3.66.39 \(\int \frac {-x^3-2 x^4+e^x (-8 x^3-2 e^6 x^3-2 x^4)+e^{2 x} (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 (12-10 x-4 x^2))+(x^3+x^4+e^x (6 x^2+2 e^6 x^2+2 x^3)+e^{2 x} (9+e^{12}+6 x+x^2+e^6 (6+2 x))) \log (\frac {x^3+x^4+e^x (6 x^2+2 e^6 x^2+2 x^3)+e^{2 x} (9+e^{12}+6 x+x^2+e^6 (6+2 x))}{x^2})}{(x^3+x^4+e^x (6 x^2+2 e^6 x^2+2 x^3)+e^{2 x} (9+e^{12}+6 x+x^2+e^6 (6+2 x))) \log ^2(\frac {x^3+x^4+e^x (6 x^2+2 e^6 x^2+2 x^3)+e^{2 x} (9+e^{12}+6 x+x^2+e^6 (6+2 x))}{x^2})} \, dx\)

Optimal. Leaf size=24 \[ \frac {x}{\log \left (x+\left (x+\frac {e^x \left (3+e^6+x\right )}{x}\right )^2\right )} \]

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Rubi [F]  time = 80.42, 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 {-x^3-2 x^4+e^x \left (-8 x^3-2 e^6 x^3-2 x^4\right )+e^{2 x} \left (18+e^{12} (2-2 x)-12 x-12 x^2-2 x^3+e^6 \left (12-10 x-4 x^2\right )\right )+\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log \left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )}{\left (x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )\right ) \log ^2\left (\frac {x^3+x^4+e^x \left (6 x^2+2 e^6 x^2+2 x^3\right )+e^{2 x} \left (9+e^{12}+6 x+x^2+e^6 (6+2 x)\right )}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^3 - 2*x^4 + E^x*(-8*x^3 - 2*E^6*x^3 - 2*x^4) + E^(2*x)*(18 + E^12*(2 - 2*x) - 12*x - 12*x^2 - 2*x^3 +
E^6*(12 - 10*x - 4*x^2)) + (x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*
(6 + 2*x)))*Log[(x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))
/x^2])/((x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))*Log[(x^
3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))/x^2]^2),x]

[Out]

