3.82.83 \(\int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} (24-24 e^{-2+x} x+e^x (-60 x+12 e^4 x-12 x^2))+e^x (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 (-20 x^2-4 x^3))}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} (12 x+e^x (-30 x+6 e^4 x-6 x^2))+e^x (25 x^2+e^8 x^2+10 x^3+x^4+e^4 (-10 x^2-2 x^3))} \, dx\)

Optimal. Leaf size=33 \[ \log \left (\left (-e^x+\frac {4}{5-e^4-\frac {3 e^{e^{-2+x}}}{x}+x}\right )^2\right ) \]

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Rubi [F]  time = 15.89, 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 {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}-20 x^2+4 e^4 x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(18*E^(2*E^(-2 + x) + x) + 8*x^2 + E^E^(-2 + x)*(24 - 24*E^(-2 + x)*x + E^x*(-60*x + 12*E^4*x - 12*x^2)) +
E^x*(50*x^2 + 2*E^8*x^2 + 20*x^3 + 2*x^4 + E^4*(-20*x^2 - 4*x^3)))/(9*E^(2*E^(-2 + x) + x) - 20*x^2 + 4*E^4*x
^2 - 4*x^3 + E^E^(-2 + x)*(12*x + E^x*(-30*x + 6*E^4*x - 6*x^2)) + E^x*(25*x^2 + E^8*x^2 + 10*x^3 + x^4 + E^4*
(-10*x^2 - 2*x^3))),x]

[Out]

