3.31.55 \(\int \frac {e^{-\frac {10}{2+e^x}} (-4 x^2+8 \log (4)+e^{2 x} (-x^2+2 \log (4))+e^x (-4 x^2+10 x^3+(8+20 x-10 x^2) \log (4))+e^{\frac {10}{2+e^x}} (4 x^4+(16 x^2-8 x^3) \log (4)+(16-16 x+4 x^2) \log ^2(4)+e^{2 x} (x^4+(4 x^2-2 x^3) \log (4)+(4-4 x+x^2) \log ^2(4))+e^x (4 x^4+(16 x^2-8 x^3) \log (4)+(16-16 x+4 x^2) \log ^2(4))))}{4 x^4+(16 x^2-8 x^3) \log (4)+(16-16 x+4 x^2) \log ^2(4)+e^{2 x} (x^4+(4 x^2-2 x^3) \log (4)+(4-4 x+x^2) \log ^2(4))+e^x (4 x^4+(16 x^2-8 x^3) \log (4)+(16-16 x+4 x^2) \log ^2(4))} \, dx\)

Optimal. Leaf size=30 \[ 5+x+\frac {e^{-\frac {10}{2+e^x}} x}{x^2+(2-x) \log (4)} \]

________________________________________________________________________________________

Rubi [F]  time = 38.65, 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^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x^2 + 8*Log[4] + E^(2*x)*(-x^2 + 2*Log[4]) + E^x*(-4*x^2 + 10*x^3 + (8 + 20*x - 10*x^2)*Log[4]) + E^(1
0/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2*x^3)*
Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2)))/(E
^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2*x^
3)*Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2)))
,x]

[Out]