-2*Defer[Int][x/Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2, x] + 2*(3 + E^6)*Defer[I
nt][1/((3 + E^6 + x)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 3*(3 + E^6)^3
*Defer[Int][1/((-9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) - 6*E^(2*x)*(1 + E^6/3)*x - E^(2*x)*x^2 - 6*E^x*(1 + E^6/3)
*x^2 - x^3 - 2*E^x*x^3 - x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(3
+ E^6)^3*(5 + 2*E^6)*Defer[Int][E^x/((-9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) - 6*E^(2*x)*(1 + E^6/3)*x - E^(2*x)*
x^2 - 6*E^x*(1 + E^6/3)*x^2 - x^3 - 2*E^x*x^3 - x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x
)^2)/x^2]^2), x] + (3 + E^6)^2*(7 + 2*E^6)*Defer[Int][x/((-9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) - 6*E^(2*x)*(1 +
E^6/3)*x - E^(2*x)*x^2 - 6*E^x*(1 + E^6/3)*x^2 - x^3 - 2*E^x*x^3 - x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E
^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 4*(3 + E^6)^2*Defer[Int][(E^x*x)/((-9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) -
6*E^(2*x)*(1 + E^6/3)*x - E^(2*x)*x^2 - 6*E^x*(1 + E^6/3)*x^2 - x^3 - 2*E^x*x^3 - x^4)*Log[x + x^2 + 2*E^x*(3
+ E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(3 + E^6)^2*Defer[Int][(E^(6 + x)*x)/((-9*E^(2*x)*(1 +
(E^6*(6 + E^6))/9) - 6*E^(2*x)*(1 + E^6/3)*x - E^(2*x)*x^2 - 6*E^x*(1 + E^6/3)*x^2 - x^3 - 2*E^x*x^3 - x^4)*Lo
g[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 3*(3 + E^6)*Defer[Int][x^2/((-9*E^(2
*x)*(1 + (E^6*(6 + E^6))/9) - 6*E^(2*x)*(1 + E^6/3)*x - E^(2*x)*x^2 - 6*E^x*(1 + E^6/3)*x^2 - x^3 - 2*E^x*x^3
- x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(3 + E^6)*(5 + 2*E^6)*Def
er[Int][(E^x*x^2)/((-9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) - 6*E^(2*x)*(1 + E^6/3)*x - E^(2*x)*x^2 - 6*E^x*(1 + E^
6/3)*x^2 - x^3 - 2*E^x*x^3 - x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] +
2*(3 + E^6)^2*Defer[Int][x^3/((-9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) - 6*E^(2*x)*(1 + E^6/3)*x - E^(2*x)*x^2 - 6*
E^x*(1 + E^6/3)*x^2 - x^3 - 2*E^x*x^3 - x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2
]^2), x] + (7 + 2*E^6)*Defer[Int][x^3/((-9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) - 6*E^(2*x)*(1 + E^6/3)*x - E^(2*x)
*x^2 - 6*E^x*(1 + E^6/3)*x^2 - x^3 - 2*E^x*x^3 - x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 +
x)^2)/x^2]^2), x] + 2*(3 + E^6)*Defer[Int][(E^x*x^3)/((-9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) - 6*E^(2*x)*(1 + E^6
/3)*x - E^(2*x)*x^2 - 6*E^x*(1 + E^6/3)*x^2 - x^3 - 2*E^x*x^3 - x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2
*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(3 + E^6)*Defer[Int][x^4/((-9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) - 6*E^(2*x)
*(1 + E^6/3)*x - E^(2*x)*x^2 - 6*E^x*(1 + E^6/3)*x^2 - x^3 - 2*E^x*x^3 - x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x
) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + (3 + E^6)^3*(7 + 2*E^6)*Defer[Int][1/((9*E^(2*x)*(1 + (E^6*(6 + E^
6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 +
2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 4*(3 + E^6)^3*Defer[Int][E^x/((9*E^(2*x)*(1 + (E
^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[
x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(3 + E^6)^4*Defer[Int][E^x/((9*E^(2*
x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 +
x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(3 + E^6)^3*Defer[Int][E^(
6 + x)/((9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x
^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + (3 + E^6)^4*
(7 + 2*E^6)*Defer[Int][1/((-3 - E^6 - x)*(9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x
)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 +
x)^2)/x^2]^2), x] + 4*(3 + E^6)^4*Defer[Int][E^x/((-3 - E^6 - x)*(9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*
x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 +
x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 3*(3 + E^6)^2*Defer[Int][x/((9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) +
6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3
+ E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] - 2*(3 + E^6)^3*Defer[Int][(E^x*x)/((9*E^(2*x)*(1 + (E^6*(
6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x +
x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(3 + E^6)^2*(5 + 2*E^6)*Defer[Int][(E^x*
x)/((9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 +
2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + (3 + E^6)*(7 + 2
*E^6)*Defer[Int][x^2/((9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 +
E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x]
+ 2*(3 + E^6)^2*Defer[Int][(E^x*x^2)/((9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x
^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)
^2)/x^2]^2), x] + 2*(3 + E^6)*Defer[Int][(E^(6 + x)*x^2)/((9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 +
E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E
^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(3 + E^6)^2*Defer[Int][x^3/((9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(
2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6
