∫ −300+270x−72x2+6x3+e4(−12+6x)+e2(−120+84x−12x2)+e 1 ___________ 5+e2−x (−50+e4(−2+x)+43x−13x2+x3+e2(−20+14x−2x2))+(−600−24e4+240x−24x2+e2(−240+48x)+e 1 ___________ 5+e2−x (−100−4e4+36x−6x2+e2(−40+8x))) log(2+x)+(−300+e4(−12−6x)−30x+48x2−6x3+e2(−120−36x+12x2)+e 1 ___________ 5+e2−x (−50+e4(−2−x)−7x+7x2−x3+e2(−20−6x+2x2))) log2(2+x) __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 1800x2+180x3−288x4+36x5+e4(72x2+36x3)+e2(720x2+216x3−72x4)+e 1 ___________ 5+e2−x (600x2+60x3−96x4+12x5+e4(24x2+12x3)+e2(240x2+72x3−24x4))+e 2 ___________ 5+e2−x (50x2+5x3−8x4+x5+e4(2x2+x3)+e2(20x2+6x3−2x4)) dx

Optimal. Leaf size=29 \[ \frac {(1+\log (2+x))^2}{x+\left (5+e^{\frac {1}{5+e^2-x}}\right ) x} \]

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Rubi [F] time = 51.78, 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 {-300+270 x-72 x^2+6 x^3+e^4 (-12+6 x)+e^2 \left (-120+84 x-12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2+x)+43 x-13 x^2+x^3+e^2 \left (-20+14 x-2 x^2\right )\right )+\left (-600-24 e^4+240 x-24 x^2+e^2 (-240+48 x)+e^{\frac {1}{5+e^2-x}} \left (-100-4 e^4+36 x-6 x^2+e^2 (-40+8 x)\right )\right ) \log (2+x)+\left (-300+e^4 (-12-6 x)-30 x+48 x^2-6 x^3+e^2 \left (-120-36 x+12 x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+e^4 (-2-x)-7 x+7 x^2-x^3+e^2 \left (-20-6 x+2 x^2\right )\right )\right ) \log ^2(2+x)}{1800 x^2+180 x^3-288 x^4+36 x^5+e^4 \left (72 x^2+36 x^3\right )+e^2 \left (720 x^2+216 x^3-72 x^4\right )+e^{\frac {1}{5+e^2-x}} \left (600 x^2+60 x^3-96 x^4+12 x^5+e^4 \left (24 x^2+12 x^3\right )+e^2 \left (240 x^2+72 x^3-24 x^4\right )\right )+e^{\frac {2}{5+e^2-x}} \left (50 x^2+5 x^3-8 x^4+x^5+e^4 \left (2 x^2+x^3\right )+e^2 \left (20 x^2+6 x^3-2 x^4\right )\right )} \, dx \end {gather*}

Int[(-300 + 270*x - 72*x^2 + 6*x^3 + E^4*(-12 + 6*x) + E^2*(-120 + 84*x - 12*x^2) + E^(5 + E^2 - x)^(-1)*(-50

+ E^4*(-2 + x) + 43*x - 13*x^2 + x^3 + E^2*(-20 + 14*x - 2*x^2)) + (-600 - 24*E^4 + 240*x - 24*x^2 + E^2*(-240

+ 48*x) + E^(5 + E^2 - x)^(-1)*(-100 - 4*E^4 + 36*x - 6*x^2 + E^2*(-40 + 8*x)))*Log[2 + x] + (-300 + E^4*(-12

- 6*x) - 30*x + 48*x^2 - 6*x^3 + E^2*(-120 - 36*x + 12*x^2) + E^(5 + E^2 - x)^(-1)*(-50 + E^4*(-2 - x) - 7*x

+ 7*x^2 - x^3 + E^2*(-20 - 6*x + 2*x^2)))*Log[2 + x]^2)/(1800*x^2 + 180*x^3 - 288*x^4 + 36*x^5 + E^4*(72*x^2 +

36*x^3) + E^2*(720*x^2 + 216*x^3 - 72*x^4) + E^(5 + E^2 - x)^(-1)*(600*x^2 + 60*x^3 - 96*x^4 + 12*x^5 + E^4*(

24*x^2 + 12*x^3) + E^2*(240*x^2 + 72*x^3 - 24*x^4)) + E^(2/(5 + E^2 - x))*(50*x^2 + 5*x^3 - 8*x^4 + x^5 + E^4*

1/((5 + E^2)*(6 + E^(5 + E^2 - x)^(-1))) + 1/(6*(5 + E^2)*(5 + E^2 - x)) - 1/(3*(7 + E^2)*(5 + E^2 - x)) + (5

+ E^2)/(6*(7 + E^2)*(5 + E^2 - x)) - (13 + 2*E^2)/(6*(7 + E^2)*(5 + E^2 - x)) + (43 + 14*E^2 + E^4)/(6*(5 + E^

2)*(7 + E^2)*(5 + E^2 - x)) - Log[6 + E^(5 + E^2 - x)^(-1)]/(6*(5 + E^2)) + Log[6 + E^(5 + E^2 - x)^(-1)]/(3*(

7 + E^2)) - ((5 + E^2)*Log[6 + E^(5 + E^2 - x)^(-1)])/(6*(7 + E^2)) + ((13 + 2*E^2)*Log[6 + E^(5 + E^2 - x)^(-

