3.24.72 \(\int \frac {8-8 e^{12}+e^8 (24-16 x)-16 x+6 x^2+e^4 (-24+32 x-6 x^2)+e^{-15+3 x^2} (8+e^{2 x} (2+4 x)+e^x (8+8 x))+e^{2 x} (2+e^4 (-6-12 x)+e^{12} (-2-4 x)+4 x+e^8 (6+12 x))+e^x (8+e^{12} (-8-8 x)-4 x^2+e^8 (24+16 x-4 x^2)+e^4 (-24-8 x+8 x^2))+e^{-10+2 x^2} (24-24 e^4-16 x+16 x^3+e^{2 x} (6+e^4 (-6-12 x)+12 x)+e^x (24+e^4 (-24-24 x)+16 x-4 x^2+8 x^3))+e^{-5+x^2} (24+24 e^8-32 x+6 x^2+16 x^3-8 x^4+e^4 (-48+32 x-16 x^3)+e^{2 x} (6+e^4 (-12-24 x)+12 x+e^8 (6+12 x))+e^x (24+8 x-8 x^2+8 x^3+e^8 (24+24 x)+e^4 (-48-32 x+8 x^2-8 x^3)))}{1-3 e^4+3 e^8-e^{12}+e^{-15+3 x^2}+e^{-10+2 x^2} (3-3 e^4)+e^{-5+x^2} (3-6 e^4+3 e^8)} \, dx\)

Optimal. Leaf size=28 \[ 2 x \left (2+e^x+\frac {x}{-1+e^4-e^{-5+x^2}}\right )^2 \]

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Rubi [F]  time = 13.25, 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 {8-8 e^{12}+e^8 (24-16 x)-16 x+6 x^2+e^4 \left (-24+32 x-6 x^2\right )+e^{-15+3 x^2} \left (8+e^{2 x} (2+4 x)+e^x (8+8 x)\right )+e^{2 x} \left (2+e^4 (-6-12 x)+e^{12} (-2-4 x)+4 x+e^8 (6+12 x)\right )+e^x \left (8+e^{12} (-8-8 x)-4 x^2+e^8 \left (24+16 x-4 x^2\right )+e^4 \left (-24-8 x+8 x^2\right )\right )+e^{-10+2 x^2} \left (24-24 e^4-16 x+16 x^3+e^{2 x} \left (6+e^4 (-6-12 x)+12 x\right )+e^x \left (24+e^4 (-24-24 x)+16 x-4 x^2+8 x^3\right )\right )+e^{-5+x^2} \left (24+24 e^8-32 x+6 x^2+16 x^3-8 x^4+e^4 \left (-48+32 x-16 x^3\right )+e^{2 x} \left (6+e^4 (-12-24 x)+12 x+e^8 (6+12 x)\right )+e^x \left (24+8 x-8 x^2+8 x^3+e^8 (24+24 x)+e^4 \left (-48-32 x+8 x^2-8 x^3\right )\right )\right )}{1-3 e^4+3 e^8-e^{12}+e^{-15+3 x^2}+e^{-10+2 x^2} \left (3-3 e^4\right )+e^{-5+x^2} \left (3-6 e^4+3 e^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(8 - 8*E^12 + E^8*(24 - 16*x) - 16*x + 6*x^2 + E^4*(-24 + 32*x - 6*x^2) + E^(-15 + 3*x^2)*(8 + E^(2*x)*(2
+ 4*x) + E^x*(8 + 8*x)) + E^(2*x)*(2 + E^4*(-6 - 12*x) + E^12*(-2 - 4*x) + 4*x + E^8*(6 + 12*x)) + E^x*(8 + E^
12*(-8 - 8*x) - 4*x^2 + E^8*(24 + 16*x - 4*x^2) + E^4*(-24 - 8*x + 8*x^2)) + E^(-10 + 2*x^2)*(24 - 24*E^4 - 16
*x + 16*x^3 + E^(2*x)*(6 + E^4*(-6 - 12*x) + 12*x) + E^x*(24 + E^4*(-24 - 24*x) + 16*x - 4*x^2 + 8*x^3)) + E^(
-5 + x^2)*(24 + 24*E^8 - 32*x + 6*x^2 + 16*x^3 - 8*x^4 + E^4*(-48 + 32*x - 16*x^3) + E^(2*x)*(6 + E^4*(-12 - 2
4*x) + 12*x + E^8*(6 + 12*x)) + E^x*(24 + 8*x - 8*x^2 + 8*x^3 + E^8*(24 + 24*x) + E^4*(-48 - 32*x + 8*x^2 - 8*
x^3))))/(1 - 3*E^4 + 3*E^8 - E^12 + E^(-15 + 3*x^2) + E^(-10 + 2*x^2)*(3 - 3*E^4) + E^(-5 + x^2)*(3 - 6*E^4 +
3*E^8)),x]

[Out]