+ x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(5 + 2*E^6)*Defer[Int][(E^x*x^3)/((9*E^(2*x)*(1 + (E^6*(6 +
E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2
+ 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*(3 + E^6)*Defer[Int][x^4/((9*E^(2*x)*(1 + (E
^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[
x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*Defer[Int][(E^x*x^4)/((9*E^(2*x)*(1
+ (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*
Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 2*Defer[Int][x^5/((9*E^(2*x)*(1 +
(E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Lo
g[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + 3*(3 + E^6)^4*Defer[Int][1/((3 + E^6
+ x)*(9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3
+ 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] - 2*(3 + E^6)^5*
Defer[Int][E^x/((3 + E^6 + x)*(9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E
^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]
^2), x] + 2*(3 + E^6)^4*(5 + 2*E^6)*Defer[Int][E^x/((3 + E^6 + x)*(9*E^(2*x)*(1 + (E^6*(6 + E^6))/9) + 6*E^(2*
x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*Log[x + x^2 + 2*E^x*(3 + E^6 +
x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] - 2*(3 + E^6)^4*Defer[Int][E^(6 + x)/((3 + E^6 + x)*(9*E^(2*x)*(1
+ (E^6*(6 + E^6))/9) + 6*E^(2*x)*(1 + E^6/3)*x + E^(2*x)*x^2 + 6*E^x*(1 + E^6/3)*x^2 + x^3 + 2*E^x*x^3 + x^4)*
Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^2), x] + Defer[Int][Log[x + x^2 + 2*E^x*(3
+ E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^3-2 x^4-2 e^x x^3 \left (4+e^6+x\right )-2 e^{2 x} \left (-9+e^{12} (-1+x)+6 x+6 x^2+x^3+e^6 \left (-6+5 x+2 x^2\right )\right )+\left (x^3+x^4+2 e^x x^2 \left (3+e^6+x\right )+e^{2 x} \left (3+e^6+x\right )^2\right ) \log \left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )}{\left (x^3+x^4+2 e^x x^2 \left (3+e^6+x\right )+e^{2 x} \left (3+e^6+x\right )^2\right ) \log ^2\left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )} \, dx\\ &=\int \left (\frac {x^2 \left (-36 e^x \left (1+\frac {1}{9} e^6 \left (6+e^6\right )\right )-9 \left (1+\frac {e^6}{3}\right ) x+6 e^{6+x} \left (1+\frac {e^6}{3}\right ) x-7 \left (1+\frac {2 e^6}{7}\right ) x^2+10 e^x \left (1+\frac {2 e^6}{5}\right ) x^2+2 e^x x^3+6 \left (1+\frac {e^6}{3}\right ) x^3+2 x^4\right )}{\left (3+e^6+x\right ) \left (9 e^{2 x} \left (1+\frac {1}{9} e^6 \left (6+e^6\right )\right )+6 e^{2 x} \left (1+\frac {e^6}{3}\right ) x+e^{2 x} x^2+6 e^x \left (1+\frac {e^6}{3}\right ) x^2+x^3+2 e^x x^3+x^4\right ) \log ^2\left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )}+\frac {6 \left (1+\frac {e^6}{3}\right )-6 \left (1+\frac {e^6}{3}\right ) x-2 x^2+3 \left (1+\frac {e^6}{3}\right ) \log \left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )+x \log \left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )}{\left (3+e^6+x\right ) \log ^2\left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )}\right ) \, dx\\ &=\int \frac {x^2 \left (-36 e^x \left (1+\frac {1}{9} e^6 \left (6+e^6\right )\right )-9 \left (1+\frac {e^6}{3}\right ) x+6 e^{6+x} \left (1+\frac {e^6}{3}\right ) x-7 \left (1+\frac {2 e^6}{7}\right ) x^2+10 e^x \left (1+\frac {2 e^6}{5}\right ) x^2+2 e^x x^3+6 \left (1+\frac {e^6}{3}\right ) x^3+2 x^4\right )}{\left (3+e^6+x\right ) \left (9 e^{2 x} \left (1+\frac {1}{9} e^6 \left (6+e^6\right )\right )+6 e^{2 x} \left (1+\frac {e^6}{3}\right ) x+e^{2 x} x^2+6 e^x \left (1+\frac {e^6}{3}\right ) x^2+x^3+2 e^x x^3+x^4\right ) \log ^2\left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )} \, dx+\int \frac {6 \left (1+\frac {e^6}{3}\right )-6 \left (1+\frac {e^6}{3}\right ) x-2 x^2+3 \left (1+\frac {e^6}{3}\right ) \log \left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )+x \log \left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )}{\left (3+e^6+x\right ) \log ^2\left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )} \, dx\\ &=\int \frac {x^2 \left (2 e^{12+x} (-2+x)+e^6 x \left (-3-2 x+2 x^2\right )+e^{6+x} \left (-24+6 x+4 x^2\right )+2 e^x \left (-18+5 x^2+x^3\right )+x \left (-9-7 x+6 x^2+2 x^3\right )\right )}{\left (3+e^6+x\right ) \left (e^{2 (6+x)}+2 e^{6+x} x^2+x^3 (1+x)+2 e^{6+2 x} (3+x)+2 e^x x^2 (3+x)+e^{2 x} (3+x)^2\right ) \log ^2\left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )} \, dx+\int \frac {-\frac {2 \left (-3+e^6 (-1+x)+3 x+x^2\right )}{3+e^6+x}+\log \left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )}{\log ^2\left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 38, normalized size = 1.58 \begin {gather*} \frac {x}{\log \left (x+x^2+2 e^x \left (3+e^6+x\right )+\frac {e^{2 x} \left (3+e^6+x\right )^2}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^3 - 2*x^4 + E^x*(-8*x^3 - 2*E^6*x^3 - 2*x^4) + E^(2*x)*(18 + E^12*(2 - 2*x) - 12*x - 12*x^2 - 2*
x^3 + E^6*(12 - 10*x - 4*x^2)) + (x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2
+ E^6*(6 + 2*x)))*Log[(x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 +
2*x)))/x^2])/((x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))*L
og[(x^3 + x^4 + E^x*(6*x^2 + 2*E^6*x^2 + 2*x^3) + E^(2*x)*(9 + E^12 + 6*x + x^2 + E^6*(6 + 2*x)))/x^2]^2),x]