1)])/(6*(7 + E^2)) - ((43 + 14*E^2 + E^4)*Log[6 + E^(5 + E^2 - x)^(-1)])/(6*(5 + E^2)*(7 + E^2)) - Defer[Int][

1/((6 + E^(5 + E^2 - x)^(-1))*x^2), x] - 2*Log[2 + x]*Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))*x^2), x] + (6*D

efer[Int][1/((6 + E^(5 + E^2 - x)^(-1))^2*x), x])/(5 + E^2)^2 + (12*Log[2 + x]*Defer[Int][1/((6 + E^(5 + E^2 -

x)^(-1))^2*x), x])/(5 + E^2)^2 + ((1 + E^2)*Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))*x), x])/(2*(5 + E^2)) +

((43 + 14*E^2 + E^4)*Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))*x), x])/(2*(5 + E^2)^2) + ((1 + E^2)*Log[2 + x]*

Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))*x), x])/(5 + E^2) + (2*(9 + 2*E^2)*Log[2 + x]*Defer[Int][1/((6 + E^(5

+ E^2 - x)^(-1))*x), x])/(5 + E^2)^2 + Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x]/(2*(5 + E^2)^2) - Defer[In

t][x/(6 + E^(5 + E^2 - x)^(-1)), x]/(2*(7 + E^2)^2) + Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x]/((5 + E^2)^2

*(7 + E^2)) + ((1 + E^2)*(13 + 2*E^2)*Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x])/(4*(5 + E^2)^3) - ((13 + 2*

E^2)*Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x])/(4*(7 + E^2)^2) - ((19 + 3*E^2)*Defer[Int][x/(6 + E^(5 + E^2

- x)^(-1)), x])/((5 + E^2)^2*(7 + E^2)^2) + ((13 + 2*E^2)*(19 + 3*E^2)*Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)

), x])/((5 + E^2)^3*(7 + E^2)^2) + (3*(1 + E^2)*Log[2 + x]*Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x])/(2*(5

+ E^2)^3) - (3*Log[2 + x]*Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x])/(2*(7 + E^2)^2) + (6*(19 + 3*E^2)*Log[2

+ x]*Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x])/((5 + E^2)^3*(7 + E^2)^2) - ((1 + E^2)*Defer[Int][x^2/(6 +

E^(5 + E^2 - x)^(-1)), x])/(4*(5 + E^2)^3) + Defer[Int][x^2/(6 + E^(5 + E^2 - x)^(-1)), x]/(4*(7 + E^2)^2) - (

(19 + 3*E^2)*Defer[Int][x^2/(6 + E^(5 + E^2 - x)^(-1)), x])/((5 + E^2)^3*(7 + E^2)^2) - (2*Defer[Int][1/((6 +

E^(5 + E^2 - x)^(-1))*(2 + x)), x])/(7 + E^2)^2 - ((5 + E^2)^2*Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))*(2 + x

)), x])/(2*(7 + E^2)^2) - ((13 + 2*E^2)*Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x])/(7 + E^2)^2 - (

(43 + 14*E^2 + E^4)*Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x])/(2*(7 + E^2)^2) - (6*Log[2 + x]*Def

er[Int][1/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x])/(7 + E^2)^2 - ((5 + E^2)^2*Log[2 + x]*Defer[Int][1/((6 + E

^(5 + E^2 - x)^(-1))*(2 + x)), x])/(7 + E^2)^2 - (2*(9 + 2*E^2)*Log[2 + x]*Defer[Int][1/((6 + E^(5 + E^2 - x)^

(-1))*(2 + x)), x])/(7 + E^2)^2 + (12*Defer[Int][Log[2 + x]/((6 + E^(5 + E^2 - x)^(-1))^2*(5 + E^2 - x)^2), x]

)/(5 + E^2) - (10*Defer[Int][Log[2 + x]/((6 + E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)^2), x])/(7 + E^2) + (4*(9 +

2*E^2)*Defer[Int][Log[2 + x]/((6 + E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)^2), x])/((5 + E^2)*(7 + E^2)) - Defer[I

nt][Log[2 + x]^2/(6 + E^(5 + E^2 - x)^(-1)), x]/(7 + E^2)^2 + ((7 + 2*E^2)*Defer[Int][Log[2 + x]^2/(6 + E^(5 +

E^2 - x)^(-1)), x])/(2*(5 + E^2)^2) - ((7 + 2*E^2)*Defer[Int][Log[2 + x]^2/(6 + E^(5 + E^2 - x)^(-1)), x])/(2

*(7 + E^2)^2) + ((7 + 2*E^2)*Defer[Int][Log[2 + x]^2/(6 + E^(5 + E^2 - x)^(-1)), x])/((5 + E^2)^2*(7 + E^2)) +

((19 + 3*E^2)*Defer[Int][Log[2 + x]^2/(6 + E^(5 + E^2 - x)^(-1)), x])/((5 + E^2)*(7 + E^2)^2) - ((7 + 2*E^2)*

(19 + 3*E^2)*Defer[Int][Log[2 + x]^2/(6 + E^(5 + E^2 - x)^(-1)), x])/((5 + E^2)^2*(7 + E^2)^2) + ((1 + E^2)*(7

