3.7.40 \(\int \frac {-3 x-2 e x^3+2 x^4+e^{2 x^2} (-27-18 e x^2+18 x^3)+(-8 x^4+16 x^6+e^{2 x^2} (36 e x^2-36 x^3)+e (8 x^3-16 x^5)) \log (\frac {1}{9} e^{-2 x^2} (9 e^{2 x^2} x+x^2))+(2 e x^3-4 x^4+e^{2 x^2} (18 e x^2-36 x^3)) \log ^2(\frac {1}{9} e^{-2 x^2} (9 e^{2 x^2} x+x^2))+(36 e^{2 x^2} x^3+8 x^4-16 x^6) \log ^3(\frac {1}{9} e^{-2 x^2} (9 e^{2 x^2} x+x^2))+(18 e^{2 x^2} x^3+2 x^4) \log ^4(\frac {1}{9} e^{-2 x^2} (9 e^{2 x^2} x+x^2))}{9 e^{2 x^2} x^2+x^3} \, dx\)

Optimal. Leaf size=35 \[ \frac {3}{x}+\left (-e+x-x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right )^2 \]

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Rubi [F]  time = 10.06, 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 {-3 x-2 e x^3+2 x^4+e^{2 x^2} \left (-27-18 e x^2+18 x^3\right )+\left (-8 x^4+16 x^6+e^{2 x^2} \left (36 e x^2-36 x^3\right )+e \left (8 x^3-16 x^5\right )\right ) \log \left (\frac {1}{9} e^{-2 x^2} \left (9 e^{2 x^2} x+x^2\right )\right )+\left (2 e x^3-4 x^4+e^{2 x^2} \left (18 e x^2-36 x^3\right )\right ) \log ^2\left (\frac {1}{9} e^{-2 x^2} \left (9 e^{2 x^2} x+x^2\right )\right )+\left (36 e^{2 x^2} x^3+8 x^4-16 x^6\right ) \log ^3\left (\frac {1}{9} e^{-2 x^2} \left (9 e^{2 x^2} x+x^2\right )\right )+\left (18 e^{2 x^2} x^3+2 x^4\right ) \log ^4\left (\frac {1}{9} e^{-2 x^2} \left (9 e^{2 x^2} x+x^2\right )\right )}{9 e^{2 x^2} x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x - 2*E*x^3 + 2*x^4 + E^(2*x^2)*(-27 - 18*E*x^2 + 18*x^3) + (-8*x^4 + 16*x^6 + E^(2*x^2)*(36*E*x^2 - 3
6*x^3) + E*(8*x^3 - 16*x^5))*Log[(9*E^(2*x^2)*x + x^2)/(9*E^(2*x^2))] + (2*E*x^3 - 4*x^4 + E^(2*x^2)*(18*E*x^2
- 36*x^3))*Log[(9*E^(2*x^2)*x + x^2)/(9*E^(2*x^2))]^2 + (36*E^(2*x^2)*x^3 + 8*x^4 - 16*x^6)*Log[(9*E^(2*x^2)*
x + x^2)/(9*E^(2*x^2))]^3 + (18*E^(2*x^2)*x^3 + 2*x^4)*Log[(9*E^(2*x^2)*x + x^2)/(9*E^(2*x^2))]^4)/(9*E^(2*x^2
)*x^2 + x^3),x]

[Out]

