3.97.81 \(\int \frac {e^{2 x^2} (-96 x^2+24 x^3-24 e^x x^4)+e^{4 x^2} (-32 x^2+8 x^3-8 e^x x^4)+e^{x^2} (32 x^2-8 x^3+8 e^x x^4)+e^{3 x^2} (96 x^2-24 x^3+24 e^x x^4)+(20-30 x+5 x^2+15 e^x x^2+e^{x^2} (-16+24 x-4 x^2-12 e^x x^2)+e^{3 x^2} (-16+24 x-4 x^2-12 e^x x^2)+e^{4 x^2} (4-6 x+x^2+3 e^x x^2)+e^{2 x^2} (24-36 x+6 x^2+18 e^x x^2)) \log (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2})+(-5 e^x+4 e^{x+x^2}-6 e^{x+2 x^2}+4 e^{x+3 x^2}-e^{x+4 x^2}) \log ^2(5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2})}{(5 e^x-4 e^{x+x^2}+6 e^{x+2 x^2}-4 e^{x+3 x^2}+e^{x+4 x^2}) \log ^2(5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2})} \, dx\)

Optimal. Leaf size=37 \[ -x+\frac {x \left (e^{-x} (4-x)+x^2\right )}{\log \left (4+\left (1-e^{x^2}\right )^4\right )} \]

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Rubi [F]  time = 51.80, 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 {e^{2 x^2} \left (-96 x^2+24 x^3-24 e^x x^4\right )+e^{4 x^2} \left (-32 x^2+8 x^3-8 e^x x^4\right )+e^{x^2} \left (32 x^2-8 x^3+8 e^x x^4\right )+e^{3 x^2} \left (96 x^2-24 x^3+24 e^x x^4\right )+\left (20-30 x+5 x^2+15 e^x x^2+e^{x^2} \left (-16+24 x-4 x^2-12 e^x x^2\right )+e^{3 x^2} \left (-16+24 x-4 x^2-12 e^x x^2\right )+e^{4 x^2} \left (4-6 x+x^2+3 e^x x^2\right )+e^{2 x^2} \left (24-36 x+6 x^2+18 e^x x^2\right )\right ) \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )+\left (-5 e^x+4 e^{x+x^2}-6 e^{x+2 x^2}+4 e^{x+3 x^2}-e^{x+4 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}{\left (5 e^x-4 e^{x+x^2}+6 e^{x+2 x^2}-4 e^{x+3 x^2}+e^{x+4 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x^2)*(-96*x^2 + 24*x^3 - 24*E^x*x^4) + E^(4*x^2)*(-32*x^2 + 8*x^3 - 8*E^x*x^4) + E^x^2*(32*x^2 - 8*x
^3 + 8*E^x*x^4) + E^(3*x^2)*(96*x^2 - 24*x^3 + 24*E^x*x^4) + (20 - 30*x + 5*x^2 + 15*E^x*x^2 + E^x^2*(-16 + 24
*x - 4*x^2 - 12*E^x*x^2) + E^(3*x^2)*(-16 + 24*x - 4*x^2 - 12*E^x*x^2) + E^(4*x^2)*(4 - 6*x + x^2 + 3*E^x*x^2)
+ E^(2*x^2)*(24 - 36*x + 6*x^2 + 18*E^x*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)] + (-5*
E^x + 4*E^(x + x^2) - 6*E^(x + 2*x^2) + 4*E^(x + 3*x^2) - E^(x + 4*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(
3*x^2) + E^(4*x^2)]^2)/((5*E^x - 4*E^(x + x^2) + 6*E^(x + 2*x^2) - 4*E^(x + 3*x^2) + E^(x + 4*x^2))*Log[5 - 4*
E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2),x]

[Out]

