\(\int \frac {-80 e^2-64 x+e^x (8 x^3+4 x^4+e^2 (4 x^2+4 x^3))+(144 e^2+136 x+e^x (-6 x^3-4 x^4+e^2 (-4 x^2-4 x^3))) \log (e^2+x)+(-100 e^2-100 x+e^x (x^3+x^4+e^2 (x^2+x^3))) \log ^2(e^2+x)+(32 e^2+32 x) \log ^3(e^2+x)+(-4 e^2-4 x) \log ^4(e^2+x)}{400 e^2+400 x+e^x (40 e^2 x^2+40 x^3)+e^{2 x} (e^2 x^4+x^5)+(-640 e^2-640 x+e^x (-32 e^2 x^2-32 x^3)) \log (e^2+x)+(416 e^2+416 x+e^x (8 e^2 x^2+8 x^3)) \log ^2(e^2+x)+(-128 e^2-128 x) \log ^3(e^2+x)+(16 e^2+16 x) \log ^4(e^2+x)} \, dx\) [6464]

Optimal result

Integrand size = 314, antiderivative size = 31 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\frac {x}{-4+\frac {\left (-e^x-\frac {4}{x^2}\right ) x^2}{\left (-2+\log \left (e^2+x\right )\right )^2}} \]

[Out]

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

Rubi [F]

\[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx \]

[In]

Int[(-80*E^2 - 64*x + E^x*(8*x^3 + 4*x^4 + E^2*(4*x^2 + 4*x^3)) + (144*E^2 + 136*x + E^x*(-6*x^3 - 4*x^4 + E^2
*(-4*x^2 - 4*x^3)))*Log[E^2 + x] + (-100*E^2 - 100*x + E^x*(x^3 + x^4 + E^2*(x^2 + x^3)))*Log[E^2 + x]^2 + (32
*E^2 + 32*x)*Log[E^2 + x]^3 + (-4*E^2 - 4*x)*Log[E^2 + x]^4)/(400*E^2 + 400*x + E^x*(40*E^2*x^2 + 40*x^3) + E^
(2*x)*(E^2*x^4 + x^5) + (-640*E^2 - 640*x + E^x*(-32*E^2*x^2 - 32*x^3))*Log[E^2 + x] + (416*E^2 + 416*x + E^x*
(8*E^2*x^2 + 8*x^3))*Log[E^2 + x]^2 + (-128*E^2 - 128*x)*Log[E^2 + x]^3 + (16*E^2 + 16*x)*Log[E^2 + x]^4),x]

[Out]