3/x - 6*E*x + 2*x^2 + 2*E^2*Log[x] - 2*(E - x)^2*Log[x + x^2/(9*E^(2*x^2))] + 2*E^2*Defer[Int][(9*E^(2*x^2) +
x)^(-1), x] - 4*E*Defer[Int][x/(9*E^(2*x^2) + x), x] + 4*E*Log[x + x^2/(9*E^(2*x^2))]*Defer[Int][x/(9*E^(2*x^2
) + x), x] + 2*(1 - 4*E^2)*Defer[Int][x^2/(9*E^(2*x^2) + x), x] - 4*Log[x + x^2/(9*E^(2*x^2))]*Defer[Int][x^2/
(9*E^(2*x^2) + x), x] + 16*E*Defer[Int][x^3/(9*E^(2*x^2) + x), x] - 16*E*Log[x + x^2/(9*E^(2*x^2))]*Defer[Int]
[x^3/(9*E^(2*x^2) + x), x] - 8*Defer[Int][x^4/(9*E^(2*x^2) + x), x] + 16*Log[x + x^2/(9*E^(2*x^2))]*Defer[Int]
[x^4/(9*E^(2*x^2) + x), x] + 2*E*Defer[Int][Log[x + x^2/(9*E^(2*x^2))]^2, x] - 4*Defer[Int][x*Log[x + x^2/(9*E
^(2*x^2))]^2, x] + 4*Defer[Int][x*Log[x + x^2/(9*E^(2*x^2))]^3, x] + 4*Defer[Int][(x^2*Log[x + x^2/(9*E^(2*x^2
))]^3)/(9*E^(2*x^2) + x), x] - 16*Defer[Int][(x^4*Log[x + x^2/(9*E^(2*x^2))]^3)/(9*E^(2*x^2) + x), x] + 2*Defe
r[Int][x*Log[x + x^2/(9*E^(2*x^2))]^4, x] - 4*E*Defer[Int][Defer[Int][x/(9*E^(2*x^2) + x), x]/x, x] - 4*E*Defe
r[Int][Defer[Int][x/(9*E^(2*x^2) + x), x]/(9*E^(2*x^2) + x), x] + 16*E*Defer[Int][(x^2*Defer[Int][x/(9*E^(2*x^
2) + x), x])/(9*E^(2*x^2) + x), x] + 4*Defer[Int][Defer[Int][x^2/(9*E^(2*x^2) + x), x]/x, x] + 4*Defer[Int][De
fer[Int][x^2/(9*E^(2*x^2) + x), x]/(9*E^(2*x^2) + x), x] - 16*Defer[Int][(x^2*Defer[Int][x^2/(9*E^(2*x^2) + x)
, x])/(9*E^(2*x^2) + x), x] + 16*E*Defer[Int][Defer[Int][x^3/(9*E^(2*x^2) + x), x]/x, x] + 16*E*Defer[Int][Def
er[Int][x^3/(9*E^(2*x^2) + x), x]/(9*E^(2*x^2) + x), x] - 64*E*Defer[Int][(x^2*Defer[Int][x^3/(9*E^(2*x^2) + x
), x])/(9*E^(2*x^2) + x), x] - 16*Defer[Int][Defer[Int][x^4/(9*E^(2*x^2) + x), x]/x, x] - 16*Defer[Int][Defer[
Int][x^4/(9*E^(2*x^2) + x), x]/(9*E^(2*x^2) + x), x] + 64*Defer[Int][(x^2*Defer[Int][x^4/(9*E^(2*x^2) + x), x]
)/(9*E^(2*x^2) + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4 x \left (-1+4 x^2\right ) \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \left (e-x+x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right )}{9 e^{2 x^2}+x}+\frac {-3-2 e x^2+2 x^3+4 e x^2 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )-4 x^3 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 e x^2 \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )-4 x^3 \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+4 x^3 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 x^3 \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{x^2}\right ) \, dx\\ &=-\left (4 \int \frac {x \left (-1+4 x^2\right ) \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \left (e-x+x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right )}{9 e^{2 x^2}+x} \, dx\right )+\int \frac {-3-2 e x^2+2 x^3+4 e x^2 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )-4 x^3 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 e x^2 \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )-4 x^3 \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+4 x^3 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 x^3 \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{x^2} \, dx\\ &=-\left (4 \int \left (-\frac {x \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \left (e-x+x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right )}{9 e^{2 x^2}+x}+\frac {4 x^3 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \left (e-x+x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right )}{9 e^{2 x^2}+x}\right ) \, dx\right )+\int \left (\frac {-3-2 e x^2+2 x^3}{x^2}+4 (e-x) \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 (e-2 x) \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+4 x \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 x \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \, dx\\ &=2 \int (e-2 x) \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+2 \int x \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int (e-x) \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int x \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int \frac {x \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \left (e-x+x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right )}{9 e^{2 x^2}+x} \, dx-16 \int \frac {x^3 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \left (e-x+x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right )}{9 e^{2 x^2}+x} \, dx+\int \frac {-3-2 e x^2+2 x^3}{x^2} \, dx\\ &=-2 (e-x)^2 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 \int \frac {(e-x)^2 \left (9 e^{2 x^2}+2 x-4 x^3\right )}{x \left (9 e^{2 x^2}+x\right )} \, dx+2 \int x \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+2 \int \left (e \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )-2 x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \, dx+4 \int x \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int \left (\frac {e x \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x}-\frac {x^2 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x}+\frac {x^2 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x}\right ) \, dx-16 \int \left (\frac {e x^3 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x}-\frac {x^4 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x}+\frac {x^4 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x}\right ) \, dx+\int \left (-2 e-\frac {3}{x^2}+2 x\right ) \, dx\\ &=\frac {3}{x}-2 e x+x^2-2 (e-x)^2 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 \int \left (\frac {(e-x)^2}{x}-\frac {(e-x)^2 \left (-1+4 x^2\right )}{9 e^{2 x^2}+x}\right ) \, dx+2 \int x \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx-4 \int \frac {x^2 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx-4 \int x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int x \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int \frac {x^2 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx+16 \int \frac {x^4 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx-16 \int \frac {x^4 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx+(2 e) \int \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+(4 e) \int \frac {x \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx-(16 e) \int \frac {x^3 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx\\ &=\frac {3}{x}-2 e x+x^2-2 (e-x)^2 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 \int \frac {(e-x)^2}{x} \, dx-2 \int \frac {(e-x)^2 \left (-1+4 x^2\right )}{9 e^{2 x^2}+x} \, dx+2 \int x \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx-4 \int x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int x \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int \frac {x^2 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx+4 \int \frac {\left (9 e^{2 x^2}+2 x-4 x^3\right ) \int \frac {x^2}{9 e^{2 x^2}+x} \, dx}{x \left (9 e^{2 x^2}+x\right )} \, dx-16 \int \frac {x^4 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx-16 \int \frac {\left (9 e^{2 x^2}+2 x-4 x^3\right ) \int \frac {x^4}{9 e^{2 x^2}+x} \, dx}{x \left (9 e^{2 x^2}+x\right )} \, dx+(2 e) \int \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx-(4 e) \int \frac {\left (9 e^{2 x^2}+2 x-4 x^3\right ) \int \frac {x}{9 e^{2 x^2}+x} \, dx}{x \left (9 e^{2 x^2}+x\right )} \, dx+(16 e) \int \frac {\left (9 e^{2 x^2}+2 x-4 x^3\right ) \int \frac {x^3}{9 e^{2 x^2}+x} \, dx}{x \left (9 e^{2 x^2}+x\right )} \, dx-\left (4 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x^2}{9 e^{2 x^2}+x} \, dx+\left (16 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x^4}{9 e^{2 x^2}+x} \, dx+\left (4 e \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x}{9 e^{2 x^2}+x} \, dx-\left (16 e \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x^3}{9 e^{2 x^2}+x} \, dx\\ &=\frac {3}{x}-2 e x+x^2-2 (e-x)^2 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 \int \left (-2 e+\frac {e^2}{x}+x\right ) \, dx-2 \int \left (-\frac {e^2}{9 e^{2 x^2}+x}+\frac {2 e x}{9 e^{2 x^2}+x}+\frac {\left (-1+4 e^2\right ) x^2}{9 e^{2 x^2}+x}-\frac {8 e x^3}{9 e^{2 x^2}+x}+\frac {4 x^4}{9 e^{2 x^2}+x}\right ) \, dx+2 \int x \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx-4 \int x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int x \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int \frac {x^2 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx+4 \int \left (\frac {\int \frac {x^2}{9 e^{2 x^2}+x} \, dx}{x}-\frac {\left (-1+4 x^2\right ) \int \frac {x^2}{9 e^{2 x^2}+x} \, dx}{9 e^{2 x^2}+x}\right ) \, dx-16 \int \frac {x^4 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx-16 \int \left (\frac {\int \frac {x^4}{9 e^{2 x^2}+x} \, dx}{x}-\frac {\left (-1+4 x^2\right ) \int \frac {x^4}{9 e^{2 x^2}+x} \, dx}{9 e^{2 x^2}+x}\right ) \, dx+(2 e) \int \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx-(4 e) \int \left (\frac {\int \frac {x}{9 e^{2 x^2}+x} \, dx}{x}-\frac {\left (-1+4 x^2\right ) \int \frac {x}{9 e^{2 x^2}+x} \, dx}{9 e^{2 x^2}+x}\right ) \, dx+(16 e) \int \left (\frac {\int \frac {x^3}{9 e^{2 x^2}+x} \, dx}{x}-\frac {\left (-1+4 x^2\right ) \int \frac {x^3}{9 e^{2 x^2}+x} \, dx}{9 e^{2 x^2}+x}\right ) \, dx-\left (4 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x^2}{9 e^{2 x^2}+x} \, dx+\left (16 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x^4}{9 e^{2 x^2}+x} \, dx+\left (4 e \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x}{9 e^{2 x^2}+x} \, dx-\left (16 e \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x^3}{9 e^{2 x^2}+x} \, dx\\ &=\frac {3}{x}-6 e x+2 x^2+2 e^2 \log (x)-2 (e-x)^2 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+2 \int x \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx-4 \int x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int x \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx+4 \int \frac {x^2 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx+4 \int \frac {\int \frac {x^2}{9 e^{2 x^2}+x} \, dx}{x} \, dx-4 \int \frac {\left (-1+4 x^2\right ) \int \frac {x^2}{9 e^{2 x^2}+x} \, dx}{9 e^{2 x^2}+x} \, dx-8 \int \frac {x^4}{9 e^{2 x^2}+x} \, dx-16 \int \frac {x^4 \log ^3\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )}{9 e^{2 x^2}+x} \, dx-16 \int \frac {\int \frac {x^4}{9 e^{2 x^2}+x} \, dx}{x} \, dx+16 \int \frac {\left (-1+4 x^2\right ) \int \frac {x^4}{9 e^{2 x^2}+x} \, dx}{9 e^{2 x^2}+x} \, dx+(2 e) \int \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \, dx-(4 e) \int \frac {x}{9 e^{2 x^2}+x} \, dx-(4 e) \int \frac {\int \frac {x}{9 e^{2 x^2}+x} \, dx}{x} \, dx+(4 e) \int \frac {\left (-1+4 x^2\right ) \int \frac {x}{9 e^{2 x^2}+x} \, dx}{9 e^{2 x^2}+x} \, dx+(16 e) \int \frac {x^3}{9 e^{2 x^2}+x} \, dx+(16 e) \int \frac {\int \frac {x^3}{9 e^{2 x^2}+x} \, dx}{x} \, dx-(16 e) \int \frac {\left (-1+4 x^2\right ) \int \frac {x^3}{9 e^{2 x^2}+x} \, dx}{9 e^{2 x^2}+x} \, dx+\left (2 e^2\right ) \int \frac {1}{9 e^{2 x^2}+x} \, dx+\left (2 \left (1-4 e^2\right )\right ) \int \frac {x^2}{9 e^{2 x^2}+x} \, dx-\left (4 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x^2}{9 e^{2 x^2}+x} \, dx+\left (16 \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x^4}{9 e^{2 x^2}+x} \, dx+\left (4 e \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x}{9 e^{2 x^2}+x} \, dx-\left (16 e \log \left (x+\frac {1}{9} e^{-2 x^2} x^2\right )\right ) \int \frac {x^3}{9 e^{2 x^2}+x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 63, normalized size = 1.80 \begin {gather*} \frac {3}{x}-2 e x+x^2+2 (e-x) x \log ^2\left (x+\frac {1}{9} e^{-2 x^2} x^2\right )+x^2 \log ^4\left (x+\frac {1}{9} e^{-2 x^2} x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x - 2*E*x^3 + 2*x^4 + E^(2*x^2)*(-27 - 18*E*x^2 + 18*x^3) + (-8*x^4 + 16*x^6 + E^(2*x^2)*(36*E*x
^2 - 36*x^3) + E*(8*x^3 - 16*x^5))*Log[(9*E^(2*x^2)*x + x^2)/(9*E^(2*x^2))] + (2*E*x^3 - 4*x^4 + E^(2*x^2)*(18
*E*x^2 - 36*x^3))*Log[(9*E^(2*x^2)*x + x^2)/(9*E^(2*x^2))]^2 + (36*E^(2*x^2)*x^3 + 8*x^4 - 16*x^6)*Log[(9*E^(2
*x^2)*x + x^2)/(9*E^(2*x^2))]^3 + (18*E^(2*x^2)*x^3 + 2*x^4)*Log[(9*E^(2*x^2)*x + x^2)/(9*E^(2*x^2))]^4)/(9*E^
(2*x^2)*x^2 + x^3),x]