-x - 32*Defer[Int][x^2/(E^x*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] + 16*Defer[Int][x^
2/(E^x*(1 + E^(2*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] + 80*Defer[Int][x^2/(E^
x*(5 - 4*E^x^2 + E^(2*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] - 32*Defer[Int][(E
^(-x + x^2)*x^2)/((5 - 4*E^x^2 + E^(2*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] +
8*Defer[Int][x^3/(E^x*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] - 4*Defer[Int][x^3/(E^x*
(1 + E^(2*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] - 20*Defer[Int][x^3/(E^x*(5 -
4*E^x^2 + E^(2*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] + 8*Defer[Int][(E^(-x + x
^2)*x^3)/((5 - 4*E^x^2 + E^(2*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] - 8*Defer[
Int][x^4/Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2, x] + 4*Defer[Int][x^4/((1 + E^(2*x^2))*Lo
g[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] + 20*Defer[Int][x^4/((5 - 4*E^x^2 + E^(2*x^2))*L
og[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] - 8*Defer[Int][(E^x^2*x^4)/((5 - 4*E^x^2 + E^(2
*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2), x] + 4*Defer[Int][1/(E^x*Log[5 - 4*E^x^2 +
6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]), x] - 6*Defer[Int][x/(E^x*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2)
+ E^(4*x^2)]), x] + 3*Defer[Int][x^2/Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)], x] + Defer[Int
][x^2/(E^x*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (e^{2 x^2} \left (-96 x^2+24 x^3-24 e^x x^4\right )+e^{4 x^2} \left (-32 x^2+8 x^3-8 e^x x^4\right )+e^{x^2} \left (32 x^2-8 x^3+8 e^x x^4\right )+e^{3 x^2} \left (96 x^2-24 x^3+24 e^x x^4\right )+\left (20-30 x+5 x^2+15 e^x x^2+e^{x^2} \left (-16+24 x-4 x^2-12 e^x x^2\right )+e^{3 x^2} \left (-16+24 x-4 x^2-12 e^x x^2\right )+e^{4 x^2} \left (4-6 x+x^2+3 e^x x^2\right )+e^{2 x^2} \left (24-36 x+6 x^2+18 e^x x^2\right )\right ) \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )+\left (-5 e^x+4 e^{x+x^2}-6 e^{x+2 x^2}+4 e^{x+3 x^2}-e^{x+4 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )\right )}{\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx\\ &=\int \frac {e^{-x} \left (8 e^{x^2} x^2 \left (4-x+e^x x^2\right )-24 e^{2 x^2} x^2 \left (4-x+e^x x^2\right )+24 e^{3 x^2} x^2 \left (4-x+e^x x^2\right )-8 e^{4 x^2} x^2 \left (4-x+e^x x^2\right )+\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right ) \left (4-6 x+\left (1+3 e^x\right ) x^2\right ) \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )-e^x \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )\right )}{\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx\\ &=\int \left (\frac {4 e^{-x} x^2 \left (4-x+e^x x^2\right )}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}-\frac {4 e^{-x} \left (-5+2 e^{x^2}\right ) x^2 \left (4-x+e^x x^2\right )}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}-\frac {e^{-x} \left (32 x^2-8 x^3+8 e^x x^4-4 \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )+6 x \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )-x^2 \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )-3 e^x x^2 \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )+e^x \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )\right )}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}\right ) \, dx\\ &=4 \int \frac {e^{-x} x^2 \left (4-x+e^x x^2\right )}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-4 \int \frac {e^{-x} \left (-5+2 e^{x^2}\right ) x^2 \left (4-x+e^x x^2\right )}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-\int \frac {e^{-x} \left (32 x^2-8 x^3+8 e^x x^4-4 \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )+6 x \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )-x^2 \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )-3 e^x x^2 \log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )+e^x \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )\right )}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx\\ &=4 \int \left (\frac {4 e^{-x} x^2}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}-\frac {e^{-x} x^3}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}+\frac {x^4}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}\right ) \, dx-4 \int \left (-\frac {20 e^{-x} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}+\frac {8 e^{-x+x^2} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}+\frac {5 e^{-x} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}-\frac {2 e^{-x+x^2} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}-\frac {5 x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}+\frac {2 e^{x^2} x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}\right ) \, dx-\int \left (1+\frac {8 e^{-x} x^2 \left (4-x+e^x x^2\right )}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}-\frac {e^{-x} \left (4-6 x+\left (1+3 e^x\right ) x^2\right )}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}\right ) \, dx\\ &=-x-4 \int \frac {e^{-x} x^3}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+4 \int \frac {x^4}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+8 \int \frac {e^{-x+x^2} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \frac {e^{x^2} x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \frac {e^{-x} x^2 \left (4-x+e^x x^2\right )}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+16 \int \frac {e^{-x} x^2}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-20 \int \frac {e^{-x} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+20 \int \frac {x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-32 \int \frac {e^{-x+x^2} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+80 \int \frac {e^{-x} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+\int \frac {e^{-x} \left (4-6 x+\left (1+3 e^x\right ) x^2\right )}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx\\ &=-x-4 \int \frac {e^{-x} x^3}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+4 \int \frac {x^4}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \left (-\frac {e^{-x} (-4+x) x^2}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}+\frac {x^4}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}\right ) \, dx+8 \int \frac {e^{-x+x^2} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \frac {e^{x^2} x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+16 \int \frac {e^{-x} x^2}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-20 \int \frac {e^{-x} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+20 \int \frac {x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-32 \int \frac {e^{-x+x^2} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+80 \int \frac {e^{-x} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+\int \left (\frac {3 x^2}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}+\frac {e^{-x} \left (4-6 x+x^2\right )}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}\right ) \, dx\\ &=-x+3 \int \frac {x^2}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-4 \int \frac {e^{-x} x^3}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+4 \int \frac {x^4}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+8 \int \frac {e^{-x} (-4+x) x^2}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+8 \int \frac {e^{-x+x^2} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \frac {x^4}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \frac {e^{x^2} x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+16 \int \frac {e^{-x} x^2}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-20 \int \frac {e^{-x} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+20 \int \frac {x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-32 \int \frac {e^{-x+x^2} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+80 \int \frac {e^{-x} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+\int \frac {e^{-x} \left (4-6 x+x^2\right )}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx\\ &=-x+3 \int \frac {x^2}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-4 \int \frac {e^{-x} x^3}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+4 \int \frac {x^4}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+8 \int \left (-\frac {4 e^{-x} x^2}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}+\frac {e^{-x} x^3}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}\right ) \, dx+8 \int \frac {e^{-x+x^2} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \frac {x^4}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \frac {e^{x^2} x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+16 \int \frac {e^{-x} x^2}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-20 \int \frac {e^{-x} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+20 \int \frac {x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-32 \int \frac {e^{-x+x^2} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+80 \int \frac {e^{-x} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+\int \left (\frac {4 e^{-x}}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}-\frac {6 e^{-x} x}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}+\frac {e^{-x} x^2}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}\right ) \, dx\\ &=-x+3 \int \frac {x^2}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-4 \int \frac {e^{-x} x^3}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+4 \int \frac {x^4}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+4 \int \frac {e^{-x}}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-6 \int \frac {e^{-x} x}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+8 \int \frac {e^{-x} x^3}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+8 \int \frac {e^{-x+x^2} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \frac {x^4}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-8 \int \frac {e^{x^2} x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+16 \int \frac {e^{-x} x^2}{\left (1+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-20 \int \frac {e^{-x} x^3}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+20 \int \frac {x^4}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-32 \int \frac {e^{-x} x^2}{\log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx-32 \int \frac {e^{-x+x^2} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+80 \int \frac {e^{-x} x^2}{\left (5-4 e^{x^2}+e^{2 x^2}\right ) \log ^2\left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx+\int \frac {e^{-x} x^2}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 59, normalized size = 1.59 \begin {gather*} x \left (-1+\frac {e^{-x} \left (4-x+e^x x^2\right )}{\log \left (5-4 e^{x^2}+6 e^{2 x^2}-4 e^{3 x^2}+e^{4 x^2}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x^2)*(-96*x^2 + 24*x^3 - 24*E^x*x^4) + E^(4*x^2)*(-32*x^2 + 8*x^3 - 8*E^x*x^4) + E^x^2*(32*x^2
- 8*x^3 + 8*E^x*x^4) + E^(3*x^2)*(96*x^2 - 24*x^3 + 24*E^x*x^4) + (20 - 30*x + 5*x^2 + 15*E^x*x^2 + E^x^2*(-1
6 + 24*x - 4*x^2 - 12*E^x*x^2) + E^(3*x^2)*(-16 + 24*x - 4*x^2 - 12*E^x*x^2) + E^(4*x^2)*(4 - 6*x + x^2 + 3*E^
x*x^2) + E^(2*x^2)*(24 - 36*x + 6*x^2 + 18*E^x*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]
+ (-5*E^x + 4*E^(x + x^2) - 6*E^(x + 2*x^2) + 4*E^(x + 3*x^2) - E^(x + 4*x^2))*Log[5 - 4*E^x^2 + 6*E^(2*x^2) -
4*E^(3*x^2) + E^(4*x^2)]^2)/((5*E^x - 4*E^(x + x^2) + 6*E^(x + 2*x^2) - 4*E^(x + 3*x^2) + E^(x + 4*x^2))*Log[
5 - 4*E^x^2 + 6*E^(2*x^2) - 4*E^(3*x^2) + E^(4*x^2)]^2),x]