-8*E^x - E^(2*x) + 8*x - (8*E^5*x^2)/(E^x^2 + E^5*(1 - E^4)) + 8*E^x*(1 + x) + E^(2*x)*(1 + 2*x) + 8*E^5*Defer
[Int][(E^x*x)/(-E^x^2 - E^5*(1 - E^4)), x] + 4*E^5*Defer[Int][(E^x*x^2)/(-E^x^2 - E^5*(1 - E^4)), x] + 6*E^10*
Defer[Int][x^2/(E^x^2 + E^5*(1 - E^4))^2, x] - 8*E^10*(1 - E^4)*Defer[Int][(E^x*x^3)/(E^x^2 + E^5*(1 - E^4))^2
, x] + 8*E^5*Defer[Int][(E^x*x^3)/(E^x^2 + E^5*(1 - E^4)), x] + 8*E^15*(1 - E^4)*Defer[Int][x^4/(E^x^2 + E^5*(
1 - E^4))^3, x] - 8*E^10*Defer[Int][x^4/(E^x^2 + E^5*(1 - E^4))^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{15} \left (8 \left (1-e^{12}\right )+e^8 (24-16 x)-16 x+6 x^2+e^4 \left (-24+32 x-6 x^2\right )+e^{-15+3 x^2} \left (8+e^{2 x} (2+4 x)+e^x (8+8 x)\right )+e^{2 x} \left (2+e^4 (-6-12 x)+e^{12} (-2-4 x)+4 x+e^8 (6+12 x)\right )+e^x \left (8+e^{12} (-8-8 x)-4 x^2+e^8 \left (24+16 x-4 x^2\right )+e^4 \left (-24-8 x+8 x^2\right )\right )+e^{-10+2 x^2} \left (24-24 e^4-16 x+16 x^3+e^{2 x} \left (6+e^4 (-6-12 x)+12 x\right )+e^x \left (24+e^4 (-24-24 x)+16 x-4 x^2+8 x^3\right )\right )+e^{-5+x^2} \left (24+24 e^8-32 x+6 x^2+16 x^3-8 x^4+e^4 \left (-48+32 x-16 x^3\right )+e^{2 x} \left (6+e^4 (-12-24 x)+12 x+e^8 (6+12 x)\right )+e^x \left (24+8 x-8 x^2+8 x^3+e^8 (24+24 x)+e^4 \left (-48-32 x+8 x^2-8 x^3\right )\right )\right )\right )}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=e^{15} \int \frac {8 \left (1-e^{12}\right )+e^8 (24-16 x)-16 x+6 x^2+e^4 \left (-24+32 x-6 x^2\right )+e^{-15+3 x^2} \left (8+e^{2 x} (2+4 x)+e^x (8+8 x)\right )+e^{2 x} \left (2+e^4 (-6-12 x)+e^{12} (-2-4 x)+4 x+e^8 (6+12 x)\right )+e^x \left (8+e^{12} (-8-8 x)-4 x^2+e^8 \left (24+16 x-4 x^2\right )+e^4 \left (-24-8 x+8 x^2\right )\right )+e^{-10+2 x^2} \left (24-24 e^4-16 x+16 x^3+e^{2 x} \left (6+e^4 (-6-12 x)+12 x\right )+e^x \left (24+e^4 (-24-24 x)+16 x-4 x^2+8 x^3\right )\right )+e^{-5+x^2} \left (24+24 e^8-32 x+6 x^2+16 x^3-8 x^4+e^4 \left (-48+32 x-16 x^3\right )+e^{2 x} \left (6+e^4 (-12-24 x)+12 x+e^8 (6+12 x)\right )+e^x \left (24+8 x-8 x^2+8 x^3+e^8 (24+24 x)+e^4 \left (-48-32 x+8 x^2-8 x^3\right )\right )\right )}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=e^{15} \int \left (\frac {8 \left (1-e^4\right ) x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3}+\frac {2 \left (2+e^x\right ) \left (2+e^x+2 e^x x\right )}{e^{15}}+\frac {2 x^2 \left (3-8 \left (1-e^4\right ) x-4 e^x \left (1-e^4\right ) x-4 x^2\right )}{e^5 \left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2}+\frac {4 x \left (-4-2 e^x-e^x x+4 x^2+2 e^x x^2\right )}{e^{10} \left (e^{x^2}+e^5 \left (1-e^4\right )\right )}\right ) \, dx\\ &=2 \int \left (2+e^x\right ) \left (2+e^x+2 e^x x\right ) \, dx+\left (4 e^5\right ) \int \frac {x \left (-4-2 e^x-e^x x+4 x^2+2 e^x x^2\right )}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (2 e^{10}\right ) \int \frac {x^2 \left (3-8 \left (1-e^4\right ) x-4 e^x \left (1-e^4\right ) x-4 x^2\right )}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=2 \int \left (4+4 e^x (1+x)+e^{2 x} (1+2 x)\right ) \, dx+\left (4 e^5\right ) \int \left (\frac {4 x}{-e^{x^2}-e^5 \left (1-e^4\right )}+\frac {2 e^x x}{-e^{x^2}-e^5 \left (1-e^4\right )}+\frac {e^x x^2}{-e^{x^2}-e^5 \left (1-e^4\right )}+\frac {4 x^3}{e^{x^2}+e^5 \left (1-e^4\right )}+\frac {2 e^x x^3}{e^{x^2}+e^5 \left (1-e^4\right )}\right ) \, dx+\left (2 e^{10}\right ) \int \frac {x^2 \left (3+4 \left (-1+e^4\right ) \left (2+e^x\right ) x-4 x^2\right )}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=8 x+2 \int e^{2 x} (1+2 x) \, dx+8 \int e^x (1+x) \, dx+\left (4 e^5\right ) \int \frac {e^x x^2}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x^3}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (16 e^5\right ) \int \frac {x}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (16 e^5\right ) \int \frac {x^3}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (2 e^{10}\right ) \int \left (\frac {3 x^2}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2}+\frac {8 \left (-1+e^4\right ) x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2}+\frac {4 e^x \left (-1+e^4\right ) x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2}-\frac {4 x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2}\right ) \, dx+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=8 x+8 e^x (1+x)+e^{2 x} (1+2 x)-2 \int e^{2 x} \, dx-8 \int e^x \, dx+\left (4 e^5\right ) \int \frac {e^x x^2}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x^3}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{-e^x-e^5 \left (1-e^4\right )} \, dx,x,x^2\right )+\left (8 e^5\right ) \operatorname {Subst}\left (\int \frac {x}{e^x+e^5 \left (1-e^4\right )} \, dx,x,x^2\right )+\left (6 e^{10}\right ) \int \frac {x^2}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10}\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10} \left (1-e^4\right )\right ) \int \frac {e^x x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (16 e^{10} \left (1-e^4\right )\right ) \int \frac {x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=-8 e^x-e^{2 x}+8 x+\frac {4 x^4}{1-e^4}+8 e^x (1+x)+e^{2 x} (1+2 x)+\left (4 e^5\right ) \int \frac {e^x x^2}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x^3}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-e^5+e^9-x\right ) x} \, dx,x,e^{x^2}\right )+\left (6 e^{10}\right ) \int \frac {x^2}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10}\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\frac {8 \operatorname {Subst}\left (\int \frac {e^x x}{e^x+e^5 \left (1-e^4\right )} \, dx,x,x^2\right )}{1-e^4}-\left (8 e^{10} \left (1-e^4\right )\right ) \int \frac {e^x x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10} \left (1-e^4\right )\right ) \operatorname {Subst}\left (\int \frac {x}{\left (e^x+e^5 \left (1-e^4\right )\right )^2} \, dx,x,x^2\right )+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=-8 e^x-e^{2 x}+8 x+\frac {4 x^4}{1-e^4}+8 e^x (1+x)+e^{2 x} (1+2 x)-\frac {8 x^2 \log \left (1+\frac {e^{-5+x^2}}{1-e^4}\right )}{1-e^4}+\left (4 e^5\right ) \int \frac {e^x x^2}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x^3}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \operatorname {Subst}\left (\int \frac {e^x x}{\left (e^x+e^5 \left (1-e^4\right )\right )^2} \, dx,x,x^2\right )-\left (8 e^5\right ) \operatorname {Subst}\left (\int \frac {x}{e^x+e^5 \left (1-e^4\right )} \, dx,x,x^2\right )+\left (6 e^{10}\right ) \int \frac {x^2}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10}\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\frac {8 \operatorname {Subst}\left (\int \frac {1}{-e^5+e^9-x} \, dx,x,e^{x^2}\right )}{1-e^4}-\frac {8 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x^2}\right )}{1-e^4}+\frac {8 \operatorname {Subst}\left (\int \log \left (1+\frac {e^{-5+x}}{1-e^4}\right ) \, dx,x,x^2\right )}{1-e^4}-\left (8 e^{10} \left (1-e^4\right )\right ) \int \frac {e^x x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=-8 e^x-e^{2 x}+8 x-\frac {8 x^2}{1-e^4}-\frac {8 e^5 x^2}{e^{x^2}+e^5 \left (1-e^4\right )}+8 e^x (1+x)+e^{2 x} (1+2 x)-\frac {8 x^2 \log \left (1+\frac {e^{-5+x^2}}{1-e^4}\right )}{1-e^4}+\frac {8 \log \left (e^{x^2}+e^5 \left (1-e^4\right )\right )}{1-e^4}+\left (4 e^5\right ) \int \frac {e^x x^2}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x^3}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{e^x+e^5 \left (1-e^4\right )} \, dx,x,x^2\right )+\left (6 e^{10}\right ) \int \frac {x^2}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10}\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\frac {8 \operatorname {Subst}\left (\int \frac {e^x x}{e^x+e^5 \left (1-e^4\right )} \, dx,x,x^2\right )}{1-e^4}+\frac {8 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{1-e^4}\right )}{x} \, dx,x,e^{-5+x^2}\right )}{1-e^4}-\left (8 e^{10} \left (1-e^4\right )\right ) \int \frac {e^x x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=-8 e^x-e^{2 x}+8 x-\frac {8 x^2}{1-e^4}-\frac {8 e^5 x^2}{e^{x^2}+e^5 \left (1-e^4\right )}+8 e^x (1+x)+e^{2 x} (1+2 x)+\frac {8 \log \left (e^{x^2}+e^5 \left (1-e^4\right )\right )}{1-e^4}-\frac {8 \text {Li}_2\left (-\frac {e^{-5+x^2}}{1-e^4}\right )}{1-e^4}+\left (4 e^5\right ) \int \frac {e^x x^2}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x^3}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (e^5-e^9+x\right )} \, dx,x,e^{x^2}\right )+\left (6 e^{10}\right ) \int \frac {x^2}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10}\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\frac {8 \operatorname {Subst}\left (\int \log \left (1+\frac {e^{-5+x}}{1-e^4}\right ) \, dx,x,x^2\right )}{1-e^4}-\left (8 e^{10} \left (1-e^4\right )\right ) \int \frac {e^x x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=-8 e^x-e^{2 x}+8 x-\frac {8 x^2}{1-e^4}-\frac {8 e^5 x^2}{e^{x^2}+e^5 \left (1-e^4\right )}+8 e^x (1+x)+e^{2 x} (1+2 x)+\frac {8 \log \left (e^{x^2}+e^5 \left (1-e^4\right )\right )}{1-e^4}-\frac {8 \text {Li}_2\left (-\frac {e^{-5+x^2}}{1-e^4}\right )}{1-e^4}+\left (4 e^5\right ) \int \frac {e^x x^2}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x^3}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (6 e^{10}\right ) \int \frac {x^2}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10}\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\frac {8 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x^2}\right )}{1-e^4}-\frac {8 \operatorname {Subst}\left (\int \frac {1}{e^5-e^9+x} \, dx,x,e^{x^2}\right )}{1-e^4}-\frac {8 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{1-e^4}\right )}{x} \, dx,x,e^{-5+x^2}\right )}{1-e^4}-\left (8 e^{10} \left (1-e^4\right )\right ) \int \frac {e^x x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ &=-8 e^x-e^{2 x}+8 x-\frac {8 e^5 x^2}{e^{x^2}+e^5 \left (1-e^4\right )}+8 e^x (1+x)+e^{2 x} (1+2 x)+\left (4 e^5\right ) \int \frac {e^x x^2}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x}{-e^{x^2}-e^5 \left (1-e^4\right )} \, dx+\left (8 e^5\right ) \int \frac {e^x x^3}{e^{x^2}+e^5 \left (1-e^4\right )} \, dx+\left (6 e^{10}\right ) \int \frac {x^2}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10}\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx-\left (8 e^{10} \left (1-e^4\right )\right ) \int \frac {e^x x^3}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^2} \, dx+\left (8 e^{15} \left (1-e^4\right )\right ) \int \frac {x^4}{\left (e^{x^2}+e^5 \left (1-e^4\right )\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.54, size = 61, normalized size = 2.18 \begin {gather*} \frac {2 \left (-2 e^9+2 e^{x^2}+e^{5+x}-e^{9+x}+e^{x+x^2}-e^5 (-2+x)\right )^2 x}{\left (e^5-e^9+e^{x^2}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8 - 8*E^12 + E^8*(24 - 16*x) - 16*x + 6*x^2 + E^4*(-24 + 32*x - 6*x^2) + E^(-15 + 3*x^2)*(8 + E^(2*
x)*(2 + 4*x) + E^x*(8 + 8*x)) + E^(2*x)*(2 + E^4*(-6 - 12*x) + E^12*(-2 - 4*x) + 4*x + E^8*(6 + 12*x)) + E^x*(
8 + E^12*(-8 - 8*x) - 4*x^2 + E^8*(24 + 16*x - 4*x^2) + E^4*(-24 - 8*x + 8*x^2)) + E^(-10 + 2*x^2)*(24 - 24*E^
4 - 16*x + 16*x^3 + E^(2*x)*(6 + E^4*(-6 - 12*x) + 12*x) + E^x*(24 + E^4*(-24 - 24*x) + 16*x - 4*x^2 + 8*x^3))
+ E^(-5 + x^2)*(24 + 24*E^8 - 32*x + 6*x^2 + 16*x^3 - 8*x^4 + E^4*(-48 + 32*x - 16*x^3) + E^(2*x)*(6 + E^4*(-
12 - 24*x) + 12*x + E^8*(6 + 12*x)) + E^x*(24 + 8*x - 8*x^2 + 8*x^3 + E^8*(24 + 24*x) + E^4*(-48 - 32*x + 8*x^
2 - 8*x^3))))/(1 - 3*E^4 + 3*E^8 - E^12 + E^(-15 + 3*x^2) + E^(-10 + 2*x^2)*(3 - 3*E^4) + E^(-5 + x^2)*(3 - 6*
E^4 + 3*E^8)),x]