80*E^2*Defer[Int][(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^(-2), x] - 16*(14 + 5*E^2)*Defer[Int][(2
0 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^(-2), x] - 80*Defer[Int][x/(20 + E^x*x^2 - 16*Log[E^2 + x] +
4*Log[E^2 + x]^2)^2, x] - 160*E^2*Defer[Int][1/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)
^2), x] - 80*E^4*Defer[Int][1/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x] + 16*E^2*(
14 + 5*E^2)*Defer[Int][1/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x] - 144*E^2*Defer
[Int][Log[E^2 + x]/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2, x] + 48*(8 + 3*E^2)*Defer[Int][Log[E
^2 + x]/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2, x] + 144*Defer[Int][(x*Log[E^2 + x])/(20 + E^x*
x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2, x] + 288*E^2*Defer[Int][Log[E^2 + x]/((E^2 + x)*(20 + E^x*x^2 - 1
6*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x] + 144*E^4*Defer[Int][Log[E^2 + x]/((E^2 + x)*(20 + E^x*x^2 - 16*Log[
E^2 + x] + 4*Log[E^2 + x]^2)^2), x] - 48*E^2*(8 + 3*E^2)*Defer[Int][Log[E^2 + x]/((E^2 + x)*(20 + E^x*x^2 - 16
*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x] + 100*E^2*Defer[Int][Log[E^2 + x]^2/(20 + E^x*x^2 - 16*Log[E^2 + x] +
4*Log[E^2 + x]^2)^2, x] - 4*(62 + 25*E^2)*Defer[Int][Log[E^2 + x]^2/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E
^2 + x]^2)^2, x] - 100*Defer[Int][(x*Log[E^2 + x]^2)/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2, x]
- 200*E^2*Defer[Int][Log[E^2 + x]^2/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x] - 1
00*E^4*Defer[Int][Log[E^2 + x]^2/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x] + 4*E^2
*(62 + 25*E^2)*Defer[Int][Log[E^2 + x]^2/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x]
- 32*E^2*Defer[Int][Log[E^2 + x]^3/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2, x] + 8*(9 + 4*E^2)*
Defer[Int][Log[E^2 + x]^3/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2, x] + 32*Defer[Int][(x*Log[E^2
+ x]^3)/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2, x] + 64*E^2*Defer[Int][Log[E^2 + x]^3/((E^2 +
x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x] + 32*E^4*Defer[Int][Log[E^2 + x]^3/((E^2 + x)*(2
0 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x] - 8*E^2*(9 + 4*E^2)*Defer[Int][Log[E^2 + x]^3/((E^2 +
x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2), x] + 4*E^2*Defer[Int][Log[E^2 + x]^4/(20 + E^x*x^2
- 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2, x] - 4*(2 + E^2)*Defer[Int][Log[E^2 + x]^4/(20 + E^x*x^2 - 16*Log[E^
2 + x] + 4*Log[E^2 + x]^2)^2, x] - 4*Defer[Int][(x*Log[E^2 + x]^4)/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2
+ x]^2)^2, x] - 8*E^2*Defer[Int][Log[E^2 + x]^4/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2
)^2), x] - 4*E^4*Defer[Int][Log[E^2 + x]^4/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^2),
x] + 4*E^2*(2 + E^2)*Defer[Int][Log[E^2 + x]^4/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^
2), x] - 4*E^2*Defer[Int][(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^(-1), x] + 4*(2 + E^2)*Defer[Int
][(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)^(-1), x] + 4*Defer[Int][x/(20 + E^x*x^2 - 16*Log[E^2 + x
] + 4*Log[E^2 + x]^2), x] + 4*E^2*Defer[Int][1/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2))
, x] + 4*E^4*Defer[Int][1/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)), x] - 4*E^2*(2 + E^2
)*Defer[Int][1/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)), x] + 4*E^2*Defer[Int][Log[E^2
+ x]/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2), x] - 2*(3 + 2*E^2)*Defer[Int][Log[E^2 + x]/(20 + E^x
*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2), x] - 4*Defer[Int][(x*Log[E^2 + x])/(20 + E^x*x^2 - 16*Log[E^2 + x]
+ 4*Log[E^2 + x]^2), x] - 4*E^2*Defer[Int][Log[E^2 + x]/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^
