∫ e8−4x−8x2+4x3+ex∕2(−4+4x)+ex(8−4ex∕2+4x−4x2) _________________________________________________________________________ ex−x (8+e2x(4−2ex∕2−8x)+8x2−8x3+ex∕2(−4+2x−2x2)+ex(−4−20x+20x2+ex∕2(2+4x))) ____________________________________________________________________________________________________________________________________________________________________________________

3.43.63 \(\int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x (8-4 e^{x/2}+4 x-4 x^2)}{e^x-x}} (8+e^{2 x} (4-2 e^{x/2}-8 x)+8 x^2-8 x^3+e^{x/2} (-4+2 x-2 x^2)+e^x (-4-20 x+20 x^2+e^{x/2} (2+4 x)))}{e^{2 x}-2 e^x x+x^2} \, dx\)

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

________________________________________________________________________________________

Rubi [F] time = 53.27, 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 {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \left (8+e^{2 x} \left (4-2 e^{x/2}-8 x\right )+8 x^2-8 x^3+e^{x/2} \left (-4+2 x-2 x^2\right )+e^x \left (-4-20 x+20 x^2+e^{x/2} (2+4 x)\right )\right )}{e^{2 x}-2 e^x x+x^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

2*x)*(4 - 2*E^(x/2) - 8*x) + 8*x^2 - 8*x^3 + E^(x/2)*(-4 + 2*x - 2*x^2) + E^x*(-4 - 20*x + 20*x^2 + E^(x/2)*(2

+ 4*x))))/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

x))*x^3)/(E^x - x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \left (8+e^{2 x} \left (4-2 e^{x/2}-8 x\right )+8 x^2-8 x^3+e^{x/2} \left (-4+2 x-2 x^2\right )+e^x \left (-4-20 x+20 x^2+e^{x/2} (2+4 x)\right )\right )}{\left (e^x-x\right )^2} \, dx\\ &=\int \left (-2 \exp \left (\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right )-4 \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) (-1+2 x)+\frac {4 \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) (-1+x) \left (-2+e^{x/2}-x+x^2\right )}{\left (e^x-x\right )^2}+\frac {2 \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \left (-2+e^{x/2}-6 x+2 x^2\right )}{e^x-x}\right ) \, dx\\ &=-\left (2 \int \exp \left (\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \, dx\right )+2 \int \frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \left (-2+e^{x/2}-6 x+2 x^2\right )}{e^x-x} \, dx-4 \int \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) (-1+2 x) \, dx+4 \int \frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) (-1+x) \left (-2+e^{x/2}-x+x^2\right )}{\left (e^x-x\right )^2} \, dx\\ &=-\left (2 \int \exp \left (\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \, dx\right )+2 \int \left (-\frac {2 \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right )}{e^x-x}+\frac {\exp \left (\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right )}{e^x-x}-\frac {6 \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) x}{e^x-x}+\frac {2 \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) x^2}{e^x-x}\right ) \, dx-4 \int \left (-\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right )+2 \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) x\right ) \, dx+4 \int \left (-\frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \left (-2+e^{x/2}-x+x^2\right )}{\left (e^x-x\right )^2}+\frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) x \left (-2+e^{x/2}-x+x^2\right )}{\left (e^x-x\right )^2}\right ) \, dx\\ &=-\left (2 \int \exp \left (\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \, dx\right )+2 \int \frac {\exp \left (\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right )}{e^x-x} \, dx+4 \int \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \, dx-4 \int \frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right )}{e^x-x} \, dx+4 \int \frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) x^2}{e^x-x} \, dx-4 \int \frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) \left (-2+e^{x/2}-x+x^2\right )}{\left (e^x-x\right )^2} \, dx+4 \int \frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) x \left (-2+e^{x/2}-x+x^2\right )}{\left (e^x-x\right )^2} \, dx-8 \int \exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) x \, dx-12 \int \frac {\exp \left (\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}\right ) x}{e^x-x} \, dx\\ &=-\left (2 \int e^{\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \, dx\right )+2 \int \frac {e^{\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}}}{e^x-x} \, dx+4 \int e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \, dx-4 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}}}{e^x-x} \, dx+4 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x^2}{e^x-x} \, dx-4 \int \left (-\frac {2 e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}}}{\left (e^x-x\right )^2}+\frac {e^{\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}}}{\left (e^x-x\right )^2}-\frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x}{\left (e^x-x\right )^2}+\frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x^2}{\left (e^x-x\right )^2}\right ) \, dx+4 \int \left (-\frac {2 e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x}{\left (e^x-x\right )^2}+\frac {e^{\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x}{\left (e^x-x\right )^2}-\frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x^2}{\left (e^x-x\right )^2}+\frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x^3}{\left (e^x-x\right )^2}\right ) \, dx-8 \int e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x \, dx-12 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x}{e^x-x} \, dx\\ &=-\left (2 \int e^{\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \, dx\right )+2 \int \frac {e^{\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}}}{e^x-x} \, dx+4 \int e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} \, dx-4 \int \frac {e^{\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}}}{\left (e^x-x\right )^2} \, dx-4 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}}}{e^x-x} \, dx+4 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x}{\left (e^x-x\right )^2} \, dx+4 \int \frac {e^{\frac {x}{2}+\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x}{\left (e^x-x\right )^2} \, dx-2 \left (4 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x^2}{\left (e^x-x\right )^2} \, dx\right )+4 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x^2}{e^x-x} \, dx+4 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x^3}{\left (e^x-x\right )^2} \, dx+8 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}}}{\left (e^x-x\right )^2} \, dx-8 \int e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x \, dx-8 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x}{\left (e^x-x\right )^2} \, dx-12 \int \frac {e^{\frac {8-4 x-8 x^2+4 x^3+e^{x/2} (-4+4 x)+e^x \left (8-4 e^{x/2}+4 x-4 x^2\right )}{e^x-x}} x}{e^x-x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A] time = 0.27, size = 36, normalized size = 1.06 \begin {gather*} e^{-\frac {4 \left (1+e^x-x\right ) \left (-2+e^{x/2}-x+x^2\right )}{e^x-x}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

