3.95.61 \(\int \frac {-18 x+72 x^2+e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)} (-72 x+e^{x/2} x+36 x^2+e^{x/4} (-12 x-3 x^2))+(36 e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)} x-36 x^2) \log (-e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)}+x)+(-36 e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)}+36 x+(18 e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)}-18 x) \log (-e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)}+x)) \log (2-\log (-e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)}+x))}{-36 e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)}+36 x+(18 e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)}-18 x) \log (-e^{\frac {1}{9} (e^{x/2}-6 e^{x/4} x+9 x^2)}+x)} \, dx\)

Optimal. Leaf size=31 \[ x \left (x+\log \left (2-\log \left (-e^{\left (-\frac {e^{x/4}}{3}+x\right )^2}+x\right )\right )\right ) \]

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Rubi [F]  time = 30.11, 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 x+72 x^2+e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} \left (-72 x+e^{x/2} x+36 x^2+e^{x/4} \left (-12 x-3 x^2\right )\right )+\left (36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )} x-36 x^2\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )+\left (-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )\right )}{-36 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+36 x+\left (18 e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}-18 x\right ) \log \left (-e^{\frac {1}{9} \left (e^{x/2}-6 e^{x/4} x+9 x^2\right )}+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-18*x + 72*x^2 + E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9)*(-72*x + E^(x/2)*x + 36*x^2 + E^(x/4)*(-12*x - 3*x
^2)) + (36*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9)*x - 36*x^2)*Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x] +
(-36*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + 36*x + (18*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) - 18*x)*Log[-E^
((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x])*Log[2 - Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x]])/(-36*E^((E
^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + 36*x + (18*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) - 18*x)*Log[-E^((E^(x/2) -
6*E^(x/4)*x + 9*x^2)/9) + x]),x]

[Out]