2*x - (8*(5 - E^4)*Defer[Int][x^2/(3*E^E^(-2 + x) - 5*(1 - E^4/5)*x - x^2)^2, x])/E^2 - (8*Defer[Int][x^3/(3*E
^E^(-2 + x) - 5*(1 - E^4/5)*x - x^2)^2, x])/E^2 + (8*Defer[Int][x/(-3*E^E^(-2 + x) + 5*(1 - E^4/5)*x + x^2), x
])/E^2 + (72*Defer[Int][E^(2 + 2*E^(-2 + x))/((3*E^E^(-2 + x) - 5*(1 - E^4/5)*x - x^2)^2*(3*E^(E^(-2 + x) + x)
+ 4*x - 5*E^x*(1 - E^4/5)*x - E^x*x^2)), x])/E^2 - (24*(5 - E^4)*Defer[Int][(E^(2 + E^(-2 + x))*x)/((3*E^E^(-
2 + x) - 5*(1 - E^4/5)*x - x^2)^2*(3*E^(E^(-2 + x) + x) + 4*x - 5*E^x*(1 - E^4/5)*x - E^x*x^2)), x])/E^2 + (48
*(2 + 5*E^2 - E^6)*Defer[Int][(E^E^(-2 + x)*x^2)/((3*E^E^(-2 + x) - 5*(1 - E^4/5)*x - x^2)^2*(3*E^(E^(-2 + x)
+ x) + 4*x - 5*E^x*(1 - E^4/5)*x - E^x*x^2)), x])/E^2 - 8*(30 - 11*E^4 + E^8)*Defer[Int][x^3/((3*E^E^(-2 + x)
- 5*(1 - E^4/5)*x - x^2)^2*(3*E^(E^(-2 + x) + x) + 4*x - 5*E^x*(1 - E^4/5)*x - E^x*x^2)), x] + (48*Defer[Int][
(E^(2 + E^(-2 + x))*x^3)/((3*E^E^(-2 + x) - 5*(1 - E^4/5)*x - x^2)^2*(3*E^(E^(-2 + x) + x) + 4*x - 5*E^x*(1 -
E^4/5)*x - E^x*x^2)), x])/E^2 - 8*(11 - 2*E^4)*Defer[Int][x^4/((3*E^E^(-2 + x) - 5*(1 - E^4/5)*x - x^2)^2*(3*E
^(E^(-2 + x) + x) + 4*x - 5*E^x*(1 - E^4/5)*x - E^x*x^2)), x] + (72*Defer[Int][(E^(2 + 2*E^(-2 + x))*x)/((3*E^
E^(-2 + x) - 5*(1 - E^4/5)*x - x^2)^2*(-3*E^(E^(-2 + x) + x) - 4*x + 5*E^x*(1 - E^4/5)*x + E^x*x^2)), x])/E^2
+ 8*Defer[Int][x^5/((3*E^E^(-2 + x) - 5*(1 - E^4/5)*x - x^2)^2*(-3*E^(E^(-2 + x) + x) - 4*x + 5*E^x*(1 - E^4/5
)*x + E^x*x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{9 e^{2 e^{-2+x}+x}+\left (-20+4 e^4\right ) x^2-4 x^3+e^{e^{-2+x}} \left (12 x+e^x \left (-30 x+6 e^4 x-6 x^2\right )\right )+e^x \left (25 x^2+e^8 x^2+10 x^3+x^4+e^4 \left (-10 x^2-2 x^3\right )\right )} \, dx\\ &=\int \frac {18 e^{2 e^{-2+x}+x}+8 x^2+e^{e^{-2+x}} \left (24-24 e^{-2+x} x+e^x \left (-60 x+12 e^4 x-12 x^2\right )\right )+e^x \left (50 x^2+2 e^8 x^2+20 x^3+2 x^4+e^4 \left (-20 x^2-4 x^3\right )\right )}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right ) \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx\\ &=\int \left (\frac {2 \left (9 e^{2+2 e^{-2+x}}-12 e^{e^{-2+x}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x-6 e^{2+e^{-2+x}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3+e^2 x^4\right )}{e^2 \left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2}+\frac {8 \left (9 e^{2+2 e^{-2+x}}-9 e^{2+2 e^{-2+x}} x-15 e^{2+e^{-2+x}} \left (1-\frac {e^4}{5}\right ) x+12 e^{e^{-2+x}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2+6 e^{2+e^{-2+x}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4-e^2 x^5\right )}{e^2 \left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )}\right ) \, dx\\ &=\frac {2 \int \frac {9 e^{2+2 e^{-2+x}}-12 e^{e^{-2+x}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x-6 e^{2+e^{-2+x}} x^2+25 e^2 \left (1+\frac {1}{25} e^4 \left (-10+e^4\right )\right ) x^2+10 e^2 \left (1-\frac {e^4}{5}\right ) x^3+e^2 x^4}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2} \, dx}{e^2}+\frac {8 \int \frac {9 e^{2+2 e^{-2+x}}-9 e^{2+2 e^{-2+x}} x-15 e^{2+e^{-2+x}} \left (1-\frac {e^4}{5}\right ) x+12 e^{e^{-2+x}} \left (1-\frac {1}{2} e^2 \left (-5+e^4\right )\right ) x^2+6 e^{2+e^{-2+x}} x^3-30 e^2 \left (1+\frac {1}{30} e^4 \left (-11+e^4\right )\right ) x^3-11 e^2 \left (1-\frac {2 e^4}{11}\right ) x^4-e^2 x^5}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}\\ &=\frac {2 \int \left (e^2+\frac {4 \left (-5+e^4-x\right ) x^2}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2}+\frac {4 x}{-3 e^{e^{-2+x}}+5 \left (1-\frac {e^4}{5}\right ) x+x^2}\right ) \, dx}{e^2}+\frac {8 \int \left (\frac {9 e^{2+2 e^{-2+x}}}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )}+\frac {3 e^{2+e^{-2+x}} \left (-5+e^4\right ) x}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )}+\frac {6 e^{e^{-2+x}} \left (2+5 e^2-e^6\right ) x^2}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )}+\frac {6 e^{2+e^{-2+x}} x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )}+\frac {e^2 \left (-30+11 e^4-e^8\right ) x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )}+\frac {e^2 \left (-11+2 e^4\right ) x^4}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )}+\frac {9 e^{2+2 e^{-2+x}} x}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (-3 e^{e^{-2+x}+x}-4 x+5 e^x \left (1-\frac {e^4}{5}\right ) x+e^x x^2\right )}+\frac {e^2 x^5}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (-3 e^{e^{-2+x}+x}-4 x+5 e^x \left (1-\frac {e^4}{5}\right ) x+e^x x^2\right )}\right ) \, dx}{e^2}\\ &=2 x+8 \int \frac {x^5}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (-3 