[Out]

x/Log[x + x^2 + 2*E^x*(3 + E^6 + x) + (E^(2*x)*(3 + E^6 + x)^2)/x^2]

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fricas [B]  time = 0.99, size = 57, normalized size = 2.38 \begin {gather*} \frac {x}{\log \left (\frac {x^{4} + x^{3} + {\left (x^{2} + 2 \, {\left (x + 3\right )} e^{6} + 6 \, x + e^{12} + 9\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{3} + x^{2} e^{6} + 3 \, x^{2}\right )} e^{x}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)*log(((exp(
6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)+((-2*x+2)*exp(6)^2+(-4
*x^2-10*x+12)*exp(6)-2*x^3-12*x^2-12*x+18)*exp(x)^2+(-2*x^3*exp(6)-2*x^4-8*x^3)*exp(x)-2*x^4-x^3)/((exp(6)^2+(
2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/log(((exp(6)^2+(2*x+6)*exp(6)+x^2
+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)^2,x, algorithm="fricas")

[Out]

x/log((x^4 + x^3 + (x^2 + 2*(x + 3)*e^6 + 6*x + e^12 + 9)*e^(2*x) + 2*(x^3 + x^2*e^6 + 3*x^2)*e^x)/x^2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)*log(((exp(
6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)+((-2*x+2)*exp(6)^2+(-4
*x^2-10*x+12)*exp(6)-2*x^3-12*x^2-12*x+18)*exp(x)^2+(-2*x^3*exp(6)-2*x^4-8*x^3)*exp(x)-2*x^4-x^3)/((exp(6)^2+(
2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/log(((exp(6)^2+(2*x+6)*exp(6)+x^2
+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)^2,x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 0.39, size = 617, normalized size = 25.71