[Out]

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

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fricas [B]  time = 0.66, size = 202, normalized size = 5.46 \begin {gather*} \frac {{\left (x^{3} e^{\left (12 \, x^{2} + 4 \, x\right )} - x e^{\left (12 \, x^{2} + 4 \, x\right )} \log \left ({\left (e^{\left (16 \, x^{2} + 4 \, x\right )} - 4 \, e^{\left (15 \, x^{2} + 4 \, x\right )} + 6 \, e^{\left (14 \, x^{2} + 4 \, x\right )} - 4 \, e^{\left (13 \, x^{2} + 4 \, x\right )} + 5 \, e^{\left (12 \, x^{2} + 4 \, x\right )}\right )} e^{\left (-12 \, x^{2} - 4 \, x\right )}\right ) - {\left (x^{2} - 4 \, x\right )} e^{\left (12 \, x^{2} + 3 \, x\right )}\right )} e^{\left (-12 \, x^{2} - 4 \, x\right )}}{\log \left ({\left (e^{\left (16 \, x^{2} + 4 \, x\right )} - 4 \, e^{\left (15 \, x^{2} + 4 \, x\right )} + 6 \, e^{\left (14 \, x^{2} + 4 \, x\right )} - 4 \, e^{\left (13 \, x^{2} + 4 \, x\right )} + 5 \, e^{\left (12 \, x^{2} + 4 \, x\right )}\right )} e^{\left (-12 \, x^{2} - 4 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)*exp(x^2)^4+4*exp(x)*exp(x^2)^3-6*exp(x)*exp(x^2)^2+4*exp(x)*exp(x^2)-5*exp(x))*log(exp(x^2
)^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)^2+((3*exp(x)*x^2+x^2-6*x+4)*exp(x^2)^4+(-12*exp(x)*x^2-4*x^2+24*x-
16)*exp(x^2)^3+(18*exp(x)*x^2+6*x^2-36*x+24)*exp(x^2)^2+(-12*exp(x)*x^2-4*x^2+24*x-16)*exp(x^2)+15*exp(x)*x^2+
5*x^2-30*x+20)*log(exp(x^2)^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)+(-8*exp(x)*x^4+8*x^3-32*x^2)*exp(x^2)^4+
(24*exp(x)*x^4-24*x^3+96*x^2)*exp(x^2)^3+(-24*exp(x)*x^4+24*x^3-96*x^2)*exp(x^2)^2+(8*exp(x)*x^4-8*x^3+32*x^2)
*exp(x^2))/(exp(x)*exp(x^2)^4-4*exp(x)*exp(x^2)^3+6*exp(x)*exp(x^2)^2-4*exp(x)*exp(x^2)+5*exp(x))/log(exp(x^2)
^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)^2,x, algorithm="fricas")

[Out]

(x^3*e^(12*x^2 + 4*x) - x*e^(12*x^2 + 4*x)*log((e^(16*x^2 + 4*x) - 4*e^(15*x^2 + 4*x) + 6*e^(14*x^2 + 4*x) - 4
*e^(13*x^2 + 4*x) + 5*e^(12*x^2 + 4*x))*e^(-12*x^2 - 4*x)) - (x^2 - 4*x)*e^(12*x^2 + 3*x))*e^(-12*x^2 - 4*x)/l
og((e^(16*x^2 + 4*x) - 4*e^(15*x^2 + 4*x) + 6*e^(14*x^2 + 4*x) - 4*e^(13*x^2 + 4*x) + 5*e^(12*x^2 + 4*x))*e^(-
12*x^2 - 4*x))