[Out]

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

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fricas [B]  time = 0.63, size = 181, normalized size = 6.46 \begin {gather*} -\frac {2 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x e^{8} + 4 \, {\left (x^{2} - 2 \, x\right )} e^{4} + {\left (x e^{\left (2 \, x\right )} + 4 \, x e^{x} + 4 \, x\right )} e^{\left (2 \, x^{2} - 10\right )} - 2 \, {\left (2 \, x^{2} + 4 \, x e^{4} + {\left (x e^{4} - x\right )} e^{\left (2 \, x\right )} + {\left (x^{2} + 4 \, x e^{4} - 4 \, x\right )} e^{x} - 4 \, x\right )} e^{\left (x^{2} - 5\right )} + {\left (x e^{8} - 2 \, x e^{4} + x\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{2} - 2 \, x e^{8} - {\left (x^{2} - 4 \, x\right )} e^{4} - 2 \, x\right )} e^{x} + 4 \, x\right )}}{2 \, {\left (e^{4} - 1\right )} e^{\left (x^{2} - 5\right )} - e^{8} + 2 \, e^{4} - e^{\left (2 \, x^{2} - 10\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+2)*exp(x)^2+(8*x+8)*exp(x)+8)*exp(x^2-5)^3+(((-12*x-6)*exp(4)+12*x+6)*exp(x)^2+((-24*x-24)*ex
p(4)+8*x^3-4*x^2+16*x+24)*exp(x)-24*exp(4)+16*x^3-16*x+24)*exp(x^2-5)^2+(((12*x+6)*exp(4)^2+(-24*x-12)*exp(4)+
12*x+6)*exp(x)^2+((24*x+24)*exp(4)^2+(-8*x^3+8*x^2-32*x-48)*exp(4)+8*x^3-8*x^2+8*x+24)*exp(x)+24*exp(4)^2+(-16
*x^3+32*x-48)*exp(4)-8*x^4+16*x^3+6*x^2-32*x+24)*exp(x^2-5)+((-4*x-2)*exp(4)^3+(12*x+6)*exp(4)^2+(-12*x-6)*exp
(4)+4*x+2)*exp(x)^2+((-8*x-8)*exp(4)^3+(-4*x^2+16*x+24)*exp(4)^2+(8*x^2-8*x-24)*exp(4)-4*x^2+8)*exp(x)-8*exp(4
)^3+(-16*x+24)*exp(4)^2+(-6*x^2+32*x-24)*exp(4)+6*x^2-16*x+8)/(exp(x^2-5)^3+(-3*exp(4)+3)*exp(x^2-5)^2+(3*exp(
4)^2-6*exp(4)+3)*exp(x^2-5)-exp(4)^3+3*exp(4)^2-3*exp(4)+1),x, algorithm="fricas")