+ 6*E^2 + E^4)*Defer[Int][Log[2 + x]^2/(6 + E^(5 + E^2 - x)^(-1)), x])/(4*(5 + E^2)^3) - ((7 + 6*E^2 + E^4)*D

efer[Int][Log[2 + x]^2/(6 + E^(5 + E^2 - x)^(-1)), x])/(4*(7 + E^2)^2) + ((19 + 3*E^2)*(7 + 6*E^2 + E^4)*Defer

[Int][Log[2 + x]^2/(6 + E^(5 + E^2 - x)^(-1)), x])/((5 + E^2)^3*(7 + E^2)^2) - (2*Defer[Int][Log[2 + x]^2/(6 +

E^(5 + E^2 - x)^(-1)), x])/(35 + 12*E^2 + E^4) + (6*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))^2*(5

+ E^2 - x)^2), x])/(5 + E^2) - (2*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)^2), x])/(7

+ E^2) - ((5 + E^2)*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)^2), x])/(7 + E^2) + ((7

+ 2*E^2)*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)^2), x])/(7 + E^2) - ((7 + 6*E^2 +

E^4)*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)^2), x])/((5 + E^2)*(7 + E^2)) + (6*Defe

r[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))^2*(5 + E^2 - x)), x])/(5 + E^2)^2 + (3*Defer[Int][Log[2 + x]^2

/((6 + E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)), x])/(7 + E^2) - (2*(7 + 2*E^2)*Defer[Int][Log[2 + x]^2/((6 + E^(5

+ E^2 - x)^(-1))*(5 + E^2 - x)), x])/((5 + E^2)*(7 + E^2)) - ((19 + 3*E^2)*Defer[Int][Log[2 + x]^2/((6 + E^(5

+ E^2 - x)^(-1))*(5 + E^2 - x)), x])/(7 + E^2)^2 - (2*(19 + 3*E^2)*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 -

x)^(-1))*(5 + E^2 - x)), x])/((5 + E^2)*(7 + E^2)^2) + ((7 + 2*E^2)*(19 + 3*E^2)*Defer[Int][Log[2 + x]^2/((6

+ E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)), x])/((5 + E^2)*(7 + E^2)^2) + ((7 + 6*E^2 + E^4)*Defer[Int][Log[2 + x]

^2/((6 + E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)), x])/((5 + E^2)^2*(7 + E^2)) - ((19 + 3*E^2)*(7 + 6*E^2 + E^4)*D

efer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*(5 + E^2 - x)), x])/((5 + E^2)^2*(7 + E^2)^2) - Defer[Int][

Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*x^2), x] + (6*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))^2*x

), x])/(5 + E^2)^2 + ((1 + E^2)*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*x), x])/(2*(5 + E^2)) - ((

7 + 6*E^2 + E^4)*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*x), x])/(2*(5 + E^2)^2) - Defer[Int][(x*L

og[2 + x]^2)/(6 + E^(5 + E^2 - x)^(-1)), x]/(2*(5 + E^2)^2) + Defer[Int][(x*Log[2 + x]^2)/(6 + E^(5 + E^2 - x)

^(-1)), x]/(2*(7 + E^2)^2) - Defer[Int][(x*Log[2 + x]^2)/(6 + E^(5 + E^2 - x)^(-1)), x]/((5 + E^2)^2*(7 + E^2)

) - ((1 + E^2)*(7 + 2*E^2)*Defer[Int][(x*Log[2 + x]^2)/(6 + E^(5 + E^2 - x)^(-1)), x])/(4*(5 + E^2)^3) + ((7 +

2*E^2)*Defer[Int][(x*Log[2 + x]^2)/(6 + E^(5 + E^2 - x)^(-1)), x])/(4*(7 + E^2)^2) + ((19 + 3*E^2)*Defer[Int]

[(x*Log[2 + x]^2)/(6 + E^(5 + E^2 - x)^(-1)), x])/((5 + E^2)^2*(7 + E^2)^2) - ((7 + 2*E^2)*(19 + 3*E^2)*Defer[

Int][(x*Log[2 + x]^2)/(6 + E^(5 + E^2 - x)^(-1)), x])/((5 + E^2)^3*(7 + E^2)^2) + ((1 + E^2)*Defer[Int][(x^2*L

og[2 + x]^2)/(6 + E^(5 + E^2 - x)^(-1)), x])/(4*(5 + E^2)^3) - Defer[Int][(x^2*Log[2 + x]^2)/(6 + E^(5 + E^2 -

x)^(-1)), x]/(4*(7 + E^2)^2) + ((19 + 3*E^2)*Defer[Int][(x^2*Log[2 + x]^2)/(6 + E^(5 + E^2 - x)^(-1)), x])/((

5 + E^2)^3*(7 + E^2)^2) + (2*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x])/(7 + E^2)^2 - (

(5 + E^2)^2*Defer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x])/(2*(7 + E^2)^2) + ((7 + 2*E^2)*D

efer[Int][Log[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x])/(7 + E^2)^2 + ((7 + 6*E^2 + E^4)*Defer[Int][L

og[2 + x]^2/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x])/(2*(7 + E^2)^2) + 2*Defer[Int][Defer[Int][1/((6 + E^(5 +

E^2 - x)^(-1))*x^2), x]/(2 + x), x] - (12*Defer[Int][Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))^2*x), x]/(2 + x

), x])/(5 + E^2)^2 - (3*(1 + E^2)*Defer[Int][Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x]/(2 + x), x])/(2*(5 +