[Out]

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

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fricas [B]  time = 0.75, size = 81, normalized size = 2.31 \begin {gather*} \frac {x^{3} \log \left (\frac {1}{9} \, {\left (x^{2} + 9 \, x e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-2 \, x^{2}\right )}\right )^{4} + x^{3} - 2 \, x^{2} e - 2 \, {\left (x^{3} - x^{2} e\right )} \log \left (\frac {1}{9} \, {\left (x^{2} + 9 \, x e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-2 \, x^{2}\right )}\right )^{2} + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3*exp(x^2)^2+2*x^4)*log(1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^4+(36*x^3*exp(x^2)^2-16*x^6+8*x^
4)*log(1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^3+((18*x^2*exp(1)-36*x^3)*exp(x^2)^2+2*x^3*exp(1)-4*x^4)*log(1/9*(
9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^2+((36*x^2*exp(1)-36*x^3)*exp(x^2)^2+(-16*x^5+8*x^3)*exp(1)+16*x^6-8*x^4)*log(
1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)+(-18*x^2*exp(1)+18*x^3-27)*exp(x^2)^2-2*x^3*exp(1)+2*x^4-3*x)/(9*x^2*exp(
x^2)^2+x^3),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 24.30, size = 104, normalized size = 2.97 \begin {gather*} \frac {x^{3} \log \left (\frac {1}{9} \, {\left (x^{2} + 9 \, x e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-2 \, x^{2}\right )}\right )^{4} - 2 \, x^{3} \log \left (\frac {1}{9} \, {\left (x^{2} + 9 \, x e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-2 \, x^{2}\right )}\right )^{2} + 2 \, x^{2} e \log \left (\frac {1}{9} \, {\left (x^{2} + 9 \, x e^{\left (2 \, x^{2}\right )}\right )} e^{\left (-2 \, x^{2}\right )}\right )^{2} + x^{3} - 2 \, x^{2} e + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3*exp(x^2)^2+2*x^4)*log(1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^4+(36*x^3*exp(x^2)^2-16*x^6+8*x^
4)*log(1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^3+((18*x^2*exp(1)-36*x^3)*exp(x^2)^2+2*x^3*exp(1)-4*x^4)*log(1/9*(
9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^2+((36*x^2*exp(1)-36*x^3)*exp(x^2)^2+(-16*x^5+8*x^3)*exp(1)+16*x^6-8*x^4)*log(
1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)+(-18*x^2*exp(1)+18*x^3-27)*exp(x^2)^2-2*x^3*exp(1)+2*x^4-3*x)/(9*x^2*exp(
x^2)^2+x^3),x, algorithm="giac")

[Out]

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

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maple [C]  time = 51.75, size = 190125, normalized size = 5432.14

