∫ 120x+116x2+36x3+58x4+54x5+18x6+2x7+e3x(4x+2x4)+e2x(36x+12x2+18x4+6x5)+ex(112x+76x2+12x3+54x4+36x5+6x6)+e4x(54x2+2e3xx2+54x3+18x4+2x5+e2x(18x2+6x3)+ex(54x2+36x3+6x4))+e2x(−60−180x−130x2−146x3−112x4−36x5−4x6+e3x(−2−4x−4x3)+e2x(−18−42x−12x2−36x3−12x4)+ex(−56−152x−78x2−120x3−72x4−12x5))_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 27x4+e3xx4+27x5+9x6+x7+e2x(9x4+3x5)+ex(27x4+18x5+3x6)+e4x(27x2+e3xx2+27x3+9x4+x5+e2x(9x2+3x3)+ex(27x2+18x3+3x4))+e2x(−54x3−2e3xx3−54x4−18x5−2x6+e2x(−18x3−6x4)+ex(−54x3−36x4−6x5)) dx [5002]

Optimal. Leaf size=37 \[ 2 x+\frac {2}{\left (e^{2 x}-x\right ) \left (x+\frac {x}{-1-\left (3+e^x+x\right )^2}\right )} \]

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

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Rubi [F]
time = 84.50, 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 {120 x+116 x^2+36 x^3+58 x^4+54 x^5+18 x^6+2 x^7+e^{3 x} \left (4 x+2 x^4\right )+e^{2 x} \left (36 x+12 x^2+18 x^4+6 x^5\right )+e^x \left (112 x+76 x^2+12 x^3+54 x^4+36 x^5+6 x^6\right )+e^{4 x} \left (54 x^2+2 e^{3 x} x^2+54 x^3+18 x^4+2 x^5+e^{2 x} \left (18 x^2+6 x^3\right )+e^x \left (54 x^2+36 x^3+6 x^4\right )\right )+e^{2 x} \left (-60-180 x-130 x^2-146 x^3-112 x^4-36 x^5-4 x^6+e^{3 x} \left (-2-4 x-4 x^3\right )+e^{2 x} \left (-18-42 x-12 x^2-36 x^3-12 x^4\right )+e^x \left (-56-152 x-78 x^2-120 x^3-72 x^4-12 x^5\right )\right )}{27 x^4+e^{3 x} x^4+27 x^5+9 x^6+x^7+e^{2 x} \left (9 x^4+3 x^5\right )+e^x \left (27 x^4+18 x^5+3 x^6\right )+e^{4 x} \left (27 x^2+e^{3 x} x^2+27 x^3+9 x^4+x^5+e^{2 x} \left (9 x^2+3 x^3\right )+e^x \left (27 x^2+18 x^3+3 x^4\right )\right )+e^{2 x} \left (-54 x^3-2 e^{3 x} x^3-54 x^4-18 x^5-2 x^6+e^{2 x} \left (-18 x^3-6 x^4\right )+e^x \left (-54 x^3-36 x^4-6 x^5\right )\right )} \, dx \end {gather*}

Int[(120*x + 116*x^2 + 36*x^3 + 58*x^4 + 54*x^5 + 18*x^6 + 2*x^7 + E^(3*x)*(4*x + 2*x^4) + E^(2*x)*(36*x + 12*

x^2 + 18*x^4 + 6*x^5) + E^x*(112*x + 76*x^2 + 12*x^3 + 54*x^4 + 36*x^5 + 6*x^6) + E^(4*x)*(54*x^2 + 2*E^(3*x)*

x^2 + 54*x^3 + 18*x^4 + 2*x^5 + E^(2*x)*(18*x^2 + 6*x^3) + E^x*(54*x^2 + 36*x^3 + 6*x^4)) + E^(2*x)*(-60 - 180

*x - 130*x^2 - 146*x^3 - 112*x^4 - 36*x^5 - 4*x^6 + E^(3*x)*(-2 - 4*x - 4*x^3) + E^(2*x)*(-18 - 42*x - 12*x^2

- 36*x^3 - 12*x^4) + E^x*(-56 - 152*x - 78*x^2 - 120*x^3 - 72*x^4 - 12*x^5)))/(27*x^4 + E^(3*x)*x^4 + 27*x^5 +

9*x^6 + x^7 + E^(2*x)*(9*x^4 + 3*x^5) + E^x*(27*x^4 + 18*x^5 + 3*x^6) + E^(4*x)*(27*x^2 + E^(3*x)*x^2 + 27*x^

3 + 9*x^4 + x^5 + E^(2*x)*(9*x^2 + 3*x^3) + E^x*(27*x^2 + 18*x^3 + 3*x^4)) + E^(2*x)*(-54*x^3 - 2*E^(3*x)*x^3

- 54*x^4 - 18*x^5 - 2*x^6 + E^(2*x)*(-18*x^3 - 6*x^4) + E^x*(-54*x^3 - 36*x^4 - 6*x^5))),x]

2*x - 4*Defer[Int][(E^(2*x) - x)^(-2), x] + (((1052*I)/9)*Defer[Int][1/((-5 + I*Sqrt[11] - 2*x)*(E^(2*x) - x)^

2), x])/Sqrt[11] - ((104*I)/9)*Sqrt[11]*Defer[Int][1/((-5 + I*Sqrt[11] - 2*x)*(E^(2*x) - x)^2), x] + (((40*I)/

27)*Defer[Int][E^x/((-5 + I*Sqrt[11] - 2*x)*(E^(2*x) - x)^2), x])/Sqrt[11] + ((4*I)/9)*Sqrt[11]*Defer[Int][1/(

(-5 + I*Sqrt[11] - 2*x)*(E^(2*x) - x)), x] - (((7280*I)/729)*Defer[Int][E^x/((-5 + I*Sqrt[11] - 2*x)*(E^(2*x)