[Out]

-2*(x^3 - 4*x^2 + 4*x*e^8 + 4*(x^2 - 2*x)*e^4 + (x*e^(2*x) + 4*x*e^x + 4*x)*e^(2*x^2 - 10) - 2*(2*x^2 + 4*x*e^
4 + (x*e^4 - x)*e^(2*x) + (x^2 + 4*x*e^4 - 4*x)*e^x - 4*x)*e^(x^2 - 5) + (x*e^8 - 2*x*e^4 + x)*e^(2*x) - 2*(x^
2 - 2*x*e^8 - (x^2 - 4*x)*e^4 - 2*x)*e^x + 4*x)/(2*(e^4 - 1)*e^(x^2 - 5) - e^8 + 2*e^4 - e^(2*x^2 - 10) - 1)

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giac [B]  time = 0.57, size = 309, normalized size = 11.04 \begin {gather*} \frac {2 \, {\left (x^{6} e^{10} + 4 \, x^{5} e^{14} - 4 \, x^{5} e^{10} - 2 \, x^{5} e^{\left (x^{2} + x + 5\right )} - 4 \, x^{5} e^{\left (x^{2} + 5\right )} + 2 \, x^{5} e^{\left (x + 14\right )} - 2 \, x^{5} e^{\left (x + 10\right )} + 4 \, x^{4} e^{18} - 8 \, x^{4} e^{14} + 4 \, x^{4} e^{10} + 4 \, x^{4} e^{\left (2 \, x^{2}\right )} + x^{4} e^{\left (2 \, x^{2} + 2 \, x\right )} + 4 \, x^{4} e^{\left (2 \, x^{2} + x\right )} - 2 \, x^{4} e^{\left (x^{2} + 2 \, x + 9\right )} + 2 \, x^{4} e^{\left (x^{2} + 2 \, x + 5\right )} - 8 \, x^{4} e^{\left (x^{2} + x + 9\right )} + 8 \, x^{4} e^{\left (x^{2} + x + 5\right )} - 8 \, x^{4} e^{\left (x^{2} + 9\right )} + 8 \, x^{4} e^{\left (x^{2} + 5\right )} + x^{4} e^{\left (2 \, x + 18\right )} - 2 \, x^{4} e^{\left (2 \, x + 14\right )} + x^{4} e^{\left (2 \, x + 10\right )} + 4 \, x^{4} e^{\left (x + 18\right )} - 8 \, x^{4} e^{\left (x + 14\right )} + 4 \, x^{4} e^{\left (x + 10\right )}\right )}}{x^{3} e^{18} - 2 \, x^{3} e^{14} + x^{3} e^{10} + x^{3} e^{\left (2 \, x^{2}\right )} - 2 \, x^{3} e^{\left (x^{2} + 9\right )} + 2 \, x^{3} e^{\left (x^{2} + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+2)*exp(x)^2+(8*x+8)*exp(x)+8)*exp(x^2-5)^3+(((-12*x-6)*exp(4)+12*x+6)*exp(x)^2+((-24*x-24)*ex
p(4)+8*x^3-4*x^2+16*x+24)*exp(x)-24*exp(4)+16*x^3-16*x+24)*exp(x^2-5)^2+(((12*x+6)*exp(4)^2+(-24*x-12)*exp(4)+
12*x+6)*exp(x)^2+((24*x+24)*exp(4)^2+(-8*x^3+8*x^2-32*x-48)*exp(4)+8*x^3-8*x^2+8*x+24)*exp(x)+24*exp(4)^2+(-16
*x^3+32*x-48)*exp(4)-8*x^4+16*x^3+6*x^2-32*x+24)*exp(x^2-5)+((-4*x-2)*exp(4)^3+(12*x+6)*exp(4)^2+(-12*x-6)*exp
(4)+4*x+2)*exp(x)^2+((-8*x-8)*exp(4)^3+(-4*x^2+16*x+24)*exp(4)^2+(8*x^2-8*x-24)*exp(4)-4*x^2+8)*exp(x)-8*exp(4
)^3+(-16*x+24)*exp(4)^2+(-6*x^2+32*x-24)*exp(4)+6*x^2-16*x+8)/(exp(x^2-5)^3+(-3*exp(4)+3)*exp(x^2-5)^2+(3*exp(
4)^2-6*exp(4)+3)*exp(x^2-5)-exp(4)^3+3*exp(4)^2-3*exp(4)+1),x, algorithm="giac")

[Out]