(32*ArcTan[(4*x - Log[16])/Sqrt[8*Log[4]^2 + (16 - 3*Log[16])*Log[16]]]*Log[16]^2)/(8*Log[4]^2 + (16 - 3*Log[1
6])*Log[16])^(3/2) - (8*ArcTan[(4*x - Log[16])/Sqrt[8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256]]]*Log[16]^3)/(8*Log[
4]^2 - 3*Log[16]^2 + 8*Log[256])^(3/2) - (4*ArcTan[(4*x - Log[16])/Sqrt[8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256]]
]*Log[16]^2*(3*Log[4]^2 - Log[16]^2 + 3*Log[256]))/(8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256])^(3/2) + (4*ArcTan[(
4*x - Log[16])/Sqrt[8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256]]]*(Log[4]^2 + Log[256])*(4*Log[4]^2 - Log[16]^2 + 4*
Log[256]))/(8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256])^(3/2) - (2*(x - Log[2])^2*Log[16])/(8*Log[4]^2 - 3*Log[16]^
2 + 8*Log[256]) + (2*x*Log[16]^2)/(8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256]) + (128*(x - Log[2])*Log[16]^2)/((16*
(x - Log[2])^2 + 8*Log[4]^2 + (16 - 3*Log[16])*Log[16])*(8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256])) - (8*x^3*(8*L
og[4]^2 - 4*(x - Log[2])*Log[16] - 3*Log[16]^2 + 8*Log[256]))/((8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256])*(16*(x
- Log[2])^2 + 8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256])) + (8*x^2*Log[16]*(8*Log[4]^2 - 4*(x - Log[2])*Log[16] -
3*Log[16]^2 + 8*Log[256]))/((8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256])*(16*(x - Log[2])^2 + 8*Log[4]^2 - 3*Log[16
]^2 + 8*Log[256])) + (8*Log[16]^2*(8*Log[4]^2 - 4*(x - Log[2])*Log[16] - 3*Log[16]^2 + 8*Log[256]))/((8*Log[4]
^2 - 3*Log[16]^2 + 8*Log[256])*(16*(x - Log[2])^2 + 8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256])) - (8*x*(Log[4]^2 +
Log[256])*(8*Log[4]^2 - 4*(x - Log[2])*Log[16] - 3*Log[16]^2 + 8*Log[256]))/((8*Log[4]^2 - 3*Log[16]^2 + 8*Lo
g[256])*(16*(x - Log[2])^2 + 8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256])) - (ArcTan[(4*x - Log[16])/Sqrt[8*Log[4]^2
- 3*Log[16]^2 + 8*Log[256]]]*(48*Log[4]^4 + 11*Log[16]^4 - 48*Log[4]^2*(Log[16]^2 - 2*Log[256]) - 48*Log[16]^
2*Log[256] + 48*Log[256]^2))/(2*(8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256])^(3/2)) + (6*x*(2*Log[4]^2 - Log[16]^2
+ Log[65536]))/(8*Log[4]^2 - 3*Log[16]^2 + 8*Log[256]) - 20*(1 - (I*Log[4])/Sqrt[-Log[4]^2 + 4*Log[16]])*Defer
[Int][1/(E^(10/(2 + E^x))*(2 + E^x)^2*(2*x - Log[4] - I*Sqrt[-Log[4]^2 + 4*Log[16]])), x] + 10*(1 - (I*Log[4])
/Sqrt[-Log[4]^2 + 4*Log[16]])*Defer[Int][1/(E^(10/(2 + E^x))*(2 + E^x)*(2*x - Log[4] - I*Sqrt[-Log[4]^2 + 4*Lo
g[16]])), x] - 20*(1 + (I*Log[4])/Sqrt[-Log[4]^2 + 4*Log[16]])*Defer[Int][1/(E^(10/(2 + E^x))*(2 + E^x)^2*(2*x
- Log[4] + I*Sqrt[-Log[4]^2 + 4*Log[16]])), x] + 10*(1 + (I*Log[4])/Sqrt[-Log[4]^2 + 4*Log[16]])*Defer[Int][1
/(E^(10/(2 + E^x))*(2 + E^x)*(2*x - Log[4] + I*Sqrt[-Log[4]^2 + 4*Log[16]])), x] + Defer[Int][x^2/(E^(10/(2 +
E^x))*(-x^4 + x^3*Log[16] - Log[16]^2 + x*Log[16]^2 - x^2*(Log[4]^2 + Log[256]))), x] + Log[16]*Defer[Int][1/(
E^(10/(2 + E^x))*(x^4 - x^3*Log[16] + Log[16]^2 - x*Log[16]^2 + x^2*(Log[4]^2 + Log[256]))), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2-e^{2 x} \left (x^2-2 \log (4)\right )+8 \log (4)+e^x \left (10 x^3+8 \log (4)+20 x \log (4)-2 x^2 (2+5 \log (4))\right )+e^{\frac {10}{2+e^x}} \left (2+e^x\right )^2 \left (x^2-x \log (4)+\log (16)\right )^2\right )}{\left (2+e^x\right )^2 \left (x^2-x \log (4)+\log (16)\right )^2} \, dx\\ &=\int \left (-\frac {20 e^{-\frac {10}{2+e^x}} x}{\left (2+e^x\right )^2 \left (x^2-x \log (4)+\log (16)\right )}+\frac {10 e^{-\frac {10}{2+e^x}} x}{\left (2+e^x\right ) \left (x^2-x \log (4)+\log (16)\right )}+\frac {e^{-\frac {10}{2+e^x}} \left (-x^2+e^{\frac {10}{2+e^x}} x^4-2 e^{\frac {10}{2+e^x}} x^3 \log (4)+\log (16)+e^{\frac {10}{2+e^x}} \log ^2(16)-e^{\frac {10}{2+e^x}} x \log ^2(16)+e^{\frac {10}{2+e^x}} x^2 \log ^2(4) \left (1+\frac {\log (256)}{\log ^2(4)}\right )\right )}{\left (x^2-x \log (4)+\log (16)\right )^2}\right ) \, dx\\ &=10 \int \frac {e^{-\frac {10}{2+e^x}} x}{\left (2+e^x\right ) \left (x^2-x \log (4)+\log (16)\right )} \, dx-20 \int \frac {e^{-\frac {10}{2+e^x}} x}{\left (2+e^x\right )^2 \left (x^2-x \log (4)+\log (16)\right )} \, dx+\int \frac {e^{-\frac {10}{2+e^x}} \left (-x^2+e^{\frac {10}{2+e^x}} x^4-2 e^{\frac {10}{2+e^x}} x^3 \log (4)+\log (16)+e^{\frac {10}{2+e^x}} \log ^2(16)-e^{\frac {10}{2+e^x}} x \log ^2(16)+e^{\frac {10}{2+e^x}} x^2 \log ^2(4) \left (1+\frac {\log (256)}{\log ^2(4)}\right )\right )}{\left (x^2-x \log (4)+\log (16)\right )^2} \, dx\\ &=10 \int \left (\frac {e^{-\frac {10}{2+e^x}} \left (1-\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )}{\left (2+e^x\right ) \left (2 x-\log (4)-i \sqrt {-\log ^2(4)+4 \log (16)}\right )}+\frac {e^{-\frac {10}{2+e^x}} \left (1+\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )}{\left (2+e^x\right ) \left (2 x-\log (4)+i \sqrt {-\log ^2(4)+4 \log (16)}\right )}\right ) \, dx-20 \int \left (\frac {e^{-\frac {10}{2+e^x}} \left (1-\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )}{\left (2+e^x\right )^2 \left (2 x-\log (4)-i \sqrt {-\log ^2(4)+4 \log (16)}\right )}+\frac {e^{-\frac {10}{2+e^x}} \left (1+\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )}{\left (2+e^x\right )^2 \left (2 x-\log (4)+i \sqrt {-\log ^2(4)+4 \log (16)}\right )}\right ) \, dx+\int \frac {x^4-e^{-\frac {10}{2+e^x}} \left (x^2-\log (16)\right )-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )}{x^4-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )} \, dx\\ &=\left (10 \left (1-\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )\right ) \int \frac {e^{-\frac {10}{2+e^x}}}{\left (2+e^x\right ) \left (2 x-\log (4)-i \sqrt {-\log ^2(4)+4 \log (16)}\right )} \, dx-\left (20 \left (1-\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )\right ) \int \frac {e^{-\frac {10}{2+e^x}}}{\left (2+e^x\right )^2 \left (2 x-\log (4)-i \sqrt {-\log ^2(4)+4 \log (16)}\right )} \, dx+\left (10 \left (1+\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )\right ) \int \frac {e^{-\frac {10}{2+e^x}}}{\left (2+e^x\right ) \left (2 x-\log (4)+i \sqrt {-\log ^2(4)+4 \log (16)}\right )} \, dx-\left (20 \left (1+\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )\right ) \int \frac {e^{-\frac {10}{2+e^x}}}{\left (2+e^x\right )^2 \left (2 x-\log (4)+i \sqrt {-\log ^2(4)+4 \log (16)}\right )} \, dx+\int \left (\frac {x^3 \log (16)}{-x^4+x^3 \log (16)-\log ^2(16)+x \log ^2(16)-x^2 \left (\log ^2(4)+\log (256)\right )}+\frac {x \log ^2(16)}{-x^4+x^3 \log (16)-\log ^2(16)+x \log ^2(16)-x^2 \left (\log ^2(4)+\log (256)\right )}+\frac {x^4}{x^4-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )}+\frac {\log ^2(16)}{x^4-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )}+\frac {e^{-\frac {10}{2+e^x}} \left (-x^2+\log (16)\right )}{x^4-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )}+\frac {x^2 \left (\log ^2(4)+\log (256)\right )}{x^4-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )}\right ) \, dx\\ &=\log (16) \int \frac {x^3}{-x^4+x^3 \log (16)-\log ^2(16)+x \log ^2(16)-x^2 \left (\log ^2(4)+\log (256)\right )} \, dx+\log ^2(16) \int \frac {x}{-x^4+x^3 \log (16)-\log ^2(16)+x \log ^2(16)-x^2 \left (\log ^2(4)+\log (256)\right )} \, dx+\log ^2(16) \int \frac {1}{x^4-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )} \, dx+\left (10 \left (1-\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )\right ) \int \frac {e^{-\frac {10}{2+e^x}}}{\left (2+e^x\right ) \left (2 x-\log (4)-i \sqrt {-\log ^2(4)+4 \log (16)}\right )} \, dx-\left (20 \left (1-\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )\right ) \int \frac {e^{-\frac {10}{2+e^x}}}{\left (2+e^x\right )^2 \left (2 x-\log (4)-i \sqrt {-\log ^2(4)+4 \log (16)}\right )} \, dx+\left (10 \left (1+\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )\right ) \int \frac {e^{-\frac {10}{2+e^x}}}{\left (2+e^x\right ) \left (2 x-\log (4)+i \sqrt {-\log ^2(4)+4 \log (16)}\right )} \, dx-\left (20 \left (1+\frac {i \log (4)}{\sqrt {-\log ^2(4)+4 \log (16)}}\right )\right ) \int \frac {e^{-\frac {10}{2+e^x}}}{\left (2+e^x\right )^2 \left (2 x-\log (4)+i \sqrt {-\log ^2(4)+4 \log (16)}\right )} \, dx+\left (\log ^2(4)+\log (256)\right ) \int \frac {x^2}{x^4-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )} \, dx+\int \frac {x^4}{x^4-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )} \, dx+\int \frac {e^{-\frac {10}{2+e^x}} \left (-x^2+\log (16)\right )}{x^4-x^3 \log (16)+\log ^2(16)-x \log ^2(16)+x^2 \left (\log ^2(4)+\log (256)\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [F]  time = 0.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-4*x^2 + 8*Log[4] + E^(2*x)*(-x^2 + 2*Log[4]) + E^x*(-4*x^2 + 10*x^3 + (8 + 20*x - 10*x^2)*Log[4])
+ E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2
*x^3)*Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2
)))/(E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2
- 2*x^3)*Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4
]^2))),x]