- x)), x])/Sqrt[11] + ((544*I)/729)*Sqrt[11]*Defer[Int][E^x/((-5 + I*Sqrt[11] - 2*x)*(E^(2*x) - x)), x] - (20*

Defer[Int][1/((E^(2*x) - x)*x^2), x])/9 + (4*Defer[Int][E^x/((E^(2*x) - x)*x^2), x])/27 + (20*Defer[Int][1/((E

^(2*x) - x)^2*x), x])/9 - (4*Defer[Int][E^x/((E^(2*x) - x)^2*x), x])/27 - (40*Defer[Int][1/((E^(2*x) - x)*x),

x])/9 + (4*Defer[Int][E^x/((E^(2*x) - x)*x), x])/27 - (((8*I)/9)*Defer[Int][1/((-5 + I*Sqrt[11] - 2*x)*(3 + E^

x + x)^3), x])/Sqrt[11] + (8*Defer[Int][1/(x*(3 + E^x + x)^3), x])/9 + (((52*I)/27)*Defer[Int][1/((-5 + I*Sqrt

[11] - 2*x)*(3 + E^x + x)^2), x])/Sqrt[11] - (2*Defer[Int][1/(x^2*(3 + E^x + x)^2), x])/9 - (4*Defer[Int][1/(x

*(3 + E^x + x)^2), x])/27 + (((16*I)/9)*Defer[Int][1/((-5 + I*Sqrt[11] - 2*x)*(3 + E^x + x)), x])/Sqrt[11] - (

4*Defer[Int][1/(x^2*(3 + E^x + x)), x])/27 - (4*Defer[Int][1/(x*(3 + E^x + x)), x])/27 - (2*(11 + (5*I)*Sqrt[1

1])*Defer[Int][1/((E^(2*x) - x)^2*(5 - I*Sqrt[11] + 2*x)), x])/99 + (4*(11 + (5*I)*Sqrt[11])*Defer[Int][E^x/((

E^(2*x) - x)^2*(5 - I*Sqrt[11] + 2*x)), x])/297 + (4*(11 + (5*I)*Sqrt[11])*Defer[Int][1/((E^(2*x) - x)*(5 - I*

Sqrt[11] + 2*x)), x])/99 - (4*(11 + (5*I)*Sqrt[11])*Defer[Int][E^x/((E^(2*x) - x)*(5 - I*Sqrt[11] + 2*x)), x])

/297 - (8*(11 + (5*I)*Sqrt[11])*Defer[Int][1/((3 + E^x + x)^3*(5 - I*Sqrt[11] + 2*x)), x])/99 + (4*(11 + (5*I)

*Sqrt[11])*Defer[Int][1/((3 + E^x + x)^2*(5 - I*Sqrt[11] + 2*x)), x])/297 + (4*(11 + (5*I)*Sqrt[11])*Defer[Int

][1/((3 + E^x + x)*(5 - I*Sqrt[11] + 2*x)), x])/297 + (((1052*I)/9)*Defer[Int][1/((E^(2*x) - x)^2*(5 + I*Sqrt[

11] + 2*x)), x])/Sqrt[11] - ((104*I)/9)*Sqrt[11]*Defer[Int][1/((E^(2*x) - x)^2*(5 + I*Sqrt[11] + 2*x)), x] - (

2*(11 - (5*I)*Sqrt[11])*Defer[Int][1/((E^(2*x) - x)^2*(5 + I*Sqrt[11] + 2*x)), x])/99 + (((40*I)/27)*Defer[Int

][E^x/((E^(2*x) - x)^2*(5 + I*Sqrt[11] + 2*x)), x])/Sqrt[11] + (4*(11 - (5*I)*Sqrt[11])*Defer[Int][E^x/((E^(2*

x) - x)^2*(5 + I*Sqrt[11] + 2*x)), x])/297 + ((4*I)/9)*Sqrt[11]*Defer[Int][1/((E^(2*x) - x)*(5 + I*Sqrt[11] +

2*x)), x] + (4*(11 - (5*I)*Sqrt[11])*Defer[Int][1/((E^(2*x) - x)*(5 + I*Sqrt[11] + 2*x)), x])/99 - (((7280*I)/

729)*Defer[Int][E^x/((E^(2*x) - x)*(5 + I*Sqrt[11] + 2*x)), x])/Sqrt[11] + ((544*I)/729)*Sqrt[11]*Defer[Int][E

^x/((E^(2*x) - x)*(5 + I*Sqrt[11] + 2*x)), x] - (4*(11 - (5*I)*Sqrt[11])*Defer[Int][E^x/((E^(2*x) - x)*(5 + I*

Sqrt[11] + 2*x)), x])/297 - (((8*I)/9)*Defer[Int][1/((3 + E^x + x)^3*(5 + I*Sqrt[11] + 2*x)), x])/Sqrt[11] - (

8*(11 - (5*I)*Sqrt[11])*Defer[Int][1/((3 + E^x + x)^3*(5 + I*Sqrt[11] + 2*x)), x])/99 + (((52*I)/27)*Defer[Int

][1/((3 + E^x + x)^2*(5 + I*Sqrt[11] + 2*x)), x])/Sqrt[11] + (4*(11 - (5*I)*Sqrt[11])*Defer[Int][1/((3 + E^x +

x)^2*(5 + I*Sqrt[11] + 2*x)), x])/297 + (((16*I)/9)*Defer[Int][1/((3 + E^x + x)*(5 + I*Sqrt[11] + 2*x)), x])/