x^2 - Defer[Int][(E^(((3 + 4*E^(x/4))*x)/6)*x^2)/((E^(E^(x/2)/9 + x^2) - E^((2*E^(x/4)*x)/3)*x)*(2 - Log[-E^((
E^(x/4) - 3*x)^2/9) + x])), x]/18 + (2*Defer[Int][(E^(((3 + 8*E^(x/4))*x)/12)*x^2)/((E^(E^(x/2)/9 + x^2) - E^(
(2*E^(x/4)*x)/3)*x)*(2 - Log[-E^((E^(x/4) - 3*x)^2/9) + x])), x])/3 + Defer[Int][(E^(((3 + 8*E^(x/4))*x)/12)*x
^3)/((E^(E^(x/2)/9 + x^2) - E^((2*E^(x/4)*x)/3)*x)*(2 - Log[-E^((E^(x/4) - 3*x)^2/9) + x])), x]/6 - (2*Defer[I
nt][(E^(x/4)*x)/(-2 + Log[-E^((E^(x/4) - 3*x)^2/9) + x]), x])/3 + Defer[Int][(E^(x/2)*x)/(-2 + Log[-E^((E^(x/4
) - 3*x)^2/9) + x]), x]/18 + 2*Defer[Int][x^2/(-2 + Log[-E^((E^(x/4) - 3*x)^2/9) + x]), x] - Defer[Int][(E^(x/
4)*x^2)/(-2 + Log[-E^((E^(x/4) - 3*x)^2/9) + x]), x]/6 + Defer[Int][(E^((2*E^(x/4)*x)/3)*x)/((-E^(E^(x/2)/9 +
x^2) + E^((2*E^(x/4)*x)/3)*x)*(-2 + Log[-E^((E^(x/4) - 3*x)^2/9) + x])), x] - 2*Defer[Int][(E^((2*E^(x/4)*x)/3
)*x^3)/((-E^(E^(x/2)/9 + x^2) + E^((2*E^(x/4)*x)/3)*x)*(-2 + Log[-E^((E^(x/4) - 3*x)^2/9) + x])), x] + 4*Defer
[Subst][Defer[Int][Log[2 - Log[-E^((E^x - 12*x)^2/9) + 4*x]], x], x, x/4]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\frac {x \left (e^{\frac {e^{x/2}}{9}+\frac {x}{2}+x^2}+36 e^{\frac {e^{x/2}}{9}+x^2} (-2+x)-3 e^{\frac {e^{x/2}}{9}+\frac {x}{4}+x^2} (4+x)+18 e^{\frac {2}{3} e^{x/4} x} (-1+4 x)\right )}{e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x}+36 \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )-18 \log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right ) \left (2 x+\log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )\right )}{18 \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx\\ &=\frac {1}{18} \int \frac {-\frac {x \left (e^{\frac {e^{x/2}}{9}+\frac {x}{2}+x^2}+36 e^{\frac {e^{x/2}}{9}+x^2} (-2+x)-3 e^{\frac {e^{x/2}}{9}+\frac {x}{4}+x^2} (4+x)+18 e^{\frac {2}{3} e^{x/4} x} (-1+4 x)\right )}{e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x}+36 \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )-18 \log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right ) \left (2 x+\log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )\right )}{2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx\\ &=\frac {1}{18} \int \left (\frac {e^{\frac {2}{3} e^{x/4} x} x \left (18+12 e^{x/4} x-e^{x/2} x-36 x^2+3 e^{x/4} x^2\right )}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}+\frac {-72 x-12 e^{x/4} x+e^{x/2} x+36 x^2-3 e^{x/4} x^2+36 x \log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )-36 \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )+18 \log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )}\right ) \, dx\\ &=\frac {1}{18} \int \frac {e^{\frac {2}{3} e^{x/4} x} x \left (18+12 e^{x/4} x-e^{x/2} x-36 x^2+3 e^{x/4} x^2\right )}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+\frac {1}{18} \int \frac {-72 x-12 e^{x/4} x+e^{x/2} x+36 x^2-3 e^{x/4} x^2+36 x \log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )-36 \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )+18 \log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx\\ &=\frac {1}{18} \int \left (\frac {18 e^{\frac {2}{3} e^{x/4} x} x}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}+\frac {12 e^{\frac {x}{4}+\frac {2}{3} e^{x/4} x} x^2}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}-\frac {e^{\frac {x}{2}+\frac {2}{3} e^{x/4} x} x^2}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}-\frac {36 e^{\frac {2}{3} e^{x/4} x} x^3}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}+\frac {3 e^{\frac {x}{4}+\frac {2}{3} e^{x/4} x} x^3}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}\right ) \, dx+\frac {1}{18} \int \left (\frac {e^{x/2} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )}-\frac {3 e^{x/4} x (4+x)}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )}+\frac {18 \left (-4 x+2 x^2+2 x \log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )-2 \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )\right )}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )}\right ) \, dx\\ &=\frac {1}{18} \int \frac {e^{x/2} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx-\frac {1}{18} \int \frac {e^{\frac {x}{2}+\frac {2}{3} e^{x/4} x} x^2}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx-\frac {1}{6} \int \frac {e^{x/4} x (4+x)}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+\frac {1}{6} \int \frac {e^{\frac {x}{4}+\frac {2}{3} e^{x/4} x} x^3}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+\frac {2}{3} \int \frac {e^{\frac {x}{4}+\frac {2}{3} e^{x/4} x} x^2}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx-2 \int \frac {e^{\frac {2}{3} e^{x/4} x} x^3}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+\int \frac {e^{\frac {2}{3} e^{x/4} x} x}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+\int \frac {-4 x+2 x^2+2 x \log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )-2 \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right ) \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx\\ &=-\left (\frac {1}{18} \int \frac {e^{\frac {1}{6} \left (3+4 e^{x/4}\right ) x} x^2}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx\right )+\frac {1}{18} \int \frac {e^{x/2} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx-\frac {1}{6} \int \left (\frac {4 e^{x/4} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )}+\frac {e^{x/4} x^2}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )}\right ) \, dx+\frac {1}{6} \int \frac {e^{\frac {1}{12} \left (3+8 e^{x/4}\right ) x} x^3}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+\frac {2}{3} \int \frac {e^{\frac {1}{12} \left (3+8 e^{x/4}\right ) x} x^2}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx-2 \int \frac {e^{\frac {2}{3} e^{x/4} x} x^3}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+\int \frac {e^{\frac {2}{3} e^{x/4} x} x}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+\int \left (\frac {2 x \left (-2+x+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )}+\log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )\right ) \, dx\\ &=-\left (\frac {1}{18} \int \frac {e^{\frac {1}{6} \left (3+4 e^{x/4}\right ) x} x^2}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx\right )+\frac {1}{18} \int \frac {e^{x/2} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+\frac {1}{6} \int \frac {e^{\frac {1}{12} \left (3+8 e^{x/4}\right ) x} x^3}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx-\frac {1}{6} \int \frac {e^{x/4} x^2}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+\frac {2}{3} \int \frac {e^{\frac {1}{12} \left (3+8 e^{x/4}\right ) x} x^2}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{x/4} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx-2 \int \frac {e^{\frac {2}{3} e^{x/4} x} x^3}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+2 \int \frac {x \left (-2+x+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+\int \frac {e^{\frac {2}{3} e^{x/4} x} x}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+\int \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right ) \, dx\\ &=-\left (\frac {1}{18} \int \frac {e^{\frac {1}{6} \left (3+4 e^{x/4}\right ) x} x^2}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx\right )+\frac {1}{18} \int \frac {e^{x/2} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+\frac {1}{6} \int \frac {e^{\frac {1}{12} \left (3+8 e^{x/4}\right ) x} x^3}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx-\frac {1}{6} \int \frac {e^{x/4} x^2}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+\frac {2}{3} \int \frac {e^{\frac {1}{12} \left (3+8 e^{x/4}\right ) x} x^2}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{x/4} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+2 \int \left (x+\frac {x^2}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )}\right ) \, dx-2 \int \frac {e^{\frac {2}{3} e^{x/4} x} x^3}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+4 \operatorname {Subst}\left (\int \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^x-12 x\right )^2}+4 x\right )\right ) \, dx,x,\frac {x}{4}\right )+\int \frac {e^{\frac {2}{3} e^{x/4} x} x}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx\\ &=x^2-\frac {1}{18} \int \frac {e^{\frac {1}{6} \left (3+4 e^{x/4}\right ) x} x^2}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+\frac {1}{18} \int \frac {e^{x/2} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+\frac {1}{6} \int \frac {e^{\frac {1}{12} \left (3+8 e^{x/4}\right ) x} x^3}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx-\frac {1}{6} \int \frac {e^{x/4} x^2}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+\frac {2}{3} \int \frac {e^{\frac {1}{12} \left (3+8 e^{x/4}\right ) x} x^2}{\left (e^{\frac {e^{x/2}}{9}+x^2}-e^{\frac {2}{3} e^{x/4} x} x\right ) \left (2-\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx-\frac {2}{3} \int \frac {e^{x/4} x}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx+2 \int \frac {x^2}{-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )} \, dx-2 \int \frac {e^{\frac {2}{3} e^{x/4} x} x^3}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx+4 \operatorname {Subst}\left (\int \log \left (2-\log \left (-e^{\frac {1}{9} \left (e^x-12 x\right )^2}+4 x\right )\right ) \, dx,x,\frac {x}{4}\right )+\int \frac {e^{\frac {2}{3} e^{x/4} x} x}{\left (-e^{\frac {e^{x/2}}{9}+x^2}+e^{\frac {2}{3} e^{x/4} x} x\right ) \left (-2+\log \left (-e^{\frac {1}{9} \left (e^{x/4}-3 x\right )^2}+x\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.97, size = 52, normalized size = 1.68 \begin {gather*} \frac {1}{18} \left (18 x^2+18 x \log \left (2-\log \left (-e^{\frac {e^{x/2}}{9}-\frac {2}{3} e^{x/4} x+x^2}+x\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18*x + 72*x^2 + E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9)*(-72*x + E^(x/2)*x + 36*x^2 + E^(x/4)*(-12*x
- 3*x^2)) + (36*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9)*x - 36*x^2)*Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9)
+ x] + (-36*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + 36*x + (18*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) - 18*x)*L
og[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x])*Log[2 - Log[-E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x]])/(-36
*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) + 36*x + (18*E^((E^(x/2) - 6*E^(x/4)*x + 9*x^2)/9) - 18*x)*Log[-E^((E^(
x/2) - 6*E^(x/4)*x + 9*x^2)/9) + x]),x]