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giac [B]  time = 0.91, size = 83, normalized size = 2.24 \begin {gather*} \frac {x^{3} e^{x} - x e^{x} \log \left (e^{\left (2 \, x^{2}\right )} - 4 \, e^{\left (x^{2}\right )} + 5\right ) - x e^{x} \log \left (e^{\left (2 \, x^{2}\right )} + 1\right ) - x^{2} + 4 \, x}{e^{x} \log \left (e^{\left (2 \, x^{2}\right )} - 4 \, e^{\left (x^{2}\right )} + 5\right ) + e^{x} \log \left (e^{\left (2 \, x^{2}\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)*exp(x^2)^4+4*exp(x)*exp(x^2)^3-6*exp(x)*exp(x^2)^2+4*exp(x)*exp(x^2)-5*exp(x))*log(exp(x^2
)^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)^2+((3*exp(x)*x^2+x^2-6*x+4)*exp(x^2)^4+(-12*exp(x)*x^2-4*x^2+24*x-
16)*exp(x^2)^3+(18*exp(x)*x^2+6*x^2-36*x+24)*exp(x^2)^2+(-12*exp(x)*x^2-4*x^2+24*x-16)*exp(x^2)+15*exp(x)*x^2+
5*x^2-30*x+20)*log(exp(x^2)^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)+(-8*exp(x)*x^4+8*x^3-32*x^2)*exp(x^2)^4+
(24*exp(x)*x^4-24*x^3+96*x^2)*exp(x^2)^3+(-24*exp(x)*x^4+24*x^3-96*x^2)*exp(x^2)^2+(8*exp(x)*x^4-8*x^3+32*x^2)
*exp(x^2))/(exp(x)*exp(x^2)^4-4*exp(x)*exp(x^2)^3+6*exp(x)*exp(x^2)^2-4*exp(x)*exp(x^2)+5*exp(x))/log(exp(x^2)
^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.20, size = 55, normalized size = 1.49

method	result	size
risch	\(-x +\frac {\left ({\mathrm e}^{x} x^{2}-x +4\right ) x \,{\mathrm e}^{-x}}{\ln \left ({\mathrm e}^{4 x^{2}}-4 \,{\mathrm e}^{3 x^{2}}+6 \,{\mathrm e}^{2 x^{2}}-4 \,{\mathrm e}^{x^{2}}+5\right )}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-exp(x)*exp(x^2)^4+4*exp(x)*exp(x^2)^3-6*exp(x)*exp(x^2)^2+4*exp(x)*exp(x^2)-5*exp(x))*ln(exp(x^2)^4-4*e
xp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)^2+((3*exp(x)*x^2+x^2-6*x+4)*exp(x^2)^4+(-12*exp(x)*x^2-4*x^2+24*x-16)*exp
(x^2)^3+(18*exp(x)*x^2+6*x^2-36*x+24)*exp(x^2)^2+(-12*exp(x)*x^2-4*x^2+24*x-16)*exp(x^2)+15*exp(x)*x^2+5*x^2-3
0*x+20)*ln(exp(x^2)^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)+(-8*exp(x)*x^4+8*x^3-32*x^2)*exp(x^2)^4+(24*exp(
x)*x^4-24*x^3+96*x^2)*exp(x^2)^3+(-24*exp(x)*x^4+24*x^3-96*x^2)*exp(x^2)^2+(8*exp(x)*x^4-8*x^3+32*x^2)*exp(x^2
))/(exp(x)*exp(x^2)^4-4*exp(x)*exp(x^2)^3+6*exp(x)*exp(x^2)^2-4*exp(x)*exp(x^2)+5*exp(x))/ln(exp(x^2)^4-4*exp(
x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)^2,x,method=_RETURNVERBOSE)

[Out]

-x+(exp(x)*x^2-x+4)*x*exp(-x)/ln(exp(4*x^2)-4*exp(3*x^2)+6*exp(2*x^2)-4*exp(x^2)+5)