Sqrt[11] + (4*(11 - (5*I)*Sqrt[11])*Defer[Int][1/((3 + E^x + x)*(5 + I*Sqrt[11] + 2*x)), x])/297 - 40*Defer[In

t][1/((E^(2*x) - x)*(9 + 5*x + x^2)^3), x] + (64*Defer[Int][E^x/((E^(2*x) - x)*(9 + 5*x + x^2)^3), x])/3 - 16*

Defer[Int][x/((E^(2*x) - x)*(9 + 5*x + x^2)^3), x] + (8*Defer[Int][(E^x*x)/((E^(2*x) - x)*(9 + 5*x + x^2)^3),

x])/3 - (64*Defer[Int][1/((3 + E^x + x)*(9 + 5*x + x^2)^3), x])/3 - (8*Defer[Int][x/((3 + E^x + x)*(9 + 5*x +

x^2)^3), x])/3 + 4*Defer[Int][1/((E^(2*x) - x)^2*(9 + 5*x + x^2)^2), x] + (80*Defer[Int][E^x/((E^(2*x) - x)^2*

(9 + 5*x + x^2)^2), x])/3 - (58*Defer[Int][1/((E^(2*x) - x)*(9 + 5*x + x^2)^2), x])/9 - (208*Defer[Int][E^x/((

E^(2*x) - x)*(9 + 5*x + x^2)^2), x])/27 - 8*Defer[Int][x/((E^(2*x) - x)^2*(9 + 5*x + x^2)^2), x] + (28*Defer[I

nt][(E^x*x)/((E^(2*x) - x)^2*(9 + 5*x + x^2)^2), x])/3 + (10*Defer[Int][x/((E^(2*x) - x)*(9 + 5*x + x^2)^2), x

])/9 - (56*Defer[Int][(E^x*x)/((E^(2*x) - x)*(9 + 5*x + x^2)^2), x])/27 + (74*Defer[Int][1/((3 + E^x + x)^2*(9

+ 5*x + x^2)^2), x])/9 + (22*Defer[Int][x/((3 + E^x + x)^2*(9 + 5*x + x^2)^2), x])/9 + (208*Defer[Int][1/((3

+ E^x + x)*(9 + 5*x + x^2)^2), x])/27 + (56*Defer[Int][x/((3 + E^x + x)*(9 + 5*x + x^2)^2), x])/27