e^{e^{-2+x}+x}-4 x+5 e^x \left (1-\frac {e^4}{5}\right ) x+e^x x^2\right )} \, dx+\frac {8 \int \frac {\left (-5+e^4-x\right ) x^2}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2} \, dx}{e^2}+\frac {8 \int \frac {x}{-3 e^{e^{-2+x}}+5 \left (1-\frac {e^4}{5}\right ) x+x^2} \, dx}{e^2}+\frac {48 \int \frac {e^{2+e^{-2+x}} x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}+\frac {72 \int \frac {e^{2+2 e^{-2+x}}}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}+\frac {72 \int \frac {e^{2+2 e^{-2+x}} x}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (-3 e^{e^{-2+x}+x}-4 x+5 e^x \left (1-\frac {e^4}{5}\right ) x+e^x x^2\right )} \, dx}{e^2}-\left (8 \left (11-2 e^4\right )\right ) \int \frac {x^4}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx-\frac {\left (24 \left (5-e^4\right )\right ) \int \frac {e^{2+e^{-2+x}} x}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}+\frac {\left (48 \left (2+5 e^2-e^6\right )\right ) \int \frac {e^{e^{-2+x}} x^2}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}-\left (8 \left (30-11 e^4+e^8\right )\right ) \int \frac {x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx\\ &=2 x+8 \int \frac {x^5}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (-3 e^{e^{-2+x}+x}-4 x+5 e^x \left (1-\frac {e^4}{5}\right ) x+e^x x^2\right )} \, dx+\frac {8 \int \frac {x}{-3 e^{e^{-2+x}}+5 \left (1-\frac {e^4}{5}\right ) x+x^2} \, dx}{e^2}+\frac {8 \int \left (\frac {\left (-5+e^4\right ) x^2}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2}-\frac {x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2}\right ) \, dx}{e^2}+\frac {48 \int \frac {e^{2+e^{-2+x}} x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}+\frac {72 \int \frac {e^{2+2 e^{-2+x}}}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}+\frac {72 \int \frac {e^{2+2 e^{-2+x}} x}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (-3 e^{e^{-2+x}+x}-4 x+5 e^x \left (1-\frac {e^4}{5}\right ) x+e^x x^2\right )} \, dx}{e^2}-\left (8 \left (11-2 e^4\right )\right ) \int \frac {x^4}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx-\frac {\left (24 \left (5-e^4\right )\right ) \int \frac {e^{2+e^{-2+x}} x}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}+\frac {\left (48 \left (2+5 e^2-e^6\right )\right ) \int \frac {e^{e^{-2+x}} x^2}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}-\left (8 \left (30-11 e^4+e^8\right )\right ) \int \frac {x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx\\ &=2 x+8 \int \frac {x^5}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (-3 e^{e^{-2+x}+x}-4 x+5 e^x \left (1-\frac {e^4}{5}\right ) x+e^x x^2\right )} \, dx-\frac {8 \int \frac {x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2} \, dx}{e^2}+\frac {8 \int \frac {x}{-3 e^{e^{-2+x}}+5 \left (1-\frac {e^4}{5}\right ) x+x^2} \, dx}{e^2}+\frac {48 \int \frac {e^{2+e^{-2+x}} x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}+\frac {72 \int \frac {e^{2+2 e^{-2+x}}}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}+\frac {72 \int \frac {e^{2+2 e^{-2+x}} x}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (-3 e^{e^{-2+x}+x}-4 x+5 e^x \left (1-\frac {e^4}{5}\right ) x+e^x x^2\right )} \, dx}{e^2}-\left (8 \left (11-2 e^4\right )\right ) \int \frac {x^4}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx-\frac {\left (8 \left (5-e^4\right )\right ) \int \frac {x^2}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2} \, dx}{e^2}-\frac {\left (24 \left (5-e^4\right )\right ) \int \frac {e^{2+e^{-2+x}} x}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}+\frac {\left (48 \left (2+5 e^2-e^6\right )\right ) \int \frac {e^{e^{-2+x}} x^2}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx}{e^2}-\left (8 \left (30-11 e^4+e^8\right )\right ) \int \frac {x^3}{\left (3 e^{e^{-2+x}}-5 \left (1-\frac {e^4}{5}\right ) x-x^2\right )^2 \left (3 e^{e^{-2+x}+x}+4 x-5 e^x \left (1-\frac {e^4}{5}\right ) x-e^x x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.18, size = 76, normalized size = 2.30 \begin {gather*} \frac {2 \left (-e^2 \log \left (3 e^{e^{-2+x}}-5 x+e^4 x-x^2\right )+e^2 \log \left (3 e^{e^{-2+x}+x}+4 x-5 e^x x+e^{4+x} x-e^x x^2\right )\right )}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18*E^(2*E^(-2 + x) + x) + 8*x^2 + E^E^(-2 + x)*(24 - 24*E^(-2 + x)*x + E^x*(-60*x + 12*E^4*x - 12*x
^2)) + E^x*(50*x^2 + 2*E^8*x^2 + 20*x^3 + 2*x^4 + E^4*(-20*x^2 - 4*x^3)))/(9*E^(2*E^(-2 + x) + x) - 20*x^2 + 4
*E^4*x^2 - 4*x^3 + E^E^(-2 + x)*(12*x + E^x*(-30*x + 6*E^4*x - 6*x^2)) + E^x*(25*x^2 + E^8*x^2 + 10*x^3 + x^4
+ E^4*(-10*x^2 - 2*x^3))),x]