+ E^(2*x)*(4 - 2*E^(x/2) - 8*x) + 8*x^2 - 8*x^3 + E^(x/2)*(-4 + 2*x - 2*x^2) + E^x*(-4 - 20*x + 20*x^2 + E^(x

/2)*(2 + 4*x))))/(E^(2*x) - 2*E^x*x + x^2),x]

[Out]

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

________________________________________________________________________________________

fricas [A] time = 0.68, size = 50, normalized size = 1.47 \begin {gather*} e^{\left (-\frac {4 \, {\left (x^{3} - 2 \, x^{2} + {\left (x - 1\right )} e^{\left (\frac {1}{2} \, x\right )} - {\left (x^{2} - x - 2\right )} e^{x} - x - e^{\left (\frac {3}{2} \, x\right )} + 2\right )}}{x - e^{x}}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1/4*x)^2-8*x+4)*exp(x)^2+((4*x+2)*exp(1/4*x)^2+20*x^2-20*x-4)*exp(x)+(-2*x^2+2*x-4)*exp(1/4

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

-x))/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B] time = 2.76, size = 57, normalized size = 1.68 \begin {gather*} e^{\left (-\frac {4 \, {\left (x^{3} - x^{2} e^{x} - 2 \, x^{2} + x e^{\left (\frac {1}{2} \, x\right )} + x e^{x} - x - e^{\left (\frac {3}{2} \, x\right )} - e^{\left (\frac {1}{2} \, x\right )} + 2 \, e^{x} + 2\right )}}{x - e^{x}}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1/4*x)^2-8*x+4)*exp(x)^2+((4*x+2)*exp(1/4*x)^2+20*x^2-20*x-4)*exp(x)+(-2*x^2+2*x-4)*exp(1/4