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (e^{7 x} x^2+3 e^{6 x} x^2 (3+x)+e^{5 x} \left (-1-2 x+27 x^2+16 x^3+3 x^4\right )+e^{4 x} \left (-9-21 x+21 x^2+9 x^3+3 x^4+x^5\right )+e^x x \left (56+38 x+6 x^2+27 x^3+18 x^4+3 x^5\right )-e^{3 x} \left (28+74 x+39 x^2+60 x^3+35 x^4+6 x^5\right )+x \left (60+58 x+18 x^2+29 x^3+27 x^4+9 x^5+x^6\right )-e^{2 x} \left (30+72 x+59 x^2+73 x^3+47 x^4+15 x^5+2 x^6\right )\right )}{\left (e^{2 x}-x\right )^2 x^2 \left (3+e^x+x\right )^3} \, dx\\ &=2 \int \frac {e^{7 x} x^2+3 e^{6 x} x^2 (3+x)+e^{5 x} \left (-1-2 x+27 x^2+16 x^3+3 x^4\right )+e^{4 x} \left (-9-21 x+21 x^2+9 x^3+3 x^4+x^5\right )+e^x x \left (56+38 x+6 x^2+27 x^3+18 x^4+3 x^5\right )-e^{3 x} \left (28+74 x+39 x^2+60 x^3+35 x^4+6 x^5\right )+x \left (60+58 x+18 x^2+29 x^3+27 x^4+9 x^5+x^6\right )-e^{2 x} \left (30+72 x+59 x^2+73 x^3+47 x^4+15 x^5+2 x^6\right )}{\left (e^{2 x}-x\right )^2 x^2 \left (3+e^x+x\right )^3} \, dx\\ &=2 \int \left (1+\frac {2 (2+x)}{x \left (3+e^x+x\right )^3 \left (9+5 x+x^2\right )}-\frac {9+16 x+3 x^2}{x^2 \left (3+e^x+x\right )^2 \left (9+5 x+x^2\right )^2}+\frac {(-1+2 x) \left (-90+6 e^x-97 x+2 e^x x-44 x^2-10 x^3-x^4\right )}{\left (e^{2 x}-x\right )^2 x \left (9+5 x+x^2\right )^2}-\frac {2 \left (27+72 x+49 x^2+12 x^3+x^4\right )}{x^2 \left (3+e^x+x\right ) \left (9+5 x+x^2\right )^3}+\frac {-810+54 e^x-2970 x+144 e^x x-3670 x^2+98 e^x x^2-2370 x^3+24 e^x x^3-919 x^4+2 e^x x^4-221 x^5-31 x^6-2 x^7}{\left (e^{2 x}-x\right ) x^2 \left (9+5 x+x^2\right )^3}\right ) \, dx\\ &=2 x-2 \int \frac {9+16 x+3 x^2}{x^2 \left (3+e^x+x\right )^2 \left (9+5 x+x^2\right )^2} \, dx+2 \int \frac {(-1+2 x) \left (-90+6 e^x-97 x+2 e^x x-44 x^2-10 x^3-x^4\right )}{\left (e^{2 x}-x\right )^2 x \left (9+5 x+x^2\right )^2} \, dx+2 \int \frac {-810+54 e^x-2970 x+144 e^x x-3670 x^2+98 e^x x^2-2370 x^3+24 e^x x^3-919 x^4+2 e^x x^4-221 x^5-31 x^6-2 x^7}{\left (e^{2 x}-x\right ) x^2 \left (9+5 x+x^2\right )^3} \, dx+4 \int \frac {2+x}{x \left (3+e^x+x\right )^3 \left (9+5 x+x^2\right )} \, dx-4 \int \frac {27+72 x+49 x^2+12 x^3+x^4}{x^2 \left (3+e^x+x\right ) \left (9+5 x+x^2\right )^3} \, dx\\ &=2 x-2 \int \left (\frac {1}{9 x^2 \left (3+e^x+x\right )^2}+\frac {2}{27 x \left (3+e^x+x\right )^2}-\frac {37+11 x}{9 \left (3+e^x+x\right )^2 \left (9+5 x+x^2\right )^2}-\frac {13+2 x}{27 \left (3+e^x+x\right )^2 \left (9+5 x+x^2\right )}\right ) \, dx+2 \int \left (-\frac {-90+6 e^x-97 x+2 e^x x-44 x^2-10 x^3-x^4}{81 \left (e^{2 x}-x\right )^2 x}+\frac {(23+x) \left (-90+6 e^x-97 x+2 e^x x-44 x^2-10 x^3-x^4\right )}{9 \left (e^{2 x}-x\right )^2 \left (9+5 x+x^2\right )^2}+\frac {(5+x) \left (-90+6 e^x-97 x+2 e^x x-44 x^2-10 x^3-x^4\right )}{81 \left (e^{2 x}-x\right )^2 \left (9+5 x+x^2\right )}\right ) \, dx+2 \int \left (\frac {-810+54 e^x-2970 x+144 e^x x-3670 x^2+98 e^x x^2-2370 x^3+24 e^x x^3-919 x^4+2 e^x x^4-221 x^5-31 x^6-2 x^7}{729 \left (e^{2 x}-x\right ) x^2}-\frac {5 \left (-810+54 e^x-2970 x+144 e^x x-3670 x^2+98 e^x x^2-2370 x^3+24 e^x x^3-919 x^4+2 e^x x^4-221 x^5-31 x^6-2 x^7\right )}{2187 \left (e^{2 x}-x\right ) x}+\frac {(16+5 x) \left (-810+54 e^x-2970 x+144 e^x x-3670 x^2+98 e^x x^2-2370 x^3+24 e^x x^3-919 x^4+2 e^x x^4-221 x^5-31 x^6-2 x^7\right )}{81 \left (e^{2 x}-x\right ) \left (9+5 x+x^2\right )^3}+\frac {(41+10 x) \left (-810+54 e^x-2970 x+144 e^x x-3670 x^2+98 e^x x^2-2370 x^3+24 e^x x^3-919 x^4+2 e^x x^4-221 x^5-31 x^6-2 x^7\right )}{729 \left (e^{2 x}-x\right ) \left (9+5 x+x^2\right )^2}+\frac {(22+5 x) \left (-810+54 e^x-2970 x+144 e^x x-3670 x^2+98 e^x x^2-2370 x^3+24 e^x x^3-919 x^4+2 e^x x^4-221 x^5-31 x^6-2 x^7\right )}{2187 \left (e^{2 x}-x\right ) \left (9+5 x+x^2\right )}\right ) \, dx-4 \int \left (\frac {1}{27 x^2 \left (3+e^x+x\right )}+\frac {1}{27 x \left (3+e^x+x\right )}+\frac {2 (8+x)}{3 \left (3+e^x+x\right ) \left (9+5 x+x^2\right )^3}-\frac {2 (26+7 x)}{27 \left (3+e^x+x\right ) \left (9+5 x+x^2\right )^2}-\frac {6+x}{27 \left (3+e^x+x\right ) \left (9+5 x+x^2\right )}\right ) \, dx+4 \int \left (\frac {2}{9 x \left (3+e^x+x\right )^3}-\frac {1+2 x}{9 \left (3+e^x+x\right )^3 \left (9+5 x+x^2\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(99\) vs. \(2(37)=74\).
time = 10.15, size = 99, normalized size = 2.68 \begin {gather*} \frac {2 \left (x^2+\frac {2 (3+x)}{\left (3+e^x+x\right ) \left (9+5 x+x^2\right )^2}+\frac {1}{\left (3+e^x+x\right )^2 \left (9+5 x+x^2\right )}+\frac {90+97 x+44 x^2+10 x^3+x^4-2 e^x (3+x)}{\left (e^{2 x}-x\right ) \left (9+5 x+x^2\right )^2}\right )}{x} \end {gather*}

Integrate[(120*x + 116*x^2 + 36*x^3 + 58*x^4 + 54*x^5 + 18*x^6 + 2*x^7 + E^(3*x)*(4*x + 2*x^4) + E^(2*x)*(36*x

+ 12*x^2 + 18*x^4 + 6*x^5) + E^x*(112*x + 76*x^2 + 12*x^3 + 54*x^4 + 36*x^5 + 6*x^6) + E^(4*x)*(54*x^2 + 2*E^

(3*x)*x^2 + 54*x^3 + 18*x^4 + 2*x^5 + E^(2*x)*(18*x^2 + 6*x^3) + E^x*(54*x^2 + 36*x^3 + 6*x^4)) + E^(2*x)*(-60

- 180*x - 130*x^2 - 146*x^3 - 112*x^4 - 36*x^5 - 4*x^6 + E^(3*x)*(-2 - 4*x - 4*x^3) + E^(2*x)*(-18 - 42*x - 1

2*x^2 - 36*x^3 - 12*x^4) + E^x*(-56 - 152*x - 78*x^2 - 120*x^3 - 72*x^4 - 12*x^5)))/(27*x^4 + E^(3*x)*x^4 + 27

*x^5 + 9*x^6 + x^7 + E^(2*x)*(9*x^4 + 3*x^5) + E^x*(27*x^4 + 18*x^5 + 3*x^6) + E^(4*x)*(27*x^2 + E^(3*x)*x^2 +