method	result	size
risch	\(\text {Expression too large to display}\)	\(190125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((18*x^3*exp(x^2)^2+2*x^4)*ln(1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^4+(36*x^3*exp(x^2)^2-16*x^6+8*x^4)*ln(1
/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^3+((18*x^2*exp(1)-36*x^3)*exp(x^2)^2+2*x^3*exp(1)-4*x^4)*ln(1/9*(9*x*exp(x
^2)^2+x^2)/exp(x^2)^2)^2+((36*x^2*exp(1)-36*x^3)*exp(x^2)^2+(-16*x^5+8*x^3)*exp(1)+16*x^6-8*x^4)*ln(1/9*(9*x*e
xp(x^2)^2+x^2)/exp(x^2)^2)+(-18*x^2*exp(1)+18*x^3-27)*exp(x^2)^2-2*x^3*exp(1)+2*x^4-3*x)/(9*x^2*exp(x^2)^2+x^3
),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [B]  time = 1.17, size = 467, normalized size = 13.34 \begin {gather*} \frac {16 \, x^{11} + 64 \, x^{9} \log \relax (3) + 8 \, {\left (12 \, \log \relax (3)^{2} - 1\right )} x^{7} + 8 \, x^{6} e + x^{3} \log \left (x + 9 \, e^{\left (2 \, x^{2}\right )}\right )^{4} + x^{3} \log \relax (x)^{4} + 16 \, {\left (4 \, \log \relax (3)^{3} - \log \relax (3)\right )} x^{5} + 16 \, x^{4} e \log \relax (3) + {\left (16 \, \log \relax (3)^{4} - 8 \, \log \relax (3)^{2} + 1\right )} x^{3} + 2 \, {\left (4 \, \log \relax (3)^{2} - 1\right )} x^{2} e - 4 \, {\left (2 \, x^{5} + 2 \, x^{3} \log \relax (3) - x^{3} \log \relax (x)\right )} \log \left (x + 9 \, e^{\left (2 \, x^{2}\right )}\right )^{3} - 8 \, {\left (x^{5} + x^{3} \log \relax (3)\right )} \log \relax (x)^{3} + 2 \, {\left (12 \, x^{7} + 24 \, x^{5} \log \relax (3) + 3 \, x^{3} \log \relax (x)^{2} + {\left (12 \, \log \relax (3)^{2} - 1\right )} x^{3} + x^{2} e - 12 \, {\left (x^{5} + x^{3} \log \relax (3)\right )} \log \relax (x)\right )} \log \left (x + 9 \, e^{\left (2 \, x^{2}\right )}\right )^{2} + 2 \, {\left (12 \, x^{7} + 24 \, x^{5} \log \relax (3) + {\left (12 \, \log \relax (3)^{2} - 1\right )} x^{3} + x^{2} e\right )} \log \relax (x)^{2} - 4 \, {\left (8 \, x^{9} + 24 \, x^{7} \log \relax (3) + 2 \, {\left (12 \, \log \relax (3)^{2} - 1\right )} x^{5} - x^{3} \log \relax (x)^{3} + 2 \, x^{4} e + 2 \, {\left (4 \, \log \relax (3)^{3} - \log \relax (3)\right )} x^{3} + 2 \, x^{2} e \log \relax (3) + 6 \, {\left (x^{5} + x^{3} \log \relax (3)\right )} \log \relax (x)^{2} - {\left (12 \, x^{7} + 24 \, x^{5} \log \relax (3) + {\left (12 \, \log \relax (3)^{2} - 1\right )} x^{3} + x^{2} e\right )} \log \relax (x)\right )} \log \left (x + 9 \, e^{\left (2 \, x^{2}\right )}\right ) - 8 \, {\left (4 \, x^{9} + 12 \, x^{7} \log \relax (3) + {\left (12 \, \log \relax (3)^{2} - 1\right )} x^{5} + x^{4} e + {\left (4 \, \log \relax (3)^{3} - \log \relax (3)\right )} x^{3} + x^{2} e \log \relax (3)\right )} \log \relax (x) + 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^3*exp(x^2)^2+2*x^4)*log(1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^4+(36*x^3*exp(x^2)^2-16*x^6+8*x^
4)*log(1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^3+((18*x^2*exp(1)-36*x^3)*exp(x^2)^2+2*x^3*exp(1)-4*x^4)*log(1/9*(
9*x*exp(x^2)^2+x^2)/exp(x^2)^2)^2+((36*x^2*exp(1)-36*x^3)*exp(x^2)^2+(-16*x^5+8*x^3)*exp(1)+16*x^6-8*x^4)*log(
1/9*(9*x*exp(x^2)^2+x^2)/exp(x^2)^2)+(-18*x^2*exp(1)+18*x^3-27)*exp(x^2)^2-2*x^3*exp(1)+2*x^4-3*x)/(9*x^2*exp(
x^2)^2+x^3),x, algorithm="maxima")

[Out]