[Out]

(18*x^2 + 18*x*Log[2 - Log[-E^(E^(x/2)/9 - (2*E^(x/4)*x)/3 + x^2) + x]])/18

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fricas [A]  time = 1.08, size = 34, normalized size = 1.10 \begin {gather*} x^{2} + x \log \left (-\log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^
2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36*x)*log(-log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)
+x)+2)+(36*x*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-36*x^2)*log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2
)+x)+(x*exp(1/4*x)^2+(-3*x^2-12*x)*exp(1/4*x)+36*x^2-72*x)*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+72*x^2-1
8*x)/((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-
36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36*x),x, algorithm="fricas")

[Out]

x^2 + x*log(-log(x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x))) + 2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {72 \, x^{2} + {\left (36 \, x^{2} + x e^{\left (\frac {1}{2} \, x\right )} - 3 \, {\left (x^{2} + 4 \, x\right )} e^{\left (\frac {1}{4} \, x\right )} - 72 \, x\right )} e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )} - 36 \, {\left (x^{2} - x e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )} \log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) - 18 \, {\left ({\left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )} \log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) - 2 \, x + 2 \, e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )} \log \left (-\log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) + 2\right ) - 18 \, x}{18 \, {\left ({\left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )} \log \left (x - e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) - 2 \, x + 2 \, e^{\left (x^{2} - \frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^
2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36*x)*log(-log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)
+x)+2)+(36*x*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-36*x^2)*log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2
)+x)+(x*exp(1/4*x)^2+(-3*x^2-12*x)*exp(1/4*x)+36*x^2-72*x)*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+72*x^2-1
8*x)/((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-
36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36*x),x, algorithm="giac")