2 + x]^2)), x] - 4*E^4*Defer[Int][Log[E^2 + x]/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2))
, x] + 2*E^2*(3 + 2*E^2)*Defer[Int][Log[E^2 + x]/((E^2 + x)*(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2
)), x] - E^2*Defer[Int][Log[E^2 + x]^2/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2), x] + (1 + E^2)*Def
er[Int][Log[E^2 + x]^2/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2), x] + Defer[Int][(x*Log[E^2 + x]^2)
/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2), x] + E^2*Defer[Int][Log[E^2 + x]^2/((E^2 + x)*(20 + E^x*
x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2)), x] + E^4*Defer[Int][Log[E^2 + x]^2/((E^2 + x)*(20 + E^x*x^2 - 16*L
og[E^2 + x] + 4*Log[E^2 + x]^2)), x] - E^2*(1 + E^2)*Defer[Int][Log[E^2 + x]^2/((E^2 + x)*(20 + E^x*x^2 - 16*L
og[E^2 + x] + 4*Log[E^2 + x]^2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (2-\log \left (e^2+x\right )\right ) \left (-40 e^2-32 x+2 e^{2+x} x^2 (1+x)+2 e^x x^3 (2+x)-\left (e^2+x\right ) \left (-52+e^x x^2 (1+x)\right ) \log \left (e^2+x\right )-24 \left (e^2+x\right ) \log ^2\left (e^2+x\right )+4 \left (e^2+x\right ) \log ^3\left (e^2+x\right )\right )}{\left (e^2+x\right ) \left (20+e^x x^2-16 \log \left (e^2+x\right )+4 \log ^2\left (e^2+x\right )\right )^2} \, dx \\ & = \int \left (\frac {\left (2-\log \left (e^2+x\right )\right ) \left (2 e^2+4 \left (1+\frac {e^2}{2}\right ) x+2 x^2-e^2 \log \left (e^2+x\right )-\left (1+e^2\right ) x \log \left (e^2+x\right )-x^2 \log \left (e^2+x\right )\right )}{\left (e^2+x\right ) \left (20+e^x x^2-16 \log \left (e^2+x\right )+4 \log ^2\left (e^2+x\right )\right )}+\frac {4 \left (2-\log \left (e^2+x\right )\right )^2 \left (-10 e^2-14 \left (1+\frac {5 e^2}{14}\right ) x-5 x^2+8 e^2 \log \left (e^2+x\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (e^2+x\right )+4 x^2 \log \left (e^2+x\right )-2 e^2 \log ^2\left (e^2+x\right )-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (e^2+x\right )-x^2 \log ^2\left (e^2+x\right )\right )}{\left (e^2+x\right ) \left (20+e^x x^2-16 \log \left (e^2+x\right )+4 \log ^2\left (e^2+x\right )\right )^2}\right ) \, dx \\ & = 4 \int \frac {\left (2-\log \left (e^2+x\right )\right )^2 \left (-10 e^2-14 \left (1+\frac {5 e^2}{14}\right ) x-5 x^2+8 e^2 \log \left (e^2+x\right )+10 \left (1+\frac {2 e^2}{5}\right ) x \log \left (e^2+x\right )+4 x^2 \log \left (e^2+x\right )-2 e^2 \log ^2\left (e^2+x\right )-2 \left (1+\frac {e^2}{2}\right ) x \log ^2\left (e^2+x\right )-x^2 \log ^2\left (e^2+x\right )\right )}{\left (e^2+x\right ) \left (20+e^x x^2-16 \log \left (e^2+x\right )+4 \log ^2\left (e^2+x\right )\right )^2} \, dx+\int \frac {\left (2-\log \left (e^2+x\right )\right ) \left (2 e^2+4 \left (1+\frac {e^2}{2}\right ) x+2 x^2-e^2 \log \left (e^2+x\right )-\left (1+e^2\right ) x \log \left (e^2+x\right )-x^2 \log \left (e^2+x\right )\right )}{\left (e^2+x\right ) \left (20+e^x x^2-16 \log \left (e^2+x\right )+4 \log ^2\left (e^2+x\right )\right )} \, dx \\ & = 4 \int \frac {\left (2-\log \left (e^2+x\right )\right )^2 \left (-5 e^2 (2+x)-x (14+5 x)+2 \left (2 e^2 (2+x)+x (5+2 x)\right ) \log \left (e^2+x\right )-(2+x) \left (e^2+x\right ) \log ^2\left (e^2+x\right )\right )}{\left (e^2+x\right ) \left (20+e^x x^2-16 \log \left (e^2+x\right )+4 \log ^2\left (e^2+x\right )\right )^2} \, dx+\int \frac {\left (2-\log \left (e^2+x\right )\right ) \left (2 \left (e^2 (1+x)+x (2+x)\right )-(1+x) \left (e^2+x\right ) \log \left (e^2+x\right )\right )}{\left (e^2+x\right ) \left (20+e^x x^2-16 \log \left (e^2+x\right )+4 \log ^2\left (e^2+x\right )\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=-\frac {x \left (-2+\log \left (e^2+x\right )\right )^2}{20+e^x x^2-16 \log \left (e^2+x\right )+4 \log ^2\left (e^2+x\right )} \]