[Out]

Integrate[(-4*x^2 + 8*Log[4] + E^(2*x)*(-x^2 + 2*Log[4]) + E^x*(-4*x^2 + 10*x^3 + (8 + 20*x - 10*x^2)*Log[4])
+ E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2
*x^3)*Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2
)))/(E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2
- 2*x^3)*Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4
]^2))), x]

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 50, normalized size = 1.67 \begin {gather*} \frac {{\left ({\left (x^{3} - 2 \, {\left (x^{2} - 2 \, x\right )} \log \relax (2)\right )} e^{\left (\frac {10}{e^{x} + 2}\right )} + x\right )} e^{\left (-\frac {10}{e^{x} + 2}\right )}}{x^{2} - 2 \, {\left (x - 2\right )} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+(4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^
3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(5/(exp(x)+2))^2+
(4*log(2)-x^2)*exp(x)^2+(2*(-10*x^2+20*x+8)*log(2)+10*x^3-4*x^2)*exp(x)+16*log(2)-4*x^2)/((4*(x^2-4*x+4)*log(2
)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+(4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4
*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)/exp(5/(exp(x)+2))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 1.30, size = 170, normalized size = 5.67 \begin {gather*} \frac {x^{3} e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} - 2 \, x^{2} e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} \log \relax (2) + 4 \, x e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} \log \relax (2) + x e^{x}}{x^{2} e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} - 2 \, x e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} \log \relax (2) + 4 \, e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+(4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^
3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(5/(exp(x)+2))^2+
(4*log(2)-x^2)*exp(x)^2+(2*(-10*x^2+20*x+8)*log(2)+10*x^3-4*x^2)*exp(x)+16*log(2)-4*x^2)/((4*(x^2-4*x+4)*log(2
)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+(4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4
*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)/exp(5/(exp(x)+2))^2,x, algorithm="giac")

[Out]

(x^3*e^((x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5) - 2*x^2*e^((x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5)*log(2) + 4*x*e^((
x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5)*log(2) + x*e^x)/(x^2*e^((x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5) - 2*x*e^((x*e
^x + 2*x - 5*e^x)/(e^x + 2) + 5)*log(2) + 4*e^((x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5)*log(2))