2*(x^6*e^10 + 4*x^5*e^14 - 4*x^5*e^10 - 2*x^5*e^(x^2 + x + 5) - 4*x^5*e^(x^2 + 5) + 2*x^5*e^(x + 14) - 2*x^5*e
^(x + 10) + 4*x^4*e^18 - 8*x^4*e^14 + 4*x^4*e^10 + 4*x^4*e^(2*x^2) + x^4*e^(2*x^2 + 2*x) + 4*x^4*e^(2*x^2 + x)
- 2*x^4*e^(x^2 + 2*x + 9) + 2*x^4*e^(x^2 + 2*x + 5) - 8*x^4*e^(x^2 + x + 9) + 8*x^4*e^(x^2 + x + 5) - 8*x^4*e
^(x^2 + 9) + 8*x^4*e^(x^2 + 5) + x^4*e^(2*x + 18) - 2*x^4*e^(2*x + 14) + x^4*e^(2*x + 10) + 4*x^4*e^(x + 18) -
8*x^4*e^(x + 14) + 4*x^4*e^(x + 10))/(x^3*e^18 - 2*x^3*e^14 + x^3*e^10 + x^3*e^(2*x^2) - 2*x^3*e^(x^2 + 9) +
2*x^3*e^(x^2 + 5))

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maple [B]  time = 0.18, size = 70, normalized size = 2.50

method	result	size
risch	\(2 x \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x +8 x +\frac {2 \left (2 \,{\mathrm e}^{4+x}-2 \,{\mathrm e}^{x^{2}+x -5}+4 \,{\mathrm e}^{4}+x -2 \,{\mathrm e}^{x}-4 \,{\mathrm e}^{x^{2}-5}-4\right ) x^{2}}{\left ({\mathrm e}^{4}-1-{\mathrm e}^{x^{2}-5}\right )^{2}}\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x+2)*exp(x)^2+(8*x+8)*exp(x)+8)*exp(x^2-5)^3+(((-12*x-6)*exp(4)+12*x+6)*exp(x)^2+((-24*x-24)*exp(4)+8
*x^3-4*x^2+16*x+24)*exp(x)-24*exp(4)+16*x^3-16*x+24)*exp(x^2-5)^2+(((12*x+6)*exp(4)^2+(-24*x-12)*exp(4)+12*x+6
)*exp(x)^2+((24*x+24)*exp(4)^2+(-8*x^3+8*x^2-32*x-48)*exp(4)+8*x^3-8*x^2+8*x+24)*exp(x)+24*exp(4)^2+(-16*x^3+3
2*x-48)*exp(4)-8*x^4+16*x^3+6*x^2-32*x+24)*exp(x^2-5)+((-4*x-2)*exp(4)^3+(12*x+6)*exp(4)^2+(-12*x-6)*exp(4)+4*
x+2)*exp(x)^2+((-8*x-8)*exp(4)^3+(-4*x^2+16*x+24)*exp(4)^2+(8*x^2-8*x-24)*exp(4)-4*x^2+8)*exp(x)-8*exp(4)^3+(-
16*x+24)*exp(4)^2+(-6*x^2+32*x-24)*exp(4)+6*x^2-16*x+8)/(exp(x^2-5)^3+(-3*exp(4)+3)*exp(x^2-5)^2+(3*exp(4)^2-6
*exp(4)+3)*exp(x^2-5)-exp(4)^3+3*exp(4)^2-3*exp(4)+1),x,method=_RETURNVERBOSE)

[Out]

2*x*exp(2*x)+8*exp(x)*x+8*x+2*(2*exp(4+x)-2*exp(x^2+x-5)+4*exp(4)+x-2*exp(x)-4*exp(x^2-5)-4)*x^2/(exp(4)-1-exp
(x^2-5))^2

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maxima [B]  time = 0.81, size = 191, normalized size = 6.82 \begin {gather*} -\frac {2 \, {\left (x^{3} e^{10} + 4 \, x^{2} {\left (e^{14} - e^{10}\right )} + x {\left (e^{18} - 2 \, e^{14} + e^{10}\right )} e^{\left (2 \, x\right )} + 4 \, x {\left (e^{18} - 2 \, e^{14} + e^{10}\right )} + {\left (x e^{\left (2 \, x\right )} + 4 \, x e^{x} + 4 \, x\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (2 \, x^{2} e^{5} + x {\left (e^{9} - e^{5}\right )} e^{\left (2 \, x\right )} + 4 \, x {\left (e^{9} - e^{5}\right )} + {\left (x^{2} e^{5} + 4 \, x {\left (e^{9} - e^{5}\right )}\right )} e^{x}\right )} e^{\left (x^{2}\right )} + 2 \, {\left (x^{2} {\left (e^{14} - e^{10}\right )} + 2 \, x {\left (e^{18} - 2 \, e^{14} + e^{10}\right )}\right )} e^{x}\right )}}{2 \, {\left (e^{9} - e^{5}\right )} e^{\left (x^{2}\right )} - e^{18} + 2 \, e^{14} - e^{10} - e^{\left (2 \, x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+2)*exp(x)^2+(8*x+8)*exp(x)+8)*exp(x^2-5)^3+(((-12*x-6)*exp(4)+12*x+6)*exp(x)^2+((-24*x-24)*ex
p(4)+8*x^3-4*x^2+16*x+24)*exp(x)-24*exp(4)+16*x^3-16*x+24)*exp(x^2-5)^2+(((12*x+6)*exp(4)^2+(-24*x-12)*exp(4)+
12*x+6)*exp(x)^2+((24*x+24)*exp(4)^2+(-8*x^3+8*x^2-32*x-48)*exp(4)+8*x^3-8*x^2+8*x+24)*exp(x)+24*exp(4)^2+(-16
*x^3+32*x-48)*exp(4)-8*x^4+16*x^3+6*x^2-32*x+24)*exp(x^2-5)+((-4*x-2)*exp(4)^3+(12*x+6)*exp(4)^2+(-12*x-6)*exp
(4)+4*x+2)*exp(x)^2+((-8*x-8)*exp(4)^3+(-4*x^2+16*x+24)*exp(4)^2+(8*x^2-8*x-24)*exp(4)-4*x^2+8)*exp(x)-8*exp(4
)^3+(-16*x+24)*exp(4)^2+(-6*x^2+32*x-24)*exp(4)+6*x^2-16*x+8)/(exp(x^2-5)^3+(-3*exp(4)+3)*exp(x^2-5)^2+(3*exp(
4)^2-6*exp(4)+3)*exp(x^2-5)-exp(4)^3+3*exp(4)^2-3*exp(4)+1),x, algorithm="maxima")