[In]

Integrate[(-80*E^2 - 64*x + E^x*(8*x^3 + 4*x^4 + E^2*(4*x^2 + 4*x^3)) + (144*E^2 + 136*x + E^x*(-6*x^3 - 4*x^4
+ E^2*(-4*x^2 - 4*x^3)))*Log[E^2 + x] + (-100*E^2 - 100*x + E^x*(x^3 + x^4 + E^2*(x^2 + x^3)))*Log[E^2 + x]^2
+ (32*E^2 + 32*x)*Log[E^2 + x]^3 + (-4*E^2 - 4*x)*Log[E^2 + x]^4)/(400*E^2 + 400*x + E^x*(40*E^2*x^2 + 40*x^3
) + E^(2*x)*(E^2*x^4 + x^5) + (-640*E^2 - 640*x + E^x*(-32*E^2*x^2 - 32*x^3))*Log[E^2 + x] + (416*E^2 + 416*x
+ E^x*(8*E^2*x^2 + 8*x^3))*Log[E^2 + x]^2 + (-128*E^2 - 128*x)*Log[E^2 + x]^3 + (16*E^2 + 16*x)*Log[E^2 + x]^4
),x]

[Out]

-((x*(-2 + Log[E^2 + x])^2)/(20 + E^x*x^2 - 16*Log[E^2 + x] + 4*Log[E^2 + x]^2))

Maple [A] (verified)

Time = 1.78 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35

method	result	size
risch	\(-\frac {x}{4}+\frac {x \left ({\mathrm e}^{x} x^{2}+4\right )}{4 \,{\mathrm e}^{x} x^{2}+16 \ln \left (x +{\mathrm e}^{2}\right )^{2}-64 \ln \left (x +{\mathrm e}^{2}\right )+80}\)	\(42\)
parallelrisch	\(\frac {2 x^{2} {\mathrm e}^{2} {\mathrm e}^{x}+8 \ln \left (x +{\mathrm e}^{2}\right )^{2} {\mathrm e}^{2}-4 \ln \left (x +{\mathrm e}^{2}\right )^{2} x -32 \ln \left (x +{\mathrm e}^{2}\right ) {\mathrm e}^{2}+16 \ln \left (x +{\mathrm e}^{2}\right ) x +40 \,{\mathrm e}^{2}-16 x}{4 \,{\mathrm e}^{x} x^{2}+16 \ln \left (x +{\mathrm e}^{2}\right )^{2}-64 \ln \left (x +{\mathrm e}^{2}\right )+80}\)	\(84\)

[In]

int(((-4*exp(2)-4*x)*ln(x+exp(2))^4+(32*exp(2)+32*x)*ln(x+exp(2))^3+(((x^3+x^2)*exp(2)+x^4+x^3)*exp(x)-100*exp
(2)-100*x)*ln(x+exp(2))^2+(((-4*x^3-4*x^2)*exp(2)-4*x^4-6*x^3)*exp(x)+144*exp(2)+136*x)*ln(x+exp(2))+((4*x^3+4
*x^2)*exp(2)+4*x^4+8*x^3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x)*ln(x+exp(2))^4+(-128*exp(2)-128*x)*ln(x+exp
(2))^3+((8*x^2*exp(2)+8*x^3)*exp(x)+416*exp(2)+416*x)*ln(x+exp(2))^2+((-32*x^2*exp(2)-32*x^3)*exp(x)-640*exp(2
)-640*x)*ln(x+exp(2))+(x^4*exp(2)+x^5)*exp(x)^2+(40*x^2*exp(2)+40*x^3)*exp(x)+400*exp(2)+400*x),x,method=_RETU
RNVERBOSE)

[Out]

-1/4*x+1/4*x*(exp(x)*x^2+4)/(exp(x)*x^2+4*ln(x+exp(2))^2-16*ln(x+exp(2))+20)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=-\frac {x \log \left (x + e^{2}\right )^{2} - 4 \, x \log \left (x + e^{2}\right ) + 4 \, x}{x^{2} e^{x} + 4 \, \log \left (x + e^{2}\right )^{2} - 16 \, \log \left (x + e^{2}\right ) + 20} \]

[In]

integrate(((-4*exp(2)-4*x)*log(x+exp(2))^4+(32*exp(2)+32*x)*log(x+exp(2))^3+(((x^3+x^2)*exp(2)+x^4+x^3)*exp(x)
-100*exp(2)-100*x)*log(x+exp(2))^2+(((-4*x^3-4*x^2)*exp(2)-4*x^4-6*x^3)*exp(x)+144*exp(2)+136*x)*log(x+exp(2))
+((4*x^3+4*x^2)*exp(2)+4*x^4+8*x^3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x)*log(x+exp(2))^4+(-128*exp(2)-128*
x)*log(x+exp(2))^3+((8*x^2*exp(2)+8*x^3)*exp(x)+416*exp(2)+416*x)*log(x+exp(2))^2+((-32*x^2*exp(2)-32*x^3)*exp
(x)-640*exp(2)-640*x)*log(x+exp(2))+(x^4*exp(2)+x^5)*exp(x)^2+(40*x^2*exp(2)+40*x^3)*exp(x)+400*exp(2)+400*x),
x, algorithm="fricas")