________________________________________________________________________________________

maple [A]  time = 0.23, size = 32, normalized size = 1.07

method	result	size
risch	\(x -\frac {x \,{\mathrm e}^{-\frac {10}{{\mathrm e}^{x}+2}}}{2 x \ln \relax (2)-x^{2}-4 \ln \relax (2)}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*(x^2-4*x+4)*ln(2)^2+2*(-2*x^3+4*x^2)*ln(2)+x^4)*exp(x)^2+(4*(4*x^2-16*x+16)*ln(2)^2+2*(-8*x^3+16*x^2)
*ln(2)+4*x^4)*exp(x)+4*(4*x^2-16*x+16)*ln(2)^2+2*(-8*x^3+16*x^2)*ln(2)+4*x^4)*exp(5/(exp(x)+2))^2+(4*ln(2)-x^2
)*exp(x)^2+(2*(-10*x^2+20*x+8)*ln(2)+10*x^3-4*x^2)*exp(x)+16*ln(2)-4*x^2)/((4*(x^2-4*x+4)*ln(2)^2+2*(-2*x^3+4*
x^2)*ln(2)+x^4)*exp(x)^2+(4*(4*x^2-16*x+16)*ln(2)^2+2*(-8*x^3+16*x^2)*ln(2)+4*x^4)*exp(x)+4*(4*x^2-16*x+16)*ln
(2)^2+2*(-8*x^3+16*x^2)*ln(2)+4*x^4)/exp(5/(exp(x)+2))^2,x,method=_RETURNVERBOSE)

[Out]

x-x/(2*x*ln(2)-x^2-4*ln(2))*exp(-10/(exp(x)+2))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} x - \int \frac {{\left (4 \, x^{2} + {\left (x^{2} - 4 \, \log \relax (2)\right )} e^{\left (2 \, x\right )} - 2 \, {\left (5 \, x^{3} - 2 \, x^{2} {\left (5 \, \log \relax (2) + 1\right )} + 20 \, x \log \relax (2) + 8 \, \log \relax (2)\right )} e^{x} - 16 \, \log \relax (2)\right )} e^{\left (-\frac {10}{e^{x} + 2}\right )}}{4 \, x^{4} - 16 \, x^{3} \log \relax (2) + 16 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2)\right )} x^{2} - 64 \, x \log \relax (2)^{2} + {\left (x^{4} - 4 \, x^{3} \log \relax (2) + 4 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2)\right )} x^{2} - 16 \, x \log \relax (2)^{2} + 16 \, \log \relax (2)^{2}\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} - 4 \, x^{3} \log \relax (2) + 4 \, {\left (\log \relax (2)^{2} + 2 \, \log \relax (2)\right )} x^{2} - 16 \, x \log \relax (2)^{2} + 16 \, \log \relax (2)^{2}\right )} e^{x} + 64 \, \log \relax (2)^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+(4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^
3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(5/(exp(x)+2))^2+
(4*log(2)-x^2)*exp(x)^2+(2*(-10*x^2+20*x+8)*log(2)+10*x^3-4*x^2)*exp(x)+16*log(2)-4*x^2)/((4*(x^2-4*x+4)*log(2
)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+(4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4
*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)/exp(5/(exp(x)+2))^2,x, algorithm="maxima")

[Out]