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

-x))/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A] time = 0.30, size = 31, normalized size = 0.91


method	result	size

risch	\({\mathrm e}^{-\frac {4 \left (x^{2}+{\mathrm e}^{\frac {x}{2}}-x -2\right ) \left (1+{\mathrm e}^{x}-x \right )}{{\mathrm e}^{x}-x}}\)	\(31\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(1/4*x)^2-8*x+4)*exp(x)^2+((4*x+2)*exp(1/4*x)^2+20*x^2-20*x-4)*exp(x)+(-2*x^2+2*x-4)*exp(1/4*x)^2-

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

exp(x)^2-2*exp(x)*x+x^2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -2 \, \int \frac {{\left (4 \, x^{3} - 4 \, x^{2} + {\left (4 \, x + e^{\left (\frac {1}{2} \, x\right )} - 2\right )} e^{\left (2 \, x\right )} + {\left (x^{2} - x + 2\right )} e^{\left (\frac {1}{2} \, x\right )} - {\left (10 \, x^{2} + {\left (2 \, x + 1\right )} e^{\left (\frac {1}{2} \, x\right )} - 10 \, x - 2\right )} e^{x} - 4\right )} e^{\left (-\frac {4 \, {\left (x^{3} - 2 \, x^{2} + {\left (x - 1\right )} e^{\left (\frac {1}{2} \, x\right )} - {\left (x^{2} - x + e^{\left (\frac {1}{2} \, x\right )} - 2\right )} e^{x} - x + 2\right )}}{x - e^{x}}\right )}}{x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1/4*x)^2-8*x+4)*exp(x)^2+((4*x+2)*exp(1/4*x)^2+20*x^2-20*x-4)*exp(x)+(-2*x^2+2*x-4)*exp(1/4

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

-x))/(exp(x)^2-2*exp(x)*x+x^2),x, algorithm="maxima")

[Out]

-2*integrate((4*x^3 - 4*x^2 + (4*x + e^(1/2*x) - 2)*e^(2*x) + (x^2 - x + 2)*e^(1/2*x) - (10*x^2 + (2*x + 1)*e^

(1/2*x) - 10*x - 2)*e^x - 4)*e^(-4*(x^3 - 2*x^2 + (x - 1)*e^(1/2*x) - (x^2 - x + e^(1/2*x) - 2)*e^x - x + 2)/(

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

________________________________________________________________________________________

mupad [B] time = 3.26, size = 143, normalized size = 4.21 \begin {gather*} {\mathrm {e}}^{-\frac {4\,x^3}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {8\,x^2}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {8\,{\mathrm {e}}^x}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {8}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{x/2}\,{\mathrm {e}}^x}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{x/2}}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {4\,x\,{\mathrm {e}}^x}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,x}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {4\,x\,{\mathrm {e}}^{x/2}}{x-{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {4\,x^2\,{\mathrm {e}}^x}{x-{\mathrm {e}}^x}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

)*(exp(x/2)*(2*x^2 - 2*x + 4) + exp(2*x)*(8*x + 2*exp(x/2) - 4) + exp(x)*(20*x - exp(x/2)*(4*x + 2) - 20*x^2 +

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

[Out]

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

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

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

________________________________________________________________________________________

sympy [A] time = 0.78, size = 51, normalized size = 1.50 \begin {gather*} e^{\frac {4 x^{3} - 8 x^{2} - 4 x + \left (4 x - 4\right ) e^{\frac {x}{2}} + \left (- 4 x^{2} + 4 x - 4 e^{\frac {x}{2}} + 8\right ) e^{x} + 8}{- x + e^{x}}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(1/4*x)**2-8*x+4)*exp(x)**2+((4*x+2)*exp(1/4*x)**2+20*x**2-20*x-4)*exp(x)+(-2*x**2+2*x-4)*ex

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

*x+8)/(exp(x)-x))/(exp(x)**2-2*exp(x)*x+x**2),x)

[Out]

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

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]