method	result	size
risch	\(-\frac {2 i x}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 x +12}+\left (2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{6}+x^{4}+\left (2 \,{\mathrm e}^{x}+1\right ) x^{3}+\left ({\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}\right ) x^{2}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x +12}+\left (2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{6}+x^{4}+\left (2 \,{\mathrm e}^{x}+1\right ) x^{3}+\left ({\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}\right ) x^{2}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}{x^{2}}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 x +12}+\left (2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{6}+x^{4}+\left (2 \,{\mathrm e}^{x}+1\right ) x^{3}+\left ({\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}\right ) x^{2}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )\right ) \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x +12}+\left (2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{6}+x^{4}+\left (2 \,{\mathrm e}^{x}+1\right ) x^{3}+\left ({\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}\right ) x^{2}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}}\right )-\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x +12}+\left (2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{6}+x^{4}+\left (2 \,{\mathrm e}^{x}+1\right ) x^{3}+\left ({\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}\right ) x^{2}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}{x^{2}}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left ({\mathrm e}^{2 x +12}+\left (2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{6}+x^{4}+\left (2 \,{\mathrm e}^{x}+1\right ) x^{3}+\left ({\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}\right ) x^{2}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}{x^{2}}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right )+4 i \ln \relax (x )-2 i \ln \left ({\mathrm e}^{2 x +12}+\left (2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{6}+x^{4}+\left (2 \,{\mathrm e}^{x}+1\right ) x^{3}+\left ({\mathrm e}^{2 x}+6 \,{\mathrm e}^{x}\right ) x^{2}+6 x \,{\mathrm e}^{2 x}+9 \,{\mathrm e}^{2 x}\right )}\)	\(617\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)*ln(((exp(6)^2+(2
*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)+((-2*x+2)*exp(6)^2+(-4*x^2-10
*x+12)*exp(6)-2*x^3-12*x^2-12*x+18)*exp(x)^2+(-2*x^3*exp(6)-2*x^4-8*x^3)*exp(x)-2*x^4-x^3)/((exp(6)^2+(2*x+6)*
exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/ln(((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*
exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)^2,x,method=_RETURNVERBOSE)

[Out]

-2*I*x/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3+Pi*csgn(I*(exp(2*x+12)+(2*exp
(x)*x^2+2*x*exp(2*x)+6*exp(2*x))*exp(6)+x^4+(2*exp(x)+1)*x^3+(exp(2*x)+6*exp(x))*x^2+6*x*exp(2*x)+9*exp(2*x)))
*csgn(I/x^2*(exp(2*x+12)+(2*exp(x)*x^2+2*x*exp(2*x)+6*exp(2*x))*exp(6)+x^4+(2*exp(x)+1)*x^3+(exp(2*x)+6*exp(x)
)*x^2+6*x*exp(2*x)+9*exp(2*x)))^2-Pi*csgn(I*(exp(2*x+12)+(2*exp(x)*x^2+2*x*exp(2*x)+6*exp(2*x))*exp(6)+x^4+(2*
exp(x)+1)*x^3+(exp(2*x)+6*exp(x))*x^2+6*x*exp(2*x)+9*exp(2*x)))*csgn(I/x^2*(exp(2*x+12)+(2*exp(x)*x^2+2*x*exp(
2*x)+6*exp(2*x))*exp(6)+x^4+(2*exp(x)+1)*x^3+(exp(2*x)+6*exp(x))*x^2+6*x*exp(2*x)+9*exp(2*x)))*csgn(I/x^2)-Pi*
csgn(I/x^2*(exp(2*x+12)+(2*exp(x)*x^2+2*x*exp(2*x)+6*exp(2*x))*exp(6)+x^4+(2*exp(x)+1)*x^3+(exp(2*x)+6*exp(x))
*x^2+6*x*exp(2*x)+9*exp(2*x)))^3+Pi*csgn(I/x^2*(exp(2*x+12)+(2*exp(x)*x^2+2*x*exp(2*x)+6*exp(2*x))*exp(6)+x^4+
(2*exp(x)+1)*x^3+(exp(2*x)+6*exp(x))*x^2+6*x*exp(2*x)+9*exp(2*x)))^2*csgn(I/x^2)+4*I*ln(x)-2*I*ln(exp(2*x+12)+
(2*exp(x)*x^2+2*x*exp(2*x)+6*exp(2*x))*exp(6)+x^4+(2*exp(x)+1)*x^3+(exp(2*x)+6*exp(x))*x^2+6*x*exp(2*x)+9*exp(
2*x)))