[Out]

-2*(x^3*e^10 + 4*x^2*(e^14 - e^10) + x*(e^18 - 2*e^14 + e^10)*e^(2*x) + 4*x*(e^18 - 2*e^14 + e^10) + (x*e^(2*x
) + 4*x*e^x + 4*x)*e^(2*x^2) - 2*(2*x^2*e^5 + x*(e^9 - e^5)*e^(2*x) + 4*x*(e^9 - e^5) + (x^2*e^5 + 4*x*(e^9 -
e^5))*e^x)*e^(x^2) + 2*(x^2*(e^14 - e^10) + 2*x*(e^18 - 2*e^14 + e^10))*e^x)/(2*(e^9 - e^5)*e^(x^2) - e^18 + 2
*e^14 - e^10 - e^(2*x^2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{3\,x^2-15}\,\left ({\mathrm {e}}^x\,\left (8\,x+8\right )+{\mathrm {e}}^{2\,x}\,\left (4\,x+2\right )+8\right )-8\,{\mathrm {e}}^{12}-16\,x-{\mathrm {e}}^4\,\left (6\,x^2-32\,x+24\right )-{\mathrm {e}}^x\,\left ({\mathrm {e}}^4\,\left (-8\,x^2+8\,x+24\right )-{\mathrm {e}}^8\,\left (-4\,x^2+16\,x+24\right )+4\,x^2+{\mathrm {e}}^{12}\,\left (8\,x+8\right )-8\right )+{\mathrm {e}}^{2\,x^2-10}\,\left ({\mathrm {e}}^{2\,x}\,\left (12\,x-{\mathrm {e}}^4\,\left (12\,x+6\right )+6\right )-24\,{\mathrm {e}}^4-16\,x+16\,x^3+{\mathrm {e}}^x\,\left (16\,x-4\,x^2+8\,x^3-{\mathrm {e}}^4\,\left (24\,x+24\right )+24\right )+24\right )+{\mathrm {e}}^{2\,x}\,\left (4\,x-{\mathrm {e}}^{12}\,\left (4\,x+2\right )-{\mathrm {e}}^4\,\left (12\,x+6\right )+{\mathrm {e}}^8\,\left (12\,x+6\right )+2\right )+6\,x^2+{\mathrm {e}}^{x^2-5}\,\left (24\,{\mathrm {e}}^8-32\,x-{\mathrm {e}}^4\,\left (16\,x^3-32\,x+48\right )+{\mathrm {e}}^x\,\left (8\,x-{\mathrm {e}}^4\,\left (8\,x^3-8\,x^2+32\,x+48\right )-8\,x^2+8\,x^3+{\mathrm {e}}^8\,\left (24\,x+24\right )+24\right )+{\mathrm {e}}^{2\,x}\,\left (12\,x+{\mathrm {e}}^8\,\left (12\,x+6\right )-{\mathrm {e}}^4\,\left (24\,x+12\right )+6\right )+6\,x^2+16\,x^3-8\,x^4+24\right )-{\mathrm {e}}^8\,\left (16\,x-24\right )+8}{3\,{\mathrm {e}}^8-3\,{\mathrm {e}}^4-{\mathrm {e}}^{12}+{\mathrm {e}}^{3\,x^2-15}-{\mathrm {e}}^{2\,x^2-10}\,\left (3\,{\mathrm {e}}^4-3\right )+{\mathrm {e}}^{x^2-5}\,\left (3\,{\mathrm {e}}^8-6\,{\mathrm {e}}^4+3\right )+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*x^2 - 15)*(exp(x)*(8*x + 8) + exp(2*x)*(4*x + 2) + 8) - 8*exp(12) - 16*x - exp(4)*(6*x^2 - 32*x + 2
4) - exp(x)*(exp(4)*(8*x - 8*x^2 + 24) - exp(8)*(16*x - 4*x^2 + 24) + 4*x^2 + exp(12)*(8*x + 8) - 8) + exp(2*x
^2 - 10)*(exp(2*x)*(12*x - exp(4)*(12*x + 6) + 6) - 24*exp(4) - 16*x + 16*x^3 + exp(x)*(16*x - 4*x^2 + 8*x^3 -
exp(4)*(24*x + 24) + 24) + 24) + exp(2*x)*(4*x - exp(12)*(4*x + 2) - exp(4)*(12*x + 6) + exp(8)*(12*x + 6) +
2) + 6*x^2 + exp(x^2 - 5)*(24*exp(8) - 32*x - exp(4)*(16*x^3 - 32*x + 48) + exp(x)*(8*x - exp(4)*(32*x - 8*x^2
+ 8*x^3 + 48) - 8*x^2 + 8*x^3 + exp(8)*(24*x + 24) + 24) + exp(2*x)*(12*x + exp(8)*(12*x + 6) - exp(4)*(24*x
+ 12) + 6) + 6*x^2 + 16*x^3 - 8*x^4 + 24) - exp(8)*(16*x - 24) + 8)/(3*exp(8) - 3*exp(4) - exp(12) + exp(3*x^2
- 15) - exp(2*x^2 - 10)*(3*exp(4) - 3) + exp(x^2 - 5)*(3*exp(8) - 6*exp(4) + 3) + 1),x)