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maxima [B]  time = 0.47, size = 83, normalized size = 2.24 \begin {gather*} \frac {x^{3} e^{x} - x e^{x} \log \left (e^{\left (2 \, x^{2}\right )} - 4 \, e^{\left (x^{2}\right )} + 5\right ) - x e^{x} \log \left (e^{\left (2 \, x^{2}\right )} + 1\right ) - x^{2} + 4 \, x}{e^{x} \log \left (e^{\left (2 \, x^{2}\right )} - 4 \, e^{\left (x^{2}\right )} + 5\right ) + e^{x} \log \left (e^{\left (2 \, x^{2}\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)*exp(x^2)^4+4*exp(x)*exp(x^2)^3-6*exp(x)*exp(x^2)^2+4*exp(x)*exp(x^2)-5*exp(x))*log(exp(x^2
)^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)^2+((3*exp(x)*x^2+x^2-6*x+4)*exp(x^2)^4+(-12*exp(x)*x^2-4*x^2+24*x-
16)*exp(x^2)^3+(18*exp(x)*x^2+6*x^2-36*x+24)*exp(x^2)^2+(-12*exp(x)*x^2-4*x^2+24*x-16)*exp(x^2)+15*exp(x)*x^2+
5*x^2-30*x+20)*log(exp(x^2)^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)+(-8*exp(x)*x^4+8*x^3-32*x^2)*exp(x^2)^4+
(24*exp(x)*x^4-24*x^3+96*x^2)*exp(x^2)^3+(-24*exp(x)*x^4+24*x^3-96*x^2)*exp(x^2)^2+(8*exp(x)*x^4-8*x^3+32*x^2)
*exp(x^2))/(exp(x)*exp(x^2)^4-4*exp(x)*exp(x^2)^3+6*exp(x)*exp(x^2)^2-4*exp(x)*exp(x^2)+5*exp(x))/log(exp(x^2)
^4-4*exp(x^2)^3+6*exp(x^2)^2-4*exp(x^2)+5)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.72, size = 343, normalized size = 9.27 \begin {gather*} \frac {x\,{\mathrm {e}}^{-x}\,\left (x^2\,{\mathrm {e}}^x-x+4\right )-\frac {{\mathrm {e}}^{-x^2-x}\,\ln \left (6\,{\mathrm {e}}^{2\,x^2}-4\,{\mathrm {e}}^{x^2}-4\,{\mathrm {e}}^{3\,x^2}+{\mathrm {e}}^{4\,x^2}+5\right )\,\left (3\,x^2\,{\mathrm {e}}^x-6\,x+x^2+4\right )\,\left (6\,{\mathrm {e}}^{2\,x^2}-4\,{\mathrm {e}}^{x^2}-4\,{\mathrm {e}}^{3\,x^2}+{\mathrm {e}}^{4\,x^2}+5\right )}{8\,x\,{\left ({\mathrm {e}}^{x^2}-1\right )}^3}}{\ln \left (6\,{\mathrm {e}}^{2\,x^2}-4\,{\mathrm {e}}^{x^2}-4\,{\mathrm {e}}^{3\,x^2}+{\mathrm {e}}^{4\,x^2}+5\right )}-\frac {5\,x}{8}-\frac {{\mathrm {e}}^{-x^2-x}\,\left (\frac {15\,x^2\,{\mathrm {e}}^x}{8}-\frac {15\,x}{4}+\frac {5\,x^2}{8}+\frac {5}{2}\right )}{x}+\frac {{\mathrm {e}}^{-x}\,\left (\frac {x^2}{8}-\frac {3\,x}{4}+\frac {1}{2}\right )}{x}+\frac {{\mathrm {e}}^{-x}\,\left (3\,x^4\,{\mathrm {e}}^x+4\,x^2-6\,x^3+x^4\right )}{2\,x^3\,\left (3\,{\mathrm {e}}^{x^2}-3\,{\mathrm {e}}^{2\,x^2}+{\mathrm {e}}^{3\,x^2}-1\right )}-\frac {{\mathrm {e}}^{-x}\,\left (3\,x^4\,{\mathrm {e}}^x+4\,x^2-6\,x^3+x^4\right )}{2\,x^3\,\left ({\mathrm {e}}^{2\,x^2}-2\,{\mathrm {e}}^{x^2}+1\right )}+\frac {{\mathrm {e}}^{-x}\,\left (3\,x^4\,{\mathrm {e}}^x+4\,x^2-6\,x^3+x^4\right )}{2\,x^3\,\left ({\mathrm {e}}^{x^2}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(6*exp(2*x^2) - 4*exp(x^2) - 4*exp(3*x^2) + exp(4*x^2) + 5)*(15*x^2*exp(x) - 30*x + exp(4*x^2)*(3*x^2*
exp(x) - 6*x + x^2 + 4) - exp(x^2)*(12*x^2*exp(x) - 24*x + 4*x^2 + 16) - exp(3*x^2)*(12*x^2*exp(x) - 24*x + 4*
x^2 + 16) + exp(2*x^2)*(18*x^2*exp(x) - 36*x + 6*x^2 + 24) + 5*x^2 + 20) - log(6*exp(2*x^2) - 4*exp(x^2) - 4*e
xp(3*x^2) + exp(4*x^2) + 5)^2*(5*exp(x) - 4*exp(x^2)*exp(x) + 6*exp(2*x^2)*exp(x) - 4*exp(3*x^2)*exp(x) + exp(
4*x^2)*exp(x)) + exp(x^2)*(8*x^4*exp(x) + 32*x^2 - 8*x^3) - exp(4*x^2)*(8*x^4*exp(x) + 32*x^2 - 8*x^3) - exp(2
*x^2)*(24*x^4*exp(x) + 96*x^2 - 24*x^3) + exp(3*x^2)*(24*x^4*exp(x) + 96*x^2 - 24*x^3))/(log(6*exp(2*x^2) - 4*
exp(x^2) - 4*exp(3*x^2) + exp(4*x^2) + 5)^2*(5*exp(x) - 4*exp(x^2)*exp(x) + 6*exp(2*x^2)*exp(x) - 4*exp(3*x^2)
*exp(x) + exp(4*x^2)*exp(x))),x)