(16*x^11 + 64*x^9*log(3) + 8*(12*log(3)^2 - 1)*x^7 + 8*x^6*e + x^3*log(x + 9*e^(2*x^2))^4 + x^3*log(x)^4 + 16*
(4*log(3)^3 - log(3))*x^5 + 16*x^4*e*log(3) + (16*log(3)^4 - 8*log(3)^2 + 1)*x^3 + 2*(4*log(3)^2 - 1)*x^2*e -
4*(2*x^5 + 2*x^3*log(3) - x^3*log(x))*log(x + 9*e^(2*x^2))^3 - 8*(x^5 + x^3*log(3))*log(x)^3 + 2*(12*x^7 + 24*
x^5*log(3) + 3*x^3*log(x)^2 + (12*log(3)^2 - 1)*x^3 + x^2*e - 12*(x^5 + x^3*log(3))*log(x))*log(x + 9*e^(2*x^2
))^2 + 2*(12*x^7 + 24*x^5*log(3) + (12*log(3)^2 - 1)*x^3 + x^2*e)*log(x)^2 - 4*(8*x^9 + 24*x^7*log(3) + 2*(12*
log(3)^2 - 1)*x^5 - x^3*log(x)^3 + 2*x^4*e + 2*(4*log(3)^3 - log(3))*x^3 + 2*x^2*e*log(3) + 6*(x^5 + x^3*log(3
))*log(x)^2 - (12*x^7 + 24*x^5*log(3) + (12*log(3)^2 - 1)*x^3 + x^2*e)*log(x))*log(x + 9*e^(2*x^2)) - 8*(4*x^9
+ 12*x^7*log(3) + (12*log(3)^2 - 1)*x^5 + x^4*e + (4*log(3)^3 - log(3))*x^3 + x^2*e*log(3))*log(x) + 3)/x

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mupad [B]  time = 0.84, size = 76, normalized size = 2.17 \begin {gather*} x^2\,{\ln \left (x+\frac {x^2\,{\mathrm {e}}^{-2\,x^2}}{9}\right )}^4-2\,x^2\,{\ln \left (x+\frac {x^2\,{\mathrm {e}}^{-2\,x^2}}{9}\right )}^2-2\,x\,\mathrm {e}+\frac {3}{x}+x^2+2\,x\,{\ln \left (x+\frac {x^2\,{\mathrm {e}}^{-2\,x^2}}{9}\right )}^2\,\mathrm {e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(exp(-2*x^2)*(x*exp(2*x^2) + x^2/9))*(exp(2*x^2)*(36*x^2*exp(1) - 36*x^3) + exp(1)*(8*x^3 - 16*x^5) -
8*x^4 + 16*x^6) - exp(2*x^2)*(18*x^2*exp(1) - 18*x^3 + 27) - 3*x + log(exp(-2*x^2)*(x*exp(2*x^2) + x^2/9))^2*(
exp(2*x^2)*(18*x^2*exp(1) - 36*x^3) + 2*x^3*exp(1) - 4*x^4) + log(exp(-2*x^2)*(x*exp(2*x^2) + x^2/9))^4*(18*x^
3*exp(2*x^2) + 2*x^4) - 2*x^3*exp(1) + log(exp(-2*x^2)*(x*exp(2*x^2) + x^2/9))^3*(36*x^3*exp(2*x^2) + 8*x^4 -
16*x^6) + 2*x^4)/(9*x^2*exp(2*x^2) + x^3),x)

[Out]

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

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sympy [B]  time = 0.65, size = 75, normalized size = 2.14 \begin {gather*} x^{2} \log {\left (\left (\frac {x^{2}}{9} + x e^{2 x^{2}}\right ) e^{- 2 x^{2}} \right )}^{4} + x^{2} - 2 e x + \left (- 2 x^{2} + 2 e x\right ) \log {\left (\left (\frac {x^{2}}{9} + x e^{2 x^{2}}\right ) e^{- 2 x^{2}} \right )}^{2} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x**3*exp(x**2)**2+2*x**4)*ln(1/9*(9*x*exp(x**2)**2+x**2)/exp(x**2)**2)**4+(36*x**3*exp(x**2)**2
-16*x**6+8*x**4)*ln(1/9*(9*x*exp(x**2)**2+x**2)/exp(x**2)**2)**3+((18*x**2*exp(1)-36*x**3)*exp(x**2)**2+2*x**3
*exp(1)-4*x**4)*ln(1/9*(9*x*exp(x**2)**2+x**2)/exp(x**2)**2)**2+((36*x**2*exp(1)-36*x**3)*exp(x**2)**2+(-16*x*
*5+8*x**3)*exp(1)+16*x**6-8*x**4)*ln(1/9*(9*x*exp(x**2)**2+x**2)/exp(x**2)**2)+(-18*x**2*exp(1)+18*x**3-27)*ex
p(x**2)**2-2*x**3*exp(1)+2*x**4-3*x)/(9*x**2*exp(x**2)**2+x**3),x)

[Out]

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

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