[Out]

int((exp(3*x^2 - 15)*(exp(x)*(8*x + 8) + exp(2*x)*(4*x + 2) + 8) - 8*exp(12) - 16*x - exp(4)*(6*x^2 - 32*x + 2
4) - exp(x)*(exp(4)*(8*x - 8*x^2 + 24) - exp(8)*(16*x - 4*x^2 + 24) + 4*x^2 + exp(12)*(8*x + 8) - 8) + exp(2*x
^2 - 10)*(exp(2*x)*(12*x - exp(4)*(12*x + 6) + 6) - 24*exp(4) - 16*x + 16*x^3 + exp(x)*(16*x - 4*x^2 + 8*x^3 -
exp(4)*(24*x + 24) + 24) + 24) + exp(2*x)*(4*x - exp(12)*(4*x + 2) - exp(4)*(12*x + 6) + exp(8)*(12*x + 6) +
2) + 6*x^2 + exp(x^2 - 5)*(24*exp(8) - 32*x - exp(4)*(16*x^3 - 32*x + 48) + exp(x)*(8*x - exp(4)*(32*x - 8*x^2
+ 8*x^3 + 48) - 8*x^2 + 8*x^3 + exp(8)*(24*x + 24) + 24) + exp(2*x)*(12*x + exp(8)*(12*x + 6) - exp(4)*(24*x
+ 12) + 6) + 6*x^2 + 16*x^3 - 8*x^4 + 24) - exp(8)*(16*x - 24) + 8)/(3*exp(8) - 3*exp(4) - exp(12) + exp(3*x^2
- 15) - exp(2*x^2 - 10)*(3*exp(4) - 3) + exp(x^2 - 5)*(3*exp(8) - 6*exp(4) + 3) + 1), x)

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sympy [B]  time = 0.61, size = 110, normalized size = 3.93 \begin {gather*} 2 x e^{2 x} + 8 x e^{x} + 8 x + \frac {2 x^{3} - 4 x^{2} e^{x} + 4 x^{2} e^{4} e^{x} - 8 x^{2} + 8 x^{2} e^{4} + \left (- 4 x^{2} e^{x} - 8 x^{2}\right ) e^{x^{2} - 5}}{\left (2 - 2 e^{4}\right ) e^{x^{2} - 5} + e^{2 x^{2} - 10} - 2 e^{4} + 1 + e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x+2)*exp(x)**2+(8*x+8)*exp(x)+8)*exp(x**2-5)**3+(((-12*x-6)*exp(4)+12*x+6)*exp(x)**2+((-24*x-24
)*exp(4)+8*x**3-4*x**2+16*x+24)*exp(x)-24*exp(4)+16*x**3-16*x+24)*exp(x**2-5)**2+(((12*x+6)*exp(4)**2+(-24*x-1
2)*exp(4)+12*x+6)*exp(x)**2+((24*x+24)*exp(4)**2+(-8*x**3+8*x**2-32*x-48)*exp(4)+8*x**3-8*x**2+8*x+24)*exp(x)+
24*exp(4)**2+(-16*x**3+32*x-48)*exp(4)-8*x**4+16*x**3+6*x**2-32*x+24)*exp(x**2-5)+((-4*x-2)*exp(4)**3+(12*x+6)
*exp(4)**2+(-12*x-6)*exp(4)+4*x+2)*exp(x)**2+((-8*x-8)*exp(4)**3+(-4*x**2+16*x+24)*exp(4)**2+(8*x**2-8*x-24)*e
xp(4)-4*x**2+8)*exp(x)-8*exp(4)**3+(-16*x+24)*exp(4)**2+(-6*x**2+32*x-24)*exp(4)+6*x**2-16*x+8)/(exp(x**2-5)**
3+(-3*exp(4)+3)*exp(x**2-5)**2+(3*exp(4)**2-6*exp(4)+3)*exp(x**2-5)-exp(4)**3+3*exp(4)**2-3*exp(4)+1),x)

[Out]

2*x*exp(2*x) + 8*x*exp(x) + 8*x + (2*x**3 - 4*x**2*exp(x) + 4*x**2*exp(4)*exp(x) - 8*x**2 + 8*x**2*exp(4) + (-
4*x**2*exp(x) - 8*x**2)*exp(x**2 - 5))/((2 - 2*exp(4))*exp(x**2 - 5) + exp(2*x**2 - 10) - 2*exp(4) + 1 + exp(8
))

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