27*x^3 + 9*x^4 + x^5 + E^(2*x)*(9*x^2 + 3*x^3) + E^x*(27*x^2 + 18*x^3 + 3*x^4)) + E^(2*x)*(-54*x^3 - 2*E^(3*x

)*x^3 - 54*x^4 - 18*x^5 - 2*x^6 + E^(2*x)*(-18*x^3 - 6*x^4) + E^x*(-54*x^3 - 36*x^4 - 6*x^5))),x]

(2*(x^2 + (2*(3 + x))/((3 + E^x + x)*(9 + 5*x + x^2)^2) + 1/((3 + E^x + x)^2*(9 + 5*x + x^2)) + (90 + 97*x + 4

4*x^2 + 10*x^3 + x^4 - 2*E^x*(3 + x))/((E^(2*x) - x)*(9 + 5*x + x^2)^2)))/x

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method	result	size

risch	\(2 x -\frac {2 \left (x^{2}+2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}+6 x +6 \,{\mathrm e}^{x}+10\right )}{x \left ({\mathrm e}^{x}+3+x \right )^{2} \left (x -{\mathrm e}^{2 x}\right )}\)	\(48\)

int(((2*x^2*exp(x)^3+(6*x^3+18*x^2)*exp(x)^2+(6*x^4+36*x^3+54*x^2)*exp(x)+2*x^5+18*x^4+54*x^3+54*x^2)*exp(2*x)

^2+((-4*x^3-4*x-2)*exp(x)^3+(-12*x^4-36*x^3-12*x^2-42*x-18)*exp(x)^2+(-12*x^5-72*x^4-120*x^3-78*x^2-152*x-56)*

exp(x)-4*x^6-36*x^5-112*x^4-146*x^3-130*x^2-180*x-60)*exp(2*x)+(2*x^4+4*x)*exp(x)^3+(6*x^5+18*x^4+12*x^2+36*x)

*exp(x)^2+(6*x^6+36*x^5+54*x^4+12*x^3+76*x^2+112*x)*exp(x)+2*x^7+18*x^6+54*x^5+58*x^4+36*x^3+116*x^2+120*x)/((

x^2*exp(x)^3+(3*x^3+9*x^2)*exp(x)^2+(3*x^4+18*x^3+27*x^2)*exp(x)+x^5+9*x^4+27*x^3+27*x^2)*exp(2*x)^2+(-2*x^3*e

xp(x)^3+(-6*x^4-18*x^3)*exp(x)^2+(-6*x^5-36*x^4-54*x^3)*exp(x)-2*x^6-18*x^5-54*x^4-54*x^3)*exp(2*x)+x^4*exp(x)

^3+(3*x^5+9*x^4)*exp(x)^2+(3*x^6+18*x^5+27*x^4)*exp(x)+x^7+9*x^6+27*x^5+27*x^4),x,method=_RETURNVERBOSE)

2*x-2/x*(x^2+2*exp(x)*x+exp(2*x)+6*x+6*exp(x)+10)/(exp(x)+3+x)^2/(x-exp(2*x))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (34) = 68\).
time = 0.58, size = 154, normalized size = 4.16 \begin {gather*} \frac {2 \, {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3} - x^{2} e^{\left (4 \, x\right )} - x^{2} - 2 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{\left (3 \, x\right )} - {\left (x^{4} + 5 \, x^{3} + 9 \, x^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} + 3 \, x^{3} - x - 3\right )} e^{x} - 6 \, x - 10\right )}}{x^{4} + 6 \, x^{3} + 9 \, x^{2} - x e^{\left (4 \, x\right )} - 2 \, {\left (x^{2} + 3 \, x\right )} e^{\left (3 \, x\right )} - {\left (x^{3} + 5 \, x^{2} + 9 \, x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{x}} \end {gather*}

integrate(((2*x^2*exp(x)^3+(6*x^3+18*x^2)*exp(x)^2+(6*x^4+36*x^3+54*x^2)*exp(x)+2*x^5+18*x^4+54*x^3+54*x^2)*ex

p(2*x)^2+((-4*x^3-4*x-2)*exp(x)^3+(-12*x^4-36*x^3-12*x^2-42*x-18)*exp(x)^2+(-12*x^5-72*x^4-120*x^3-78*x^2-152*

x-56)*exp(x)-4*x^6-36*x^5-112*x^4-146*x^3-130*x^2-180*x-60)*exp(2*x)+(2*x^4+4*x)*exp(x)^3+(6*x^5+18*x^4+12*x^2

+36*x)*exp(x)^2+(6*x^6+36*x^5+54*x^4+12*x^3+76*x^2+112*x)*exp(x)+2*x^7+18*x^6+54*x^5+58*x^4+36*x^3+116*x^2+120

*x)/((x^2*exp(x)^3+(3*x^3+9*x^2)*exp(x)^2+(3*x^4+18*x^3+27*x^2)*exp(x)+x^5+9*x^4+27*x^3+27*x^2)*exp(2*x)^2+(-2

*x^3*exp(x)^3+(-6*x^4-18*x^3)*exp(x)^2+(-6*x^5-36*x^4-54*x^3)*exp(x)-2*x^6-18*x^5-54*x^4-54*x^3)*exp(2*x)+x^4*

exp(x)^3+(3*x^5+9*x^4)*exp(x)^2+(3*x^6+18*x^5+27*x^4)*exp(x)+x^7+9*x^6+27*x^5+27*x^4),x, algorithm="maxima")

2*(x^5 + 6*x^4 + 9*x^3 - x^2*e^(4*x) - x^2 - 2*(x^3 + 3*x^2)*e^(3*x) - (x^4 + 5*x^3 + 9*x^2 + 1)*e^(2*x) + 2*(

x^4 + 3*x^3 - x - 3)*e^x - 6*x - 10)/(x^4 + 6*x^3 + 9*x^2 - x*e^(4*x) - 2*(x^2 + 3*x)*e^(3*x) - (x^3 + 5*x^2 +