E^2)^3) + (3*Defer[Int][Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x]/(2 + x), x])/(2*(7 + E^2)^2) - (6*(19 + 3*

E^2)*Defer[Int][Defer[Int][x/(6 + E^(5 + E^2 - x)^(-1)), x]/(2 + x), x])/((5 + E^2)^3*(7 + E^2)^2) + (6*Defer[

Int][Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x]/(2 + x), x])/(7 + E^2)^2 + ((5 + E^2)^2*Defer[Int][

Defer[Int][1/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x]/(2 + x), x])/(7 + E^2)^2 + (2*(9 + 2*E^2)*Defer[Int][Def

er[Int][1/((6 + E^(5 + E^2 - x)^(-1))*(2 + x)), x]/(2 + x), x])/(7 + E^2)^2 - ((1 + E^2)*Defer[Int][Defer[Int]

[(6*x + E^(5 + E^2 - x)^(-1)*x)^(-1), x]/(2 + x), x])/(5 + E^2) - (2*(9 + 2*E^2)*Defer[Int][Defer[Int][(6*x +

E^(5 + E^2 - x)^(-1)*x)^(-1), x]/(2 + x), x])/(5 + E^2)^2 - (3*Defer[Int][Defer[Subst][Defer[Int][(6 + E^x^(-1

))^(-1), x], x, 5 + E^2 - x]/(2 + x), x])/(5 + E^2)^2 + (3*Defer[Int][Defer[Subst][Defer[Int][(6 + E^x^(-1))^(

-1), x], x, 5 + E^2 - x]/(2 + x), x])/(7 + E^2)^2 - (6*Defer[Int][Defer[Subst][Defer[Int][(6 + E^x^(-1))^(-1),

x], x, 5 + E^2 - x]/(2 + x), x])/((5 + E^2)^2*(7 + E^2)) - ((1 + E^2)*(9 + 2*E^2)*Defer[Int][Defer[Subst][Def

er[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x]/(2 + x), x])/(5 + E^2)^3 + ((9 + 2*E^2)*Defer[Int][Defer[Subs

t][Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x]/(2 + x), x])/(7 + E^2)^2 + (6*(19 + 3*E^2)*Defer[Int][D

efer[Subst][Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x]/(2 + x), x])/((5 + E^2)^2*(7 + E^2)^2) - (4*(9

+ 2*E^2)*(19 + 3*E^2)*Defer[Int][Defer[Subst][Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x]/(2 + x), x]

)/((5 + E^2)^3*(7 + E^2)^2) - (12*Defer[Int][Defer[Subst][Defer[Int][1/((6 + E^x)^2*x), x], x, (5 + E^2 - x)^(

-1)]/(2 + x), x])/(5 + E^2)^2 + (4*(9 + 2*E^2)*Defer[Int][Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (

5 + E^2 - x)^(-1)]/(2 + x), x])/((5 + E^2)^2*(7 + E^2)) + (10*(19 + 3*E^2)*Defer[Int][Defer[Subst][Defer[Int][

(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1)]/(2 + x), x])/((5 + E^2)*(7 + E^2)^2) - (4*(9 + 2*E^2)*(19 + 3*E

^2)*Defer[Int][Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1)]/(2 + x), x])/((5 + E^2)^

2*(7 + E^2)^2) - (12*Defer[Int][Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1)]/(2 + x)

, x])/(35 + 12*E^2 + E^4) - Defer[Subst][Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x]/(7 + E^2)^2 + ((1

3 + 2*E^2)*Defer[Subst][Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/(2*(5 + E^2)^2) - ((13 + 2*E^2)*D

efer[Subst][Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/(2*(7 + E^2)^2) + ((13 + 2*E^2)*Defer[Subst][

Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/((5 + E^2)^2*(7 + E^2)) + ((19 + 3*E^2)*Defer[Subst][Defe

r[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/((5 + E^2)*(7 + E^2)^2) - ((13 + 2*E^2)*(19 + 3*E^2)*Defer[Su

bst][Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/((5 + E^2)^2*(7 + E^2)^2) - (2*Defer[Subst][Defer[In

t][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/(35 + 12*E^2 + E^4) + ((1 + E^2)*(43 + 14*E^2 + E^4)*Defer[Subst]

[Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/(4*(5 + E^2)^3) - ((43 + 14*E^2 + E^4)*Defer[Subst][Defe

r[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/(4*(7 + E^2)^2) + ((19 + 3*E^2)*(43 + 14*E^2 + E^4)*Defer[Sub

st][Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/((5 + E^2)^3*(7 + E^2)^2) + (3*Log[2 + x]*Defer[Subst

][Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/(5 + E^2)^2 - (3*Log[2 + x]*Defer[Subst][Defer[Int][(6

+ E^x^(-1))^(-1), x], x, 5 + E^2 - x])/(7 + E^2)^2 + (6*Log[2 + x]*Defer[Subst][Defer[Int][(6 + E^x^(-1))^(-1)

, x], x, 5 + E^2 - x])/((5 + E^2)^2*(7 + E^2)) + ((1 + E^2)*(9 + 2*E^2)*Log[2 + x]*Defer[Subst][Defer[Int][(6

+ E^x^(-1))^(-1), x], x, 5 + E^2 - x])/(5 + E^2)^3 - ((9 + 2*E^2)*Log[2 + x]*Defer[Subst][Defer[Int][(6 + E^x^