[Out]

integrate(-1/18*(72*x^2 + (36*x^2 + x*e^(1/2*x) - 3*(x^2 + 4*x)*e^(1/4*x) - 72*x)*e^(x^2 - 2/3*x*e^(1/4*x) + 1
/9*e^(1/2*x)) - 36*(x^2 - x*e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x)))*log(x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*
e^(1/2*x))) - 18*((x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x)))*log(x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/
2*x))) - 2*x + 2*e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x)))*log(-log(x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2
*x))) + 2) - 18*x)/((x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x)))*log(x - e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(
1/2*x))) - 2*x + 2*e^(x^2 - 2/3*x*e^(1/4*x) + 1/9*e^(1/2*x))), x)

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maple [A]  time = 0.27, size = 35, normalized size = 1.13

method	result	size
risch	\(x^{2}+\ln \left (-\ln \left (-{\mathrm e}^{\frac {{\mathrm e}^{\frac {x}{2}}}{9}-\frac {2 x \,{\mathrm e}^{\frac {x}{4}}}{3}+x^{2}}+x \right )+2\right ) x\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*ln(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-3
6*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36*x)*ln(-ln(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)+2)+(3
6*x*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-36*x^2)*ln(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)+(x*ex
p(1/4*x)^2+(-3*x^2-12*x)*exp(1/4*x)+36*x^2-72*x)*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+72*x^2-18*x)/((18*
exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*ln(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-36*exp(1/9*
exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36*x),x,method=_RETURNVERBOSE)

[Out]

x^2+ln(-ln(-exp(1/9*exp(1/2*x)-2/3*x*exp(1/4*x)+x^2)+x)+2)*x

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maxima [A]  time = 0.63, size = 48, normalized size = 1.55 \begin {gather*} x^{2} - x \log \relax (3) + x \log \left (2 \, x e^{\left (\frac {1}{4} \, x\right )} - 3 \, \log \left (x e^{\left (\frac {2}{3} \, x e^{\left (\frac {1}{4} \, x\right )}\right )} - e^{\left (x^{2} + \frac {1}{9} \, e^{\left (\frac {1}{2} \, x\right )}\right )}\right ) + 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^
2)+x)-36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36*x)*log(-log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)
+x)+2)+(36*x*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-36*x^2)*log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2
)+x)+(x*exp(1/4*x)^2+(-3*x^2-12*x)*exp(1/4*x)+36*x^2-72*x)*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+72*x^2-1
8*x)/((18*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)-18*x)*log(-exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+x)-
36*exp(1/9*exp(1/4*x)^2-2/3*x*exp(1/4*x)+x^2)+36*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 8.22, size = 33, normalized size = 1.06 \begin {gather*} x\,\left (x+\ln \left (2-\ln \left (x-{\mathrm {e}}^{\frac {{\mathrm {e}}^{x/2}}{9}}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {2\,x\,{\mathrm {e}}^{x/4}}{3}}\right )\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x - log(x - exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2))*(36*x*exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2) -
36*x^2) + exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2)*(72*x + exp(x/4)*(12*x + 3*x^2) - x*exp(x/2) - 36*x^2) + l
og(2 - log(x - exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2)))*(36*exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2) - 36*x
+ log(x - exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2))*(18*x - 18*exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2))) - 72
*x^2)/(36*exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2) - 36*x + log(x - exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2))*
(18*x - 18*exp(exp(x/2)/9 - (2*x*exp(x/4))/3 + x^2))),x)

[Out]

x*(x + log(2 - log(x - exp(exp(x/2)/9)*exp(x^2)*exp(-(2*x*exp(x/4))/3))))

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sympy [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(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)-18*x)*ln(-exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+
x**2)+x)-36*exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)+36*x)*ln(-ln(-exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+
x**2)+x)+2)+(36*x*exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)-36*x**2)*ln(-exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/
4*x)+x**2)+x)+(x*exp(1/4*x)**2+(-3*x**2-12*x)*exp(1/4*x)+36*x**2-72*x)*exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+
x**2)+72*x**2-18*x)/((18*exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)-18*x)*ln(-exp(1/9*exp(1/4*x)**2-2/3*x*ex
p(1/4*x)+x**2)+x)-36*exp(1/9*exp(1/4*x)**2-2/3*x*exp(1/4*x)+x**2)+36*x),x)

[Out]

Timed out

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