[Out]

(2*(-(E^2*Log[3*E^E^(-2 + x) - 5*x + E^4*x - x^2]) + E^2*Log[3*E^(E^(-2 + x) + x) + 4*x - 5*E^x*x + E^(4 + x)*
x - E^x*x^2]))/E^2

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fricas [B]  time = 0.97, size = 64, normalized size = 1.94 \begin {gather*} 2 \, x - 2 \, \log \left (-x^{2} + x e^{4} - 5 \, x + 3 \, e^{\left (e^{\left (x - 2\right )}\right )}\right ) + 2 \, \log \left (-{\left ({\left (x^{2} - x e^{4} + 5 \, x\right )} e^{x} - 4 \, x - 3 \, e^{\left (x + e^{\left (x - 2\right )}\right )}\right )} e^{\left (-x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(x)*exp(exp(x-2))^2+((12*x*exp(4)-12*x^2-60*x)*exp(x)-24*x*exp(x-2)+24)*exp(exp(x-2))+(2*x^2*
exp(4)^2+(-4*x^3-20*x^2)*exp(4)+2*x^4+20*x^3+50*x^2)*exp(x)+8*x^2)/(9*exp(x)*exp(exp(x-2))^2+((6*x*exp(4)-6*x^
2-30*x)*exp(x)+12*x)*exp(exp(x-2))+(x^2*exp(4)^2+(-2*x^3-10*x^2)*exp(4)+x^4+10*x^3+25*x^2)*exp(x)+4*x^2*exp(4)
-4*x^3-20*x^2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(x)*exp(exp(x-2))^2+((12*x*exp(4)-12*x^2-60*x)*exp(x)-24*x*exp(x-2)+24)*exp(exp(x-2))+(2*x^2*
exp(4)^2+(-4*x^3-20*x^2)*exp(4)+2*x^4+20*x^3+50*x^2)*exp(x)+8*x^2)/(9*exp(x)*exp(exp(x-2))^2+((6*x*exp(4)-6*x^
2-30*x)*exp(x)+12*x)*exp(exp(x-2))+(x^2*exp(4)^2+(-2*x^3-10*x^2)*exp(4)+x^4+10*x^3+25*x^2)*exp(x)+4*x^2*exp(4)
-4*x^3-20*x^2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 1.58, size = 61, normalized size = 1.85