(-1))^(-1), x], x, 5 + E^2 - x])/(7 + E^2)^2 - (6*(19 + 3*E^2)*Log[2 + x]*Defer[Subst][Defer[Int][(6 + E^x^(-1

))^(-1), x], x, 5 + E^2 - x])/((5 + E^2)^2*(7 + E^2)^2) + (4*(9 + 2*E^2)*(19 + 3*E^2)*Log[2 + x]*Defer[Subst][

Defer[Int][(6 + E^x^(-1))^(-1), x], x, 5 + E^2 - x])/((5 + E^2)^3*(7 + E^2)^2) + (6*Defer[Subst][Defer[Int][1/

((6 + E^x)^2*x), x], x, (5 + E^2 - x)^(-1)])/(5 + E^2)^2 + (12*Log[2 + x]*Defer[Subst][Defer[Int][1/((6 + E^x)

^2*x), x], x, (5 + E^2 - x)^(-1)])/(5 + E^2)^2 - (3*Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^

2 - x)^(-1)])/(7 + E^2) + (2*(13 + 2*E^2)*Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1

)])/((5 + E^2)*(7 + E^2)) + ((19 + 3*E^2)*Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1

)])/(7 + E^2)^2 - (2*(19 + 3*E^2)*Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1)])/((5

+ E^2)*(7 + E^2)^2) - ((13 + 2*E^2)*(19 + 3*E^2)*Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 -

x)^(-1)])/((5 + E^2)*(7 + E^2)^2) - ((43 + 14*E^2 + E^4)*Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (

5 + E^2 - x)^(-1)])/((5 + E^2)^2*(7 + E^2)) + ((19 + 3*E^2)*(43 + 14*E^2 + E^4)*Defer[Subst][Defer[Int][(6*x +

E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1)])/((5 + E^2)^2*(7 + E^2)^2) - (4*(9 + 2*E^2)*Log[2 + x]*Defer[Subst][D

efer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1)])/((5 + E^2)^2*(7 + E^2)) - (10*(19 + 3*E^2)*Log[2 + x

]*Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1)])/((5 + E^2)*(7 + E^2)^2) + (4*(9 + 2*

E^2)*(19 + 3*E^2)*Log[2 + x]*Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1)])/((5 + E^2

)^2*(7 + E^2)^2) + (12*Log[2 + x]*Defer[Subst][Defer[Int][(6*x + E^x*x)^(-1), x], x, (5 + E^2 - x)^(-1)])/(35

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(1+\log (2+x)) \left (6 e^4 (-2+x)+e^{4+\frac {1}{5+e^2-x}} (-2+x)+6 (-5+x)^2 (-2+x)-12 e^2 \left (10-7 x+x^2\right )-2 e^{2+\frac {1}{5+e^2-x}} \left (10-7 x+x^2\right )+e^{\frac {1}{5+e^2-x}} \left (-50+43 x-13 x^2+x^3\right )-(2+x) \left (6 e^4+e^{4+\frac {1}{5+e^2-x}}-12 e^2 (-5+x)-2 e^{2+\frac {1}{5+e^2-x}} (-5+x)+6 (-5+x)^2+e^{\frac {1}{5+e^2-x}} \left (25-9 x+x^2\right )\right ) \log (2+x)\right )}{\left (6+e^{\frac {1}{5+e^2-x}}\right )^2 \left (5+e^2-x\right )^2 x^2 (2+x)} \, dx\\ &=\int \left (\frac {6 (1+\log (2+x))^2}{\left (6+e^{\frac {1}{5+e^2-x}}\right )^2 \left (5+e^2-x\right )^2 x}+\frac {-50 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right )+43 \left (1+\frac {1}{43} e^2 \left (14+e^2\right )\right ) x-13 \left (1+\frac {2 e^2}{13}\right ) x^2+x^3-100 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right ) \log (2+x)+36 \left (1+\frac {2 e^2}{9}\right ) x \log (2+x)-6 x^2 \log (2+x)-50 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right ) \log ^2(2+x)-7 \left (1+\frac {1}{7} e^2 \left (6+e^2\right )\right ) x \log ^2(2+x)+7 \left (1+\frac {2 e^2}{7}\right ) x^2 \log ^2(2+x)-x^3 \log ^2(2+x)}{\left (6+e^{\frac {1}{5+e^2-x}}\right ) \left (5+e^2-x\right )^2 x^2 (2+x)}\right ) \, dx\\ &=6 \int \frac {(1+\log (2+x))^2}{\left (6+e^{\frac {1}{5+e^2-x}}\right )^2 \left (5+e^2-x\right )^2 x} \, dx+\int \frac {-50 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right )+43 \left (1+\frac {1}{43} e^2 \left (14+e^2\right )\right ) x-13 \left (1+\frac {2 e^2}{13}\right ) x^2+x^3-100 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right ) \log (2+x)+36 \left (1+\frac {2 e^2}{9}\right ) x \log (2+x)-6 x^2 \log (2+x)-50 \left (1+\frac {1}{25} e^2 \left (10+e^2\right )\right ) \log ^2(2+x)-7 \left (1+\frac {1}{7} e^2 \left (6+e^2\right )\right ) x \log ^2(2+x)+7 \left (1+\frac {2 e^2}{7}\right ) x^2 \log ^2(2+x)-x^3 \log ^2(2+x)}{\left (6+e^{\frac {1}{5+e^2-x}}\right ) \left (5+e^2-x\right )^2 x^2 (2+x)} \, dx\\ &=6 \int \left (\frac {(1+\log (2+x))^2}{\left (5+e^2\right ) \left (6+e^{\frac {1}{5+e^2-x}}\right )^2 \left (5+e^2-x\right )^2}+\frac {(1+\log (2+x))^2}{\left (5+e^2\right )^2 \left (6+e^{\frac {1}{5+e^2-x}}\right )^2 \left (5+e^2-x\right )}+\frac {(1+\log (2+x))^2}{\left (5+e^2\right )^2 \left (6+e^{\frac {1}{5+e^2-x}}\right )^2 x}\right ) \, dx+\int \frac {-50+e^4 (-2+x)+43 x-13 x^2+x^3-2 e^2 \left (10-7 x+x^2\right )-2 \left (50+2 e^4-4 e^2 (-5+x)-18 x+3 x^2\right ) \log (2+x)-\left (50+7 x-7 x^2+x^3+e^4 (2+x)+e^2 \left (20+6 x-2 x^2\right )\right ) \log ^2(2+x)}{\left (6+e^{\frac {1}{5+e^2-x}}\right ) \left (5+e^2-x\right )^2 x^2 (2+x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 0.50, size = 40, normalized size = 1.38 \begin {gather*} \frac {e^{\frac {1}{-5-e^2+x}} (1+\log (2+x))^2}{x+6 e^{\frac {1}{-5-e^2+x}} x} \end {gather*}

