-4*x)*exp(x)+3*x^3+4*x^2)*log(x)-2*exp(x)^2+(2*x+8)*exp(x)-8*x)/(2*exp(x
)-2*x))/(2*exp(x)^2-4*exp(x)*x+2*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.44, size = 98, normalized size = 2.58 \begin {gather*} x^{\frac {x^2\,{\mathrm {e}}^x+4\,x\,{\mathrm {e}}^x-4\,x^2-3\,x^3}{2\,\left (x-{\mathrm {e}}^x\right )}}\,{\mathrm {e}}^{\frac {8\,x}{2\,x-2\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {8\,{\mathrm {e}}^x}{2\,x-2\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{2\,x}}{2\,x-2\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {2\,x\,{\mathrm {e}}^x}{2\,x-2\,{\mathrm {e}}^x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((8*x + 2*exp(2*x) - log(x)*(4*x^2 - exp(x)*(4*x + x^2) + 3*x^3) - exp(x)*(2*x + 8))/(2*x - 2*exp(x))
)*(2*exp(3*x) - exp(x)*(8*x + 2*x^2) + log(x)*(exp(2*x)*(2*x + 4) + 4*x^2 + 6*x^3 - exp(x)*(8*x + 10*x^2 - 2*x
^3)) - exp(2*x)*(3*x - 4) + 4*x^2 + 3*x^3))/(2*exp(2*x) - 4*x*exp(x) + 2*x^2),x)

[Out]

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

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sympy [B]  time = 1.23, size = 53, normalized size = 1.39 \begin {gather*} e^{\frac {- 8 x + \left (2 x + 8\right ) e^{x} + \left (3 x^{3} + 4 x^{2} + \left (- x^{2} - 4 x\right ) e^{x}\right ) \log {\relax (x )} - 2 e^{2 x}}{- 2 x + 2 e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-4)*exp(x)**2+(-2*x**3+10*x**2+8*x)*exp(x)-6*x**3-4*x**2)*ln(x)-2*exp(x)**3+(3*x-4)*exp(x)**2
+(2*x**2+8*x)*exp(x)-3*x**3-4*x**2)*exp((((-x**2-4*x)*exp(x)+3*x**3+4*x**2)*ln(x)-2*exp(x)**2+(2*x+8)*exp(x)-8
*x)/(2*exp(x)-2*x))/(2*exp(x)**2-4*exp(x)*x+2*x**2),x)

[Out]

exp((-8*x + (2*x + 8)*exp(x) + (3*x**3 + 4*x**2 + (-x**2 - 4*x)*exp(x))*log(x) - 2*exp(2*x))/(-2*x + 2*exp(x))
)

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