method	result	size
norman	\(-2 \ln \left (x \,{\mathrm e}^{4}-x^{2}-5 x +3 \,{\mathrm e}^{{\mathrm e}^{x} {\mathrm e}^{-2}}\right )+2 \ln \left (x \,{\mathrm e}^{4} {\mathrm e}^{x}-{\mathrm e}^{x} x^{2}-5 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{{\mathrm e}^{x} {\mathrm e}^{-2}} {\mathrm e}^{x}+4 x \right )\)	\(61\)
risch	\(2 x +2 \ln \left ({\mathrm e}^{{\mathrm e}^{x -2}}-\frac {\left (x \,{\mathrm e}^{x -4}+5 \,{\mathrm e}^{x -4}-4 \,{\mathrm e}^{-4}-{\mathrm e}^{x}\right ) x \,{\mathrm e}^{-x +4}}{3}\right )-2 \ln \left ({\mathrm e}^{{\mathrm e}^{x -2}}-\frac {x \left (x \,{\mathrm e}^{-4}+5 \,{\mathrm e}^{-4}-1\right ) {\mathrm e}^{4}}{3}\right )\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*exp(x)*exp(exp(x-2))^2+((12*x*exp(4)-12*x^2-60*x)*exp(x)-24*x*exp(x-2)+24)*exp(exp(x-2))+(2*x^2*exp(4)
^2+(-4*x^3-20*x^2)*exp(4)+2*x^4+20*x^3+50*x^2)*exp(x)+8*x^2)/(9*exp(x)*exp(exp(x-2))^2+((6*x*exp(4)-6*x^2-30*x
)*exp(x)+12*x)*exp(exp(x-2))+(x^2*exp(4)^2+(-2*x^3-10*x^2)*exp(4)+x^4+10*x^3+25*x^2)*exp(x)+4*x^2*exp(4)-4*x^3
-20*x^2),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x*exp(4)-x^2-5*x+3*exp(exp(x)*exp(-2)))+2*ln(x*exp(4)*exp(x)-exp(x)*x^2-5*exp(x)*x+3*exp(exp(x)*exp(-2))
*exp(x)+4*x)

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maxima [B]  time = 0.51, size = 61, normalized size = 1.85 \begin {gather*} 2 \, x - 2 \, \log \left (-\frac {1}{3} \, x^{2} + \frac {1}{3} \, x {\left (e^{4} - 5\right )} + e^{\left (e^{\left (x - 2\right )}\right )}\right ) + 2 \, \log \left (-\frac {1}{3} \, {\left ({\left (x^{2} - x {\left (e^{4} - 5\right )}\right )} e^{x} - 4 \, x - 3 \, e^{\left (x + e^{\left (x - 2\right )}\right )}\right )} e^{\left (-x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(x)*exp(exp(x-2))^2+((12*x*exp(4)-12*x^2-60*x)*exp(x)-24*x*exp(x-2)+24)*exp(exp(x-2))+(2*x^2*
exp(4)^2+(-4*x^3-20*x^2)*exp(4)+2*x^4+20*x^3+50*x^2)*exp(x)+8*x^2)/(9*exp(x)*exp(exp(x-2))^2+((6*x*exp(4)-6*x^
2-30*x)*exp(x)+12*x)*exp(exp(x-2))+(x^2*exp(4)^2+(-2*x^3-10*x^2)*exp(4)+x^4+10*x^3+25*x^2)*exp(x)+4*x^2*exp(4)
-4*x^3-20*x^2),x, algorithm="maxima")

[Out]