Integrate[(-300 + 270*x - 72*x^2 + 6*x^3 + E^4*(-12 + 6*x) + E^2*(-120 + 84*x - 12*x^2) + E^(5 + E^2 - x)^(-1)

*(-50 + E^4*(-2 + x) + 43*x - 13*x^2 + x^3 + E^2*(-20 + 14*x - 2*x^2)) + (-600 - 24*E^4 + 240*x - 24*x^2 + E^2

*(-240 + 48*x) + E^(5 + E^2 - x)^(-1)*(-100 - 4*E^4 + 36*x - 6*x^2 + E^2*(-40 + 8*x)))*Log[2 + x] + (-300 + E^

4*(-12 - 6*x) - 30*x + 48*x^2 - 6*x^3 + E^2*(-120 - 36*x + 12*x^2) + E^(5 + E^2 - x)^(-1)*(-50 + E^4*(-2 - x)

- 7*x + 7*x^2 - x^3 + E^2*(-20 - 6*x + 2*x^2)))*Log[2 + x]^2)/(1800*x^2 + 180*x^3 - 288*x^4 + 36*x^5 + E^4*(72

*x^2 + 36*x^3) + E^2*(720*x^2 + 216*x^3 - 72*x^4) + E^(5 + E^2 - x)^(-1)*(600*x^2 + 60*x^3 - 96*x^4 + 12*x^5 +

E^4*(24*x^2 + 12*x^3) + E^2*(240*x^2 + 72*x^3 - 24*x^4)) + E^(2/(5 + E^2 - x))*(50*x^2 + 5*x^3 - 8*x^4 + x^5

(E^(-5 - E^2 + x)^(-1)*(1 + Log[2 + x])^2)/(x + 6*E^(-5 - E^2 + x)^(-1)*x)

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fricas [A] time = 0.70, size = 35, normalized size = 1.21 \begin {gather*} \frac {\log \left (x + 2\right )^{2} + 2 \, \log \left (x + 2\right ) + 1}{x e^{\left (-\frac {1}{x - e^{2} - 5}\right )} + 6 \, x} \end {gather*}

integrate(((((-x-2)*exp(2)^2+(2*x^2-6*x-20)*exp(2)-x^3+7*x^2-7*x-50)*exp(1/(exp(2)+5-x))+(-6*x-12)*exp(2)^2+(1

2*x^2-36*x-120)*exp(2)-6*x^3+48*x^2-30*x-300)*log(2+x)^2+((-4*exp(2)^2+(8*x-40)*exp(2)-6*x^2+36*x-100)*exp(1/(

exp(2)+5-x))-24*exp(2)^2+(48*x-240)*exp(2)-24*x^2+240*x-600)*log(2+x)+((x-2)*exp(2)^2+(-2*x^2+14*x-20)*exp(2)+

x^3-13*x^2+43*x-50)*exp(1/(exp(2)+5-x))+(6*x-12)*exp(2)^2+(-12*x^2+84*x-120)*exp(2)+6*x^3-72*x^2+270*x-300)/((

(x^3+2*x^2)*exp(2)^2+(-2*x^4+6*x^3+20*x^2)*exp(2)+x^5-8*x^4+5*x^3+50*x^2)*exp(1/(exp(2)+5-x))^2+((12*x^3+24*x^

2)*exp(2)^2+(-24*x^4+72*x^3+240*x^2)*exp(2)+12*x^5-96*x^4+60*x^3+600*x^2)*exp(1/(exp(2)+5-x))+(36*x^3+72*x^2)*

exp(2)^2+(-72*x^4+216*x^3+720*x^2)*exp(2)+36*x^5-288*x^4+180*x^3+1800*x^2),x, algorithm="fricas")

(log(x + 2)^2 + 2*log(x + 2) + 1)/(x*e^(-1/(x - e^2 - 5)) + 6*x)