[Out]

-(x*log(x + e^2)^2 - 4*x*log(x + e^2) + 4*x)/(x^2*e^x + 4*log(x + e^2)^2 - 16*log(x + e^2) + 20)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\frac {- x \log {\left (x + e^{2} \right )}^{2} + 4 x \log {\left (x + e^{2} \right )} - 4 x}{x^{2} e^{x} + 4 \log {\left (x + e^{2} \right )}^{2} - 16 \log {\left (x + e^{2} \right )} + 20} \]

[In]

integrate(((-4*exp(2)-4*x)*ln(x+exp(2))**4+(32*exp(2)+32*x)*ln(x+exp(2))**3+(((x**3+x**2)*exp(2)+x**4+x**3)*ex
p(x)-100*exp(2)-100*x)*ln(x+exp(2))**2+(((-4*x**3-4*x**2)*exp(2)-4*x**4-6*x**3)*exp(x)+144*exp(2)+136*x)*ln(x+
exp(2))+((4*x**3+4*x**2)*exp(2)+4*x**4+8*x**3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x)*ln(x+exp(2))**4+(-128*
exp(2)-128*x)*ln(x+exp(2))**3+((8*x**2*exp(2)+8*x**3)*exp(x)+416*exp(2)+416*x)*ln(x+exp(2))**2+((-32*x**2*exp(
2)-32*x**3)*exp(x)-640*exp(2)-640*x)*ln(x+exp(2))+(x**4*exp(2)+x**5)*exp(x)**2+(40*x**2*exp(2)+40*x**3)*exp(x)
+400*exp(2)+400*x),x)

[Out]

(-x*log(x + exp(2))**2 + 4*x*log(x + exp(2)) - 4*x)/(x**2*exp(x) + 4*log(x + exp(2))**2 - 16*log(x + exp(2)) +
20)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=-\frac {x \log \left (x + e^{2}\right )^{2} - 4 \, x \log \left (x + e^{2}\right ) + 4 \, x}{x^{2} e^{x} + 4 \, \log \left (x + e^{2}\right )^{2} - 16 \, \log \left (x + e^{2}\right ) + 20} \]

[In]

integrate(((-4*exp(2)-4*x)*log(x+exp(2))^4+(32*exp(2)+32*x)*log(x+exp(2))^3+(((x^3+x^2)*exp(2)+x^4+x^3)*exp(x)
-100*exp(2)-100*x)*log(x+exp(2))^2+(((-4*x^3-4*x^2)*exp(2)-4*x^4-6*x^3)*exp(x)+144*exp(2)+136*x)*log(x+exp(2))
+((4*x^3+4*x^2)*exp(2)+4*x^4+8*x^3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x)*log(x+exp(2))^4+(-128*exp(2)-128*
x)*log(x+exp(2))^3+((8*x^2*exp(2)+8*x^3)*exp(x)+416*exp(2)+416*x)*log(x+exp(2))^2+((-32*x^2*exp(2)-32*x^3)*exp
(x)-640*exp(2)-640*x)*log(x+exp(2))+(x^4*exp(2)+x^5)*exp(x)^2+(40*x^2*exp(2)+40*x^3)*exp(x)+400*exp(2)+400*x),
x, algorithm="maxima")

[Out]

-(x*log(x + e^2)^2 - 4*x*log(x + e^2) + 4*x)/(x^2*e^x + 4*log(x + e^2)^2 - 16*log(x + e^2) + 20)

Giac [F(-2)]