x - integrate((4*x^2 + (x^2 - 4*log(2))*e^(2*x) - 2*(5*x^3 - 2*x^2*(5*log(2) + 1) + 20*x*log(2) + 8*log(2))*e^
x - 16*log(2))*e^(-10/(e^x + 2))/(4*x^4 - 16*x^3*log(2) + 16*(log(2)^2 + 2*log(2))*x^2 - 64*x*log(2)^2 + (x^4
- 4*x^3*log(2) + 4*(log(2)^2 + 2*log(2))*x^2 - 16*x*log(2)^2 + 16*log(2)^2)*e^(2*x) + 4*(x^4 - 4*x^3*log(2) +
4*(log(2)^2 + 2*log(2))*x^2 - 16*x*log(2)^2 + 16*log(2)^2)*e^x + 64*log(2)^2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {10}{{\mathrm {e}}^x+2}}\,\left (16\,\ln \relax (2)+{\mathrm {e}}^{\frac {10}{{\mathrm {e}}^x+2}}\,\left ({\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (16\,x^2-8\,x^3\right )+4\,{\ln \relax (2)}^2\,\left (4\,x^2-16\,x+16\right )+4\,x^4\right )+2\,\ln \relax (2)\,\left (16\,x^2-8\,x^3\right )+4\,{\ln \relax (2)}^2\,\left (4\,x^2-16\,x+16\right )+{\mathrm {e}}^{2\,x}\,\left (4\,{\ln \relax (2)}^2\,\left (x^2-4\,x+4\right )+2\,\ln \relax (2)\,\left (4\,x^2-2\,x^3\right )+x^4\right )+4\,x^4\right )+{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (-10\,x^2+20\,x+8\right )-4\,x^2+10\,x^3\right )-4\,x^2+{\mathrm {e}}^{2\,x}\,\left (4\,\ln \relax (2)-x^2\right )\right )}{{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (16\,x^2-8\,x^3\right )+4\,{\ln \relax (2)}^2\,\left (4\,x^2-16\,x+16\right )+4\,x^4\right )+2\,\ln \relax (2)\,\left (16\,x^2-8\,x^3\right )+4\,{\ln \relax (2)}^2\,\left (4\,x^2-16\,x+16\right )+{\mathrm {e}}^{2\,x}\,\left (4\,{\ln \relax (2)}^2\,\left (x^2-4\,x+4\right )+2\,\ln \relax (2)\,\left (4\,x^2-2\,x^3\right )+x^4\right )+4\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-10/(exp(x) + 2))*(16*log(2) + exp(10/(exp(x) + 2))*(exp(x)*(2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(
4*x^2 - 16*x + 16) + 4*x^4) + 2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + exp(2*x)*(4*log(2)^
2*(x^2 - 4*x + 4) + 2*log(2)*(4*x^2 - 2*x^3) + x^4) + 4*x^4) + exp(x)*(2*log(2)*(20*x - 10*x^2 + 8) - 4*x^2 +
10*x^3) - 4*x^2 + exp(2*x)*(4*log(2) - x^2)))/(exp(x)*(2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x +
16) + 4*x^4) + 2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + exp(2*x)*(4*log(2)^2*(x^2 - 4*x +
4) + 2*log(2)*(4*x^2 - 2*x^3) + x^4) + 4*x^4),x)

[Out]

int((exp(-10/(exp(x) + 2))*(16*log(2) + exp(10/(exp(x) + 2))*(exp(x)*(2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(
4*x^2 - 16*x + 16) + 4*x^4) + 2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + exp(2*x)*(4*log(2)^
2*(x^2 - 4*x + 4) + 2*log(2)*(4*x^2 - 2*x^3) + x^4) + 4*x^4) + exp(x)*(2*log(2)*(20*x - 10*x^2 + 8) - 4*x^2 +
10*x^3) - 4*x^2 + exp(2*x)*(4*log(2) - x^2)))/(exp(x)*(2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x +
16) + 4*x^4) + 2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + exp(2*x)*(4*log(2)^2*(x^2 - 4*x +
4) + 2*log(2)*(4*x^2 - 2*x^3) + x^4) + 4*x^4), x)

________________________________________________________________________________________

sympy [A]  time = 0.60, size = 26, normalized size = 0.87 \begin {gather*} x + \frac {x e^{- \frac {10}{e^{x} + 2}}}{x^{2} - 2 x \log {\relax (2 )} + 4 \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*(x**2-4*x+4)*ln(2)**2+2*(-2*x**3+4*x**2)*ln(2)+x**4)*exp(x)**2+(4*(4*x**2-16*x+16)*ln(2)**2+2*(
-8*x**3+16*x**2)*ln(2)+4*x**4)*exp(x)+4*(4*x**2-16*x+16)*ln(2)**2+2*(-8*x**3+16*x**2)*ln(2)+4*x**4)*exp(5/(exp
(x)+2))**2+(4*ln(2)-x**2)*exp(x)**2+(2*(-10*x**2+20*x+8)*ln(2)+10*x**3-4*x**2)*exp(x)+16*ln(2)-4*x**2)/((4*(x*
*2-4*x+4)*ln(2)**2+2*(-2*x**3+4*x**2)*ln(2)+x**4)*exp(x)**2+(4*(4*x**2-16*x+16)*ln(2)**2+2*(-8*x**3+16*x**2)*l
n(2)+4*x**4)*exp(x)+4*(4*x**2-16*x+16)*ln(2)**2+2*(-8*x**3+16*x**2)*ln(2)+4*x**4)/exp(5/(exp(x)+2))**2,x)

[Out]

x + x*exp(-10/(exp(x) + 2))/(x**2 - 2*x*log(2) + 4*log(2))

________________________________________________________________________________________