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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

integrate(((((-x-2)*exp(2)^2+(2*x^2-6*x-20)*exp(2)-x^3+7*x^2-7*x-50)*exp(1/(exp(2)+5-x))+(-6*x-12)*exp(2)^2+(1

2*x^2-36*x-120)*exp(2)-6*x^3+48*x^2-30*x-300)*log(2+x)^2+((-4*exp(2)^2+(8*x-40)*exp(2)-6*x^2+36*x-100)*exp(1/(

exp(2)+5-x))-24*exp(2)^2+(48*x-240)*exp(2)-24*x^2+240*x-600)*log(2+x)+((x-2)*exp(2)^2+(-2*x^2+14*x-20)*exp(2)+

x^3-13*x^2+43*x-50)*exp(1/(exp(2)+5-x))+(6*x-12)*exp(2)^2+(-12*x^2+84*x-120)*exp(2)+6*x^3-72*x^2+270*x-300)/((

(x^3+2*x^2)*exp(2)^2+(-2*x^4+6*x^3+20*x^2)*exp(2)+x^5-8*x^4+5*x^3+50*x^2)*exp(1/(exp(2)+5-x))^2+((12*x^3+24*x^

2)*exp(2)^2+(-24*x^4+72*x^3+240*x^2)*exp(2)+12*x^5-96*x^4+60*x^3+600*x^2)*exp(1/(exp(2)+5-x))+(36*x^3+72*x^2)*

exp(2)^2+(-72*x^4+216*x^3+720*x^2)*exp(2)+36*x^5-288*x^4+180*x^3+1800*x^2),x, algorithm="giac")

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method	result	size

risch	\(\frac {\ln \left (2+x \right )^{2}}{x \left ({\mathrm e}^{\frac {1}{{\mathrm e}^{2}+5-x}}+6\right )}+\frac {2 \ln \left (2+x \right )}{x \left ({\mathrm e}^{\frac {1}{{\mathrm e}^{2}+5-x}}+6\right )}+\frac {1}{x \left ({\mathrm e}^{\frac {1}{{\mathrm e}^{2}+5-x}}+6\right )}\)	\(67\)

int(((((-x-2)*exp(2)^2+(2*x^2-6*x-20)*exp(2)-x^3+7*x^2-7*x-50)*exp(1/(exp(2)+5-x))+(-6*x-12)*exp(2)^2+(12*x^2-

36*x-120)*exp(2)-6*x^3+48*x^2-30*x-300)*ln(2+x)^2+((-4*exp(2)^2+(8*x-40)*exp(2)-6*x^2+36*x-100)*exp(1/(exp(2)+

5-x))-24*exp(2)^2+(48*x-240)*exp(2)-24*x^2+240*x-600)*ln(2+x)+((x-2)*exp(2)^2+(-2*x^2+14*x-20)*exp(2)+x^3-13*x

^2+43*x-50)*exp(1/(exp(2)+5-x))+(6*x-12)*exp(2)^2+(-12*x^2+84*x-120)*exp(2)+6*x^3-72*x^2+270*x-300)/(((x^3+2*x

^2)*exp(2)^2+(-2*x^4+6*x^3+20*x^2)*exp(2)+x^5-8*x^4+5*x^3+50*x^2)*exp(1/(exp(2)+5-x))^2+((12*x^3+24*x^2)*exp(2

)^2+(-24*x^4+72*x^3+240*x^2)*exp(2)+12*x^5-96*x^4+60*x^3+600*x^2)*exp(1/(exp(2)+5-x))+(36*x^3+72*x^2)*exp(2)^2

+(-72*x^4+216*x^3+720*x^2)*exp(2)+36*x^5-288*x^4+180*x^3+1800*x^2),x,method=_RETURNVERBOSE)

1/x/(exp(1/(exp(2)+5-x))+6)*ln(2+x)^2+2/x/(exp(1/(exp(2)+5-x))+6)*ln(2+x)+1/x/(exp(1/(exp(2)+5-x))+6)

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maxima [A] time = 0.88, size = 42, normalized size = 1.45 \begin {gather*} \frac {{\left (\log \left (x + 2\right )^{2} + 2 \, \log \left (x + 2\right ) + 1\right )} e^{\left (\frac {1}{x - e^{2} - 5}\right )}}{6 \, x e^{\left (\frac {1}{x - e^{2} - 5}\right )} + x} \end {gather*}

integrate(((((-x-2)*exp(2)^2+(2*x^2-6*x-20)*exp(2)-x^3+7*x^2-7*x-50)*exp(1/(exp(2)+5-x))+(-6*x-12)*exp(2)^2+(1

2*x^2-36*x-120)*exp(2)-6*x^3+48*x^2-30*x-300)*log(2+x)^2+((-4*exp(2)^2+(8*x-40)*exp(2)-6*x^2+36*x-100)*exp(1/(

exp(2)+5-x))-24*exp(2)^2+(48*x-240)*exp(2)-24*x^2+240*x-600)*log(2+x)+((x-2)*exp(2)^2+(-2*x^2+14*x-20)*exp(2)+

x^3-13*x^2+43*x-50)*exp(1/(exp(2)+5-x))+(6*x-12)*exp(2)^2+(-12*x^2+84*x-120)*exp(2)+6*x^3-72*x^2+270*x-300)/((