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (34) = 68\).
time = 0.37, size = 154, normalized size = 4.16 \begin {gather*} \frac {2 \, {\left (x^{5} + 6 \, x^{4} + 9 \, x^{3} - x^{2} e^{\left (4 \, x\right )} - x^{2} - 2 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{\left (3 \, x\right )} - {\left (x^{4} + 5 \, x^{3} + 9 \, x^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} + 3 \, x^{3} - x - 3\right )} e^{x} - 6 \, x - 10\right )}}{x^{4} + 6 \, x^{3} + 9 \, x^{2} - x e^{\left (4 \, x\right )} - 2 \, {\left (x^{2} + 3 \, x\right )} e^{\left (3 \, x\right )} - {\left (x^{3} + 5 \, x^{2} + 9 \, x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{x}} \end {gather*}

integrate(((2*x^2*exp(x)^3+(6*x^3+18*x^2)*exp(x)^2+(6*x^4+36*x^3+54*x^2)*exp(x)+2*x^5+18*x^4+54*x^3+54*x^2)*ex

p(2*x)^2+((-4*x^3-4*x-2)*exp(x)^3+(-12*x^4-36*x^3-12*x^2-42*x-18)*exp(x)^2+(-12*x^5-72*x^4-120*x^3-78*x^2-152*

x-56)*exp(x)-4*x^6-36*x^5-112*x^4-146*x^3-130*x^2-180*x-60)*exp(2*x)+(2*x^4+4*x)*exp(x)^3+(6*x^5+18*x^4+12*x^2

+36*x)*exp(x)^2+(6*x^6+36*x^5+54*x^4+12*x^3+76*x^2+112*x)*exp(x)+2*x^7+18*x^6+54*x^5+58*x^4+36*x^3+116*x^2+120

*x)/((x^2*exp(x)^3+(3*x^3+9*x^2)*exp(x)^2+(3*x^4+18*x^3+27*x^2)*exp(x)+x^5+9*x^4+27*x^3+27*x^2)*exp(2*x)^2+(-2

*x^3*exp(x)^3+(-6*x^4-18*x^3)*exp(x)^2+(-6*x^5-36*x^4-54*x^3)*exp(x)-2*x^6-18*x^5-54*x^4-54*x^3)*exp(2*x)+x^4*

exp(x)^3+(3*x^5+9*x^4)*exp(x)^2+(3*x^6+18*x^5+27*x^4)*exp(x)+x^7+9*x^6+27*x^5+27*x^4),x, algorithm="fricas")

2*(x^5 + 6*x^4 + 9*x^3 - x^2*e^(4*x) - x^2 - 2*(x^3 + 3*x^2)*e^(3*x) - (x^4 + 5*x^3 + 9*x^2 + 1)*e^(2*x) + 2*(

x^4 + 3*x^3 - x - 3)*e^x - 6*x - 10)/(x^4 + 6*x^3 + 9*x^2 - x*e^(4*x) - 2*(x^2 + 3*x)*e^(3*x) - (x^3 + 5*x^2 +

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (26) = 52\).
time = 0.31, size = 94, normalized size = 2.54 \begin {gather*} 2 x + \frac {2 x^{2} + 12 x + \left (4 x + 12\right ) e^{x} + 2 e^{2 x} + 20}{- x^{4} - 6 x^{3} - 9 x^{2} + x e^{4 x} + \left (2 x^{2} + 6 x\right ) e^{3 x} + \left (- 2 x^{3} - 6 x^{2}\right ) e^{x} + \left (x^{3} + 5 x^{2} + 9 x\right ) e^{2 x}} \end {gather*}

integrate(((2*x**2*exp(x)**3+(6*x**3+18*x**2)*exp(x)**2+(6*x**4+36*x**3+54*x**2)*exp(x)+2*x**5+18*x**4+54*x**3

+54*x**2)*exp(2*x)**2+((-4*x**3-4*x-2)*exp(x)**3+(-12*x**4-36*x**3-12*x**2-42*x-18)*exp(x)**2+(-12*x**5-72*x**

4-120*x**3-78*x**2-152*x-56)*exp(x)-4*x**6-36*x**5-112*x**4-146*x**3-130*x**2-180*x-60)*exp(2*x)+(2*x**4+4*x)*

exp(x)**3+(6*x**5+18*x**4+12*x**2+36*x)*exp(x)**2+(6*x**6+36*x**5+54*x**4+12*x**3+76*x**2+112*x)*exp(x)+2*x**7

+18*x**6+54*x**5+58*x**4+36*x**3+116*x**2+120*x)/((x**2*exp(x)**3+(3*x**3+9*x**2)*exp(x)**2+(3*x**4+18*x**3+27

*x**2)*exp(x)+x**5+9*x**4+27*x**3+27*x**2)*exp(2*x)**2+(-2*x**3*exp(x)**3+(-6*x**4-18*x**3)*exp(x)**2+(-6*x**5

-36*x**4-54*x**3)*exp(x)-2*x**6-18*x**5-54*x**4-54*x**3)*exp(2*x)+x**4*exp(x)**3+(3*x**5+9*x**4)*exp(x)**2+(3*

x**6+18*x**5+27*x**4)*exp(x)+x**7+9*x**6+27*x**5+27*x**4),x)

2*x + (2*x**2 + 12*x + (4*x + 12)*exp(x) + 2*exp(2*x) + 20)/(-x**4 - 6*x**3 - 9*x**2 + x*exp(4*x) + (2*x**2 +