Exception generated. \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((-4*exp(2)-4*x)*log(x+exp(2))^4+(32*exp(2)+32*x)*log(x+exp(2))^3+(((x^3+x^2)*exp(2)+x^4+x^3)*exp(x)
-100*exp(2)-100*x)*log(x+exp(2))^2+(((-4*x^3-4*x^2)*exp(2)-4*x^4-6*x^3)*exp(x)+144*exp(2)+136*x)*log(x+exp(2))
+((4*x^3+4*x^2)*exp(2)+4*x^4+8*x^3)*exp(x)-80*exp(2)-64*x)/((16*exp(2)+16*x)*log(x+exp(2))^4+(-128*exp(2)-128*
x)*log(x+exp(2))^3+((8*x^2*exp(2)+8*x^3)*exp(x)+416*exp(2)+416*x)*log(x+exp(2))^2+((-32*x^2*exp(2)-32*x^3)*exp
(x)-640*exp(2)-640*x)*log(x+exp(2))+(x^4*exp(2)+x^5)*exp(x)^2+(40*x^2*exp(2)+40*x^3)*exp(x)+400*exp(2)+400*x),
x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Not invertible Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 13.45 (sec) , antiderivative size = 454, normalized size of antiderivative = 14.65 \[ \int \frac {-80 e^2-64 x+e^x \left (8 x^3+4 x^4+e^2 \left (4 x^2+4 x^3\right )\right )+\left (144 e^2+136 x+e^x \left (-6 x^3-4 x^4+e^2 \left (-4 x^2-4 x^3\right )\right )\right ) \log \left (e^2+x\right )+\left (-100 e^2-100 x+e^x \left (x^3+x^4+e^2 \left (x^2+x^3\right )\right )\right ) \log ^2\left (e^2+x\right )+\left (32 e^2+32 x\right ) \log ^3\left (e^2+x\right )+\left (-4 e^2-4 x\right ) \log ^4\left (e^2+x\right )}{400 e^2+400 x+e^x \left (40 e^2 x^2+40 x^3\right )+e^{2 x} \left (e^2 x^4+x^5\right )+\left (-640 e^2-640 x+e^x \left (-32 e^2 x^2-32 x^3\right )\right ) \log \left (e^2+x\right )+\left (416 e^2+416 x+e^x \left (8 e^2 x^2+8 x^3\right )\right ) \log ^2\left (e^2+x\right )+\left (-128 e^2-128 x\right ) \log ^3\left (e^2+x\right )+\left (16 e^2+16 x\right ) \log ^4\left (e^2+x\right )} \, dx=\frac {32\,x^4\,{\mathrm {e}}^x+64\,x\,{\mathrm {e}}^2+8\,x^6\,{\mathrm {e}}^{2\,x}+4\,x^7\,{\mathrm {e}}^{2\,x}+x^8\,{\mathrm {e}}^{2\,x}+x^8\,{\mathrm {e}}^{3\,x}+x^9\,{\mathrm {e}}^{3\,x}+\frac {x^{10}\,{\mathrm {e}}^{3\,x}}{4}+32\,x^3\,{\mathrm {e}}^{x+2}+16\,x^5\,{\mathrm {e}}^{2\,x+2}+12\,x^4\,{\mathrm {e}}^{2\,x+4}+12\,x^6\,{\mathrm {e}}^{2\,x+2}+4\,x^3\,{\mathrm {e}}^{2\,x+6}+12\,x^5\,{\mathrm {e}}^{2\,x+4}+3\,x^7\,{\mathrm {e}}^{2\,x+2}+4\,x^4\,{\mathrm {e}}^{2\,x+6}+3\,x^6\,{\mathrm {e}}^{2\,x+4}+3\,x^7\,{\mathrm {e}}^{3\,x+2}+x^5\,{\mathrm {e}}^{2\,x+6}+3\,x^6\,{\mathrm {e}}^{3\,x+4}+3\,x^8\,{\mathrm {e}}^{3\,x+2}+x^5\,{\mathrm {e}}^{3\,x+6}+3\,x^7\,{\mathrm {e}}^{3\,x+4}+\frac {3\,x^9\,{\mathrm {e}}^{3\,x+2}}{4}+x^6\,{\mathrm {e}}^{3\,x+6}+\frac {3\,x^8\,{\mathrm {e}}^{3\,x+4}}{4}+\frac {x^7\,{\mathrm {e}}^{3\,x+6}}{4}+64\,x^2}{\left (4\,{\ln \left (x+{\mathrm {e}}^2\right )}^2-16\,\ln \left (x+{\mathrm {e}}^2\right )+x^2\,{\mathrm {e}}^x+20\right )\,\left (64\,x+64\,{\mathrm {e}}^2+16\,x^3\,{\mathrm {e}}^x+4\,x^5\,{\mathrm {e}}^{2\,x}+4\,x^6\,{\mathrm {e}}^{2\,x}+x^7\,{\mathrm {e}}^{2\,x}+16\,x^2\,{\mathrm {e}}^{x+2}+12\,x^4\,{\mathrm {e}}^{2\,x+2}+12\,x^3\,{\mathrm {e}}^{2\,x+4}+12\,x^5\,{\mathrm {e}}^{2\,x+2}+4\,x^2\,{\mathrm {e}}^{2\,x+6}+12\,x^4\,{\mathrm {e}}^{2\,x+4}+3\,x^6\,{\mathrm {e}}^{2\,x+2}+4\,x^3\,{\mathrm {e}}^{2\,x+6}+3\,x^5\,{\mathrm {e}}^{2\,x+4}+x^4\,{\mathrm {e}}^{2\,x+6}\right )}-\frac {x}{4} \]