(x^3+2*x^2)*exp(2)^2+(-2*x^4+6*x^3+20*x^2)*exp(2)+x^5-8*x^4+5*x^3+50*x^2)*exp(1/(exp(2)+5-x))^2+((12*x^3+24*x^

2)*exp(2)^2+(-24*x^4+72*x^3+240*x^2)*exp(2)+12*x^5-96*x^4+60*x^3+600*x^2)*exp(1/(exp(2)+5-x))+(36*x^3+72*x^2)*

exp(2)^2+(-72*x^4+216*x^3+720*x^2)*exp(2)+36*x^5-288*x^4+180*x^3+1800*x^2),x, algorithm="maxima")

(log(x + 2)^2 + 2*log(x + 2) + 1)*e^(1/(x - e^2 - 5))/(6*x*e^(1/(x - e^2 - 5)) + x)

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mupad [B] time = 2.66, size = 60, normalized size = 2.07 \begin {gather*} \frac {x\,\left (2\,{\ln \left (x+2\right )}^2+4\,\ln \left (x+2\right )+2\right )+x^2\,\left ({\ln \left (x+2\right )}^2+2\,\ln \left (x+2\right )+1\right )}{x^2\,\left ({\mathrm {e}}^{\frac {1}{{\mathrm {e}}^2-x+5}}+6\right )\,\left (x+2\right )} \end {gather*}

int(-(exp(2)*(12*x^2 - 84*x + 120) - 270*x + log(x + 2)^2*(30*x + exp(2)*(36*x - 12*x^2 + 120) + exp(1/(exp(2)

- x + 5))*(7*x + exp(2)*(6*x - 2*x^2 + 20) + exp(4)*(x + 2) - 7*x^2 + x^3 + 50) - 48*x^2 + 6*x^3 + exp(4)*(6*

x + 12) + 300) + log(x + 2)*(24*exp(4) - 240*x + 24*x^2 + exp(1/(exp(2) - x + 5))*(4*exp(4) - 36*x + 6*x^2 - e

xp(2)*(8*x - 40) + 100) - exp(2)*(48*x - 240) + 600) - exp(1/(exp(2) - x + 5))*(43*x - exp(2)*(2*x^2 - 14*x +

20) + exp(4)*(x - 2) - 13*x^2 + x^3 - 50) + 72*x^2 - 6*x^3 - exp(4)*(6*x - 12) + 300)/(exp(1/(exp(2) - x + 5))

*(exp(4)*(24*x^2 + 12*x^3) + exp(2)*(240*x^2 + 72*x^3 - 24*x^4) + 600*x^2 + 60*x^3 - 96*x^4 + 12*x^5) + exp(4)

*(72*x^2 + 36*x^3) + exp(2)*(720*x^2 + 216*x^3 - 72*x^4) + 1800*x^2 + 180*x^3 - 288*x^4 + 36*x^5 + exp(2/(exp(

2) - x + 5))*(exp(4)*(2*x^2 + x^3) + exp(2)*(20*x^2 + 6*x^3 - 2*x^4) + 50*x^2 + 5*x^3 - 8*x^4 + x^5)),x)

(x*(4*log(x + 2) + 2*log(x + 2)^2 + 2) + x^2*(2*log(x + 2) + log(x + 2)^2 + 1))/(x^2*(exp(1/(exp(2) - x + 5))

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sympy [A] time = 0.81, size = 29, normalized size = 1.00 \begin {gather*} \frac {\log {\left (x + 2 \right )}^{2} + 2 \log {\left (x + 2 \right )} + 1}{x e^{\frac {1}{- x + 5 + e^{2}}} + 6 x} \end {gather*}

integrate(((((-x-2)*exp(2)**2+(2*x**2-6*x-20)*exp(2)-x**3+7*x**2-7*x-50)*exp(1/(exp(2)+5-x))+(-6*x-12)*exp(2)*

*2+(12*x**2-36*x-120)*exp(2)-6*x**3+48*x**2-30*x-300)*ln(2+x)**2+((-4*exp(2)**2+(8*x-40)*exp(2)-6*x**2+36*x-10

0)*exp(1/(exp(2)+5-x))-24*exp(2)**2+(48*x-240)*exp(2)-24*x**2+240*x-600)*ln(2+x)+((x-2)*exp(2)**2+(-2*x**2+14*

x-20)*exp(2)+x**3-13*x**2+43*x-50)*exp(1/(exp(2)+5-x))+(6*x-12)*exp(2)**2+(-12*x**2+84*x-120)*exp(2)+6*x**3-72

*x**2+270*x-300)/(((x**3+2*x**2)*exp(2)**2+(-2*x**4+6*x**3+20*x**2)*exp(2)+x**5-8*x**4+5*x**3+50*x**2)*exp(1/(

exp(2)+5-x))**2+((12*x**3+24*x**2)*exp(2)**2+(-24*x**4+72*x**3+240*x**2)*exp(2)+12*x**5-96*x**4+60*x**3+600*x*

*2)*exp(1/(exp(2)+5-x))+(36*x**3+72*x**2)*exp(2)**2+(-72*x**4+216*x**3+720*x**2)*exp(2)+36*x**5-288*x**4+180*x

(log(x + 2)**2 + 2*log(x + 2) + 1)/(x*exp(1/(-x + 5 + exp(2))) + 6*x)

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