6*x)*exp(3*x) + (-2*x**3 - 6*x**2)*exp(x) + (x**3 + 5*x**2 + 9*x)*exp(2*x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (34) = 68\).
time = 1.03, size = 525, normalized size = 14.19 \begin {gather*} \frac {2 \, {\left (x^{9} - x^{8} e^{\left (2 \, x\right )} + 2 \, x^{8} e^{x} + 16 \, x^{8} - 2 \, x^{7} e^{\left (3 \, x\right )} - 15 \, x^{7} e^{\left (2 \, x\right )} + 26 \, x^{7} e^{x} + 112 \, x^{7} - x^{6} e^{\left (4 \, x\right )} - 26 \, x^{6} e^{\left (3 \, x\right )} - 102 \, x^{6} e^{\left (2 \, x\right )} + 146 \, x^{6} e^{x} + 436 \, x^{6} - 10 \, x^{5} e^{\left (4 \, x\right )} - 146 \, x^{5} e^{\left (3 \, x\right )} - 395 \, x^{5} e^{\left (2 \, x\right )} + 434 \, x^{5} e^{x} + 976 \, x^{5} - 43 \, x^{4} e^{\left (4 \, x\right )} - 438 \, x^{4} e^{\left (3 \, x\right )} - 920 \, x^{4} e^{\left (2 \, x\right )} + 650 \, x^{4} e^{x} + 1070 \, x^{4} - 90 \, x^{3} e^{\left (4 \, x\right )} - 702 \, x^{3} e^{\left (3 \, x\right )} - 1235 \, x^{3} e^{\left (2 \, x\right )} + 194 \, x^{3} e^{x} - 170 \, x^{3} - 81 \, x^{2} e^{\left (4 \, x\right )} - 486 \, x^{2} e^{\left (3 \, x\right )} - 812 \, x^{2} e^{\left (2 \, x\right )} - 878 \, x^{2} e^{x} - 2119 \, x^{2} + 2 \, x e^{\left (3 \, x\right )} - 163 \, x e^{\left (2 \, x\right )} - 1410 \, x e^{x} - 2799 \, x + 6 \, e^{\left (3 \, x\right )} - 135 \, e^{\left (2 \, x\right )} - 972 \, e^{x} - 1620\right )}}{x^{8} - x^{7} e^{\left (2 \, x\right )} + 2 \, x^{7} e^{x} + 16 \, x^{7} - 2 \, x^{6} e^{\left (3 \, x\right )} - 15 \, x^{6} e^{\left (2 \, x\right )} + 26 \, x^{6} e^{x} + 112 \, x^{6} - x^{5} e^{\left (4 \, x\right )} - 26 \, x^{5} e^{\left (3 \, x\right )} - 102 \, x^{5} e^{\left (2 \, x\right )} + 146 \, x^{5} e^{x} + 438 \, x^{5} - 10 \, x^{4} e^{\left (4 \, x\right )} - 146 \, x^{4} e^{\left (3 \, x\right )} - 395 \, x^{4} e^{\left (2 \, x\right )} + 438 \, x^{4} e^{x} + 1008 \, x^{4} - 43 \, x^{3} e^{\left (4 \, x\right )} - 438 \, x^{3} e^{\left (3 \, x\right )} - 918 \, x^{3} e^{\left (2 \, x\right )} + 702 \, x^{3} e^{x} + 1296 \, x^{3} - 90 \, x^{2} e^{\left (4 \, x\right )} - 702 \, x^{2} e^{\left (3 \, x\right )} - 1215 \, x^{2} e^{\left (2 \, x\right )} + 486 \, x^{2} e^{x} + 729 \, x^{2} - 81 \, x e^{\left (4 \, x\right )} - 486 \, x e^{\left (3 \, x\right )} - 729 \, x e^{\left (2 \, x\right )}} \end {gather*}

integrate(((2*x^2*exp(x)^3+(6*x^3+18*x^2)*exp(x)^2+(6*x^4+36*x^3+54*x^2)*exp(x)+2*x^5+18*x^4+54*x^3+54*x^2)*ex

p(2*x)^2+((-4*x^3-4*x-2)*exp(x)^3+(-12*x^4-36*x^3-12*x^2-42*x-18)*exp(x)^2+(-12*x^5-72*x^4-120*x^3-78*x^2-152*

x-56)*exp(x)-4*x^6-36*x^5-112*x^4-146*x^3-130*x^2-180*x-60)*exp(2*x)+(2*x^4+4*x)*exp(x)^3+(6*x^5+18*x^4+12*x^2

+36*x)*exp(x)^2+(6*x^6+36*x^5+54*x^4+12*x^3+76*x^2+112*x)*exp(x)+2*x^7+18*x^6+54*x^5+58*x^4+36*x^3+116*x^2+120

*x)/((x^2*exp(x)^3+(3*x^3+9*x^2)*exp(x)^2+(3*x^4+18*x^3+27*x^2)*exp(x)+x^5+9*x^4+27*x^3+27*x^2)*exp(2*x)^2+(-2

*x^3*exp(x)^3+(-6*x^4-18*x^3)*exp(x)^2+(-6*x^5-36*x^4-54*x^3)*exp(x)-2*x^6-18*x^5-54*x^4-54*x^3)*exp(2*x)+x^4*

exp(x)^3+(3*x^5+9*x^4)*exp(x)^2+(3*x^6+18*x^5+27*x^4)*exp(x)+x^7+9*x^6+27*x^5+27*x^4),x, algorithm="giac")

2*(x^9 - x^8*e^(2*x) + 2*x^8*e^x + 16*x^8 - 2*x^7*e^(3*x) - 15*x^7*e^(2*x) + 26*x^7*e^x + 112*x^7 - x^6*e^(4*x

) - 26*x^6*e^(3*x) - 102*x^6*e^(2*x) + 146*x^6*e^x + 436*x^6 - 10*x^5*e^(4*x) - 146*x^5*e^(3*x) - 395*x^5*e^(2