[In]

int(-(64*x + 80*exp(2) + log(x + exp(2))^4*(4*x + 4*exp(2)) - log(x + exp(2))^3*(32*x + 32*exp(2)) + log(x + e
xp(2))^2*(100*x + 100*exp(2) - exp(x)*(exp(2)*(x^2 + x^3) + x^3 + x^4)) - exp(x)*(exp(2)*(4*x^2 + 4*x^3) + 8*x
^3 + 4*x^4) - log(x + exp(2))*(136*x + 144*exp(2) - exp(x)*(exp(2)*(4*x^2 + 4*x^3) + 6*x^3 + 4*x^4)))/(400*x +
400*exp(2) + log(x + exp(2))^4*(16*x + 16*exp(2)) - log(x + exp(2))^3*(128*x + 128*exp(2)) + exp(2*x)*(x^4*ex
p(2) + x^5) + exp(x)*(40*x^2*exp(2) + 40*x^3) - log(x + exp(2))*(640*x + 640*exp(2) + exp(x)*(32*x^2*exp(2) +
32*x^3)) + log(x + exp(2))^2*(416*x + 416*exp(2) + exp(x)*(8*x^2*exp(2) + 8*x^3))),x)

[Out]

(32*x^4*exp(x) + 64*x*exp(2) + 8*x^6*exp(2*x) + 4*x^7*exp(2*x) + x^8*exp(2*x) + x^8*exp(3*x) + x^9*exp(3*x) +
(x^10*exp(3*x))/4 + 32*x^3*exp(x + 2) + 16*x^5*exp(2*x + 2) + 12*x^4*exp(2*x + 4) + 12*x^6*exp(2*x + 2) + 4*x^
3*exp(2*x + 6) + 12*x^5*exp(2*x + 4) + 3*x^7*exp(2*x + 2) + 4*x^4*exp(2*x + 6) + 3*x^6*exp(2*x + 4) + 3*x^7*ex
p(3*x + 2) + x^5*exp(2*x + 6) + 3*x^6*exp(3*x + 4) + 3*x^8*exp(3*x + 2) + x^5*exp(3*x + 6) + 3*x^7*exp(3*x + 4
) + (3*x^9*exp(3*x + 2))/4 + x^6*exp(3*x + 6) + (3*x^8*exp(3*x + 4))/4 + (x^7*exp(3*x + 6))/4 + 64*x^2)/((4*lo
g(x + exp(2))^2 - 16*log(x + exp(2)) + x^2*exp(x) + 20)*(64*x + 64*exp(2) + 16*x^3*exp(x) + 4*x^5*exp(2*x) + 4
*x^6*exp(2*x) + x^7*exp(2*x) + 16*x^2*exp(x + 2) + 12*x^4*exp(2*x + 2) + 12*x^3*exp(2*x + 4) + 12*x^5*exp(2*x
+ 2) + 4*x^2*exp(2*x + 6) + 12*x^4*exp(2*x + 4) + 3*x^6*exp(2*x + 2) + 4*x^3*exp(2*x + 6) + 3*x^5*exp(2*x + 4)
+ x^4*exp(2*x + 6))) - x/4