2*x - 2*log(-1/3*x^2 + 1/3*x*(e^4 - 5) + e^(e^(x - 2))) + 2*log(-1/3*((x^2 - x*(e^4 - 5))*e^x - 4*x - 3*e^(x +
e^(x - 2)))*e^(-x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {18\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^{x-2}}-{\mathrm {e}}^{{\mathrm {e}}^{x-2}}\,\left (24\,x\,{\mathrm {e}}^{x-2}+{\mathrm {e}}^x\,\left (60\,x-12\,x\,{\mathrm {e}}^4+12\,x^2\right )-24\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^8-{\mathrm {e}}^4\,\left (4\,x^3+20\,x^2\right )+50\,x^2+20\,x^3+2\,x^4\right )+8\,x^2}{9\,{\mathrm {e}}^{x+2\,{\mathrm {e}}^{x-2}}+{\mathrm {e}}^x\,\left (x^2\,{\mathrm {e}}^8-{\mathrm {e}}^4\,\left (2\,x^3+10\,x^2\right )+25\,x^2+10\,x^3+x^4\right )+{\mathrm {e}}^{{\mathrm {e}}^{x-2}}\,\left (12\,x-{\mathrm {e}}^x\,\left (30\,x-6\,x\,{\mathrm {e}}^4+6\,x^2\right )\right )+4\,x^2\,{\mathrm {e}}^4-20\,x^2-4\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*exp(2*exp(x - 2))*exp(x) - exp(exp(x - 2))*(24*x*exp(x - 2) + exp(x)*(60*x - 12*x*exp(4) + 12*x^2) - 2
4) + exp(x)*(2*x^2*exp(8) - exp(4)*(20*x^2 + 4*x^3) + 50*x^2 + 20*x^3 + 2*x^4) + 8*x^2)/(exp(x)*(x^2*exp(8) -
exp(4)*(10*x^2 + 2*x^3) + 25*x^2 + 10*x^3 + x^4) + exp(exp(x - 2))*(12*x - exp(x)*(30*x - 6*x*exp(4) + 6*x^2))
+ 9*exp(2*exp(x - 2))*exp(x) + 4*x^2*exp(4) - 20*x^2 - 4*x^3),x)

[Out]

int((18*exp(x + 2*exp(x - 2)) - exp(exp(x - 2))*(24*x*exp(x - 2) + exp(x)*(60*x - 12*x*exp(4) + 12*x^2) - 24)
+ exp(x)*(2*x^2*exp(8) - exp(4)*(20*x^2 + 4*x^3) + 50*x^2 + 20*x^3 + 2*x^4) + 8*x^2)/(9*exp(x + 2*exp(x - 2))
+ exp(x)*(x^2*exp(8) - exp(4)*(10*x^2 + 2*x^3) + 25*x^2 + 10*x^3 + x^4) + exp(exp(x - 2))*(12*x - exp(x)*(30*x
- 6*x*exp(4) + 6*x^2)) + 4*x^2*exp(4) - 20*x^2 - 4*x^3), x)

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sympy [B]  time = 1.89, size = 83, normalized size = 2.52 \begin {gather*} 2 x - 2 \log {\left (\frac {\left (- x^{2} e^{x} - 5 x e^{x} + x e^{4} e^{x}\right ) e^{- x}}{3} + e^{\frac {e^{x}}{e^{2}}} \right )} + 2 \log {\left (\frac {\left (- x^{2} e^{x} - 5 x e^{x} + x e^{4} e^{x} + 4 x\right ) e^{- x}}{3} + e^{\frac {e^{x}}{e^{2}}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*exp(x)*exp(exp(x-2))**2+((12*x*exp(4)-12*x**2-60*x)*exp(x)-24*x*exp(x-2)+24)*exp(exp(x-2))+(2*x*
*2*exp(4)**2+(-4*x**3-20*x**2)*exp(4)+2*x**4+20*x**3+50*x**2)*exp(x)+8*x**2)/(9*exp(x)*exp(exp(x-2))**2+((6*x*
exp(4)-6*x**2-30*x)*exp(x)+12*x)*exp(exp(x-2))+(x**2*exp(4)**2+(-2*x**3-10*x**2)*exp(4)+x**4+10*x**3+25*x**2)*
exp(x)+4*x**2*exp(4)-4*x**3-20*x**2),x)

[Out]

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

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