*x) + 434*x^5*e^x + 976*x^5 - 43*x^4*e^(4*x) - 438*x^4*e^(3*x) - 920*x^4*e^(2*x) + 650*x^4*e^x + 1070*x^4 - 90

*x^3*e^(4*x) - 702*x^3*e^(3*x) - 1235*x^3*e^(2*x) + 194*x^3*e^x - 170*x^3 - 81*x^2*e^(4*x) - 486*x^2*e^(3*x) -

812*x^2*e^(2*x) - 878*x^2*e^x - 2119*x^2 + 2*x*e^(3*x) - 163*x*e^(2*x) - 1410*x*e^x - 2799*x + 6*e^(3*x) - 13

5*e^(2*x) - 972*e^x - 1620)/(x^8 - x^7*e^(2*x) + 2*x^7*e^x + 16*x^7 - 2*x^6*e^(3*x) - 15*x^6*e^(2*x) + 26*x^6*

e^x + 112*x^6 - x^5*e^(4*x) - 26*x^5*e^(3*x) - 102*x^5*e^(2*x) + 146*x^5*e^x + 438*x^5 - 10*x^4*e^(4*x) - 146*

x^4*e^(3*x) - 395*x^4*e^(2*x) + 438*x^4*e^x + 1008*x^4 - 43*x^3*e^(4*x) - 438*x^3*e^(3*x) - 918*x^3*e^(2*x) +

702*x^3*e^x + 1296*x^3 - 90*x^2*e^(4*x) - 702*x^2*e^(3*x) - 1215*x^2*e^(2*x) + 486*x^2*e^x + 729*x^2 - 81*x*e^

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {120\,x+{\mathrm {e}}^{3\,x}\,\left (2\,x^4+4\,x\right )+{\mathrm {e}}^{2\,x}\,\left (6\,x^5+18\,x^4+12\,x^2+36\,x\right )+{\mathrm {e}}^x\,\left (6\,x^6+36\,x^5+54\,x^4+12\,x^3+76\,x^2+112\,x\right )-{\mathrm {e}}^{2\,x}\,\left (180\,x+{\mathrm {e}}^{3\,x}\,\left (4\,x^3+4\,x+2\right )+{\mathrm {e}}^x\,\left (12\,x^5+72\,x^4+120\,x^3+78\,x^2+152\,x+56\right )+{\mathrm {e}}^{2\,x}\,\left (12\,x^4+36\,x^3+12\,x^2+42\,x+18\right )+130\,x^2+146\,x^3+112\,x^4+36\,x^5+4\,x^6+60\right )+116\,x^2+36\,x^3+58\,x^4+54\,x^5+18\,x^6+2\,x^7+{\mathrm {e}}^{4\,x}\,\left ({\mathrm {e}}^x\,\left (6\,x^4+36\,x^3+54\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (6\,x^3+18\,x^2\right )+2\,x^2\,{\mathrm {e}}^{3\,x}+54\,x^2+54\,x^3+18\,x^4+2\,x^5\right )}{{\mathrm {e}}^x\,\left (3\,x^6+18\,x^5+27\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (3\,x^5+9\,x^4\right )+x^4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\,\left ({\mathrm {e}}^x\,\left (3\,x^4+18\,x^3+27\,x^2\right )+{\mathrm {e}}^{2\,x}\,\left (3\,x^3+9\,x^2\right )+x^2\,{\mathrm {e}}^{3\,x}+27\,x^2+27\,x^3+9\,x^4+x^5\right )+27\,x^4+27\,x^5+9\,x^6+x^7-{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^x\,\left (6\,x^5+36\,x^4+54\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (6\,x^4+18\,x^3\right )+2\,x^3\,{\mathrm {e}}^{3\,x}+54\,x^3+54\,x^4+18\,x^5+2\,x^6\right )} \,d x \end {gather*}

int((120*x + exp(3*x)*(4*x + 2*x^4) + exp(2*x)*(36*x + 12*x^2 + 18*x^4 + 6*x^5) + exp(x)*(112*x + 76*x^2 + 12*

x^3 + 54*x^4 + 36*x^5 + 6*x^6) - exp(2*x)*(180*x + exp(3*x)*(4*x + 4*x^3 + 2) + exp(x)*(152*x + 78*x^2 + 120*x

^3 + 72*x^4 + 12*x^5 + 56) + exp(2*x)*(42*x + 12*x^2 + 36*x^3 + 12*x^4 + 18) + 130*x^2 + 146*x^3 + 112*x^4 + 3

6*x^5 + 4*x^6 + 60) + 116*x^2 + 36*x^3 + 58*x^4 + 54*x^5 + 18*x^6 + 2*x^7 + exp(4*x)*(exp(x)*(54*x^2 + 36*x^3

+ 6*x^4) + exp(2*x)*(18*x^2 + 6*x^3) + 2*x^2*exp(3*x) + 54*x^2 + 54*x^3 + 18*x^4 + 2*x^5))/(exp(x)*(27*x^4 + 1

8*x^5 + 3*x^6) + exp(2*x)*(9*x^4 + 3*x^5) + x^4*exp(3*x) + exp(4*x)*(exp(x)*(27*x^2 + 18*x^3 + 3*x^4) + exp(2*

x)*(9*x^2 + 3*x^3) + x^2*exp(3*x) + 27*x^2 + 27*x^3 + 9*x^4 + x^5) + 27*x^4 + 27*x^5 + 9*x^6 + x^7 - exp(2*x)*

(exp(x)*(54*x^3 + 36*x^4 + 6*x^5) + exp(2*x)*(18*x^3 + 6*x^4) + 2*x^3*exp(3*x) + 54*x^3 + 54*x^4 + 18*x^5 + 2*