[Out]

(x*exp(-x)*(x^2*exp(x) - x + 4) - (exp(- x - x^2)*log(6*exp(2*x^2) - 4*exp(x^2) - 4*exp(3*x^2) + exp(4*x^2) +
5)*(3*x^2*exp(x) - 6*x + x^2 + 4)*(6*exp(2*x^2) - 4*exp(x^2) - 4*exp(3*x^2) + exp(4*x^2) + 5))/(8*x*(exp(x^2)
- 1)^3))/log(6*exp(2*x^2) - 4*exp(x^2) - 4*exp(3*x^2) + exp(4*x^2) + 5) - (5*x)/8 - (exp(- x - x^2)*((15*x^2*e
xp(x))/8 - (15*x)/4 + (5*x^2)/8 + 5/2))/x + (exp(-x)*(x^2/8 - (3*x)/4 + 1/2))/x + (exp(-x)*(3*x^4*exp(x) + 4*x
^2 - 6*x^3 + x^4))/(2*x^3*(3*exp(x^2) - 3*exp(2*x^2) + exp(3*x^2) - 1)) - (exp(-x)*(3*x^4*exp(x) + 4*x^2 - 6*x
^3 + x^4))/(2*x^3*(exp(2*x^2) - 2*exp(x^2) + 1)) + (exp(-x)*(3*x^4*exp(x) + 4*x^2 - 6*x^3 + x^4))/(2*x^3*(exp(
x^2) - 1))

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sympy [B]  time = 0.91, size = 51, normalized size = 1.38 \begin {gather*} - x + \frac {\left (x^{3} e^{x} - x^{2} + 4 x\right ) e^{- x}}{\log {\left (e^{4 x^{2}} - 4 e^{3 x^{2}} + 6 e^{2 x^{2}} - 4 e^{x^{2}} + 5 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)*exp(x**2)**4+4*exp(x)*exp(x**2)**3-6*exp(x)*exp(x**2)**2+4*exp(x)*exp(x**2)-5*exp(x))*ln(e
xp(x**2)**4-4*exp(x**2)**3+6*exp(x**2)**2-4*exp(x**2)+5)**2+((3*exp(x)*x**2+x**2-6*x+4)*exp(x**2)**4+(-12*exp(
x)*x**2-4*x**2+24*x-16)*exp(x**2)**3+(18*exp(x)*x**2+6*x**2-36*x+24)*exp(x**2)**2+(-12*exp(x)*x**2-4*x**2+24*x
-16)*exp(x**2)+15*exp(x)*x**2+5*x**2-30*x+20)*ln(exp(x**2)**4-4*exp(x**2)**3+6*exp(x**2)**2-4*exp(x**2)+5)+(-8
*exp(x)*x**4+8*x**3-32*x**2)*exp(x**2)**4+(24*exp(x)*x**4-24*x**3+96*x**2)*exp(x**2)**3+(-24*exp(x)*x**4+24*x*
*3-96*x**2)*exp(x**2)**2+(8*exp(x)*x**4-8*x**3+32*x**2)*exp(x**2))/(exp(x)*exp(x**2)**4-4*exp(x)*exp(x**2)**3+
6*exp(x)*exp(x**2)**2-4*exp(x)*exp(x**2)+5*exp(x))/ln(exp(x**2)**4-4*exp(x**2)**3+6*exp(x**2)**2-4*exp(x**2)+5
)**2,x)

[Out]

-x + (x**3*exp(x) - x**2 + 4*x)*exp(-x)/log(exp(4*x**2) - 4*exp(3*x**2) + 6*exp(2*x**2) - 4*exp(x**2) + 5)

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