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maxima [B]  time = 6.47, size = 56, normalized size = 2.33 \begin {gather*} \frac {x}{\log \left (x^{4} + x^{3} + {\left (x^{2} + 2 \, x {\left (e^{6} + 3\right )} + e^{12} + 6 \, e^{6} + 9\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{3} + x^{2} {\left (e^{6} + 3\right )}\right )} e^{x}\right ) - 2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)*log(((exp(
6)^2+(2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)+((-2*x+2)*exp(6)^2+(-4
*x^2-10*x+12)*exp(6)-2*x^3-12*x^2-12*x+18)*exp(x)^2+(-2*x^3*exp(6)-2*x^4-8*x^3)*exp(x)-2*x^4-x^3)/((exp(6)^2+(
2*x+6)*exp(6)+x^2+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/log(((exp(6)^2+(2*x+6)*exp(6)+x^2
+6*x+9)*exp(x)^2+(2*x^2*exp(6)+2*x^3+6*x^2)*exp(x)+x^4+x^3)/x^2)^2,x, algorithm="maxima")

[Out]

x/(log(x^4 + x^3 + (x^2 + 2*x*(e^6 + 3) + e^12 + 6*e^6 + 9)*e^(2*x) + 2*(x^3 + x^2*(e^6 + 3))*e^x) - 2*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {{\mathrm {e}}^{2\,x}\,\left (12\,x+{\mathrm {e}}^6\,\left (4\,x^2+10\,x-12\right )+12\,x^2+2\,x^3+{\mathrm {e}}^{12}\,\left (2\,x-2\right )-18\right )-\ln \left (\frac {{\mathrm {e}}^{2\,x}\,\left (6\,x+{\mathrm {e}}^{12}+x^2+{\mathrm {e}}^6\,\left (2\,x+6\right )+9\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^6+6\,x^2+2\,x^3\right )+x^3+x^4}{x^2}\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (6\,x+{\mathrm {e}}^{12}+x^2+{\mathrm {e}}^6\,\left (2\,x+6\right )+9\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^6+6\,x^2+2\,x^3\right )+x^3+x^4\right )+{\mathrm {e}}^x\,\left (2\,x^3\,{\mathrm {e}}^6+8\,x^3+2\,x^4\right )+x^3+2\,x^4}{{\ln \left (\frac {{\mathrm {e}}^{2\,x}\,\left (6\,x+{\mathrm {e}}^{12}+x^2+{\mathrm {e}}^6\,\left (2\,x+6\right )+9\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^6+6\,x^2+2\,x^3\right )+x^3+x^4}{x^2}\right )}^2\,\left ({\mathrm {e}}^{2\,x}\,\left (6\,x+{\mathrm {e}}^{12}+x^2+{\mathrm {e}}^6\,\left (2\,x+6\right )+9\right )+{\mathrm {e}}^x\,\left (2\,x^2\,{\mathrm {e}}^6+6\,x^2+2\,x^3\right )+x^3+x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(12*x + exp(6)*(10*x + 4*x^2 - 12) + 12*x^2 + 2*x^3 + exp(12)*(2*x - 2) - 18) - log((exp(2*x)*(
6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)/x^2)*(exp(2*x
)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4) + exp(x)*(
2*x^3*exp(6) + 8*x^3 + 2*x^4) + x^3 + 2*x^4)/(log((exp(2*x)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp
(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)/x^2)^2*(exp(2*x)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9)
+ exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)),x)

[Out]

-int((exp(2*x)*(12*x + exp(6)*(10*x + 4*x^2 - 12) + 12*x^2 + 2*x^3 + exp(12)*(2*x - 2) - 18) - log((exp(2*x)*(
6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)/x^2)*(exp(2*x
)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4) + exp(x)*(
2*x^3*exp(6) + 8*x^3 + 2*x^4) + x^3 + 2*x^4)/(log((exp(2*x)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9) + exp
(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)/x^2)^2*(exp(2*x)*(6*x + exp(12) + x^2 + exp(6)*(2*x + 6) + 9)
+ exp(x)*(2*x^2*exp(6) + 6*x^2 + 2*x^3) + x^3 + x^4)), x)

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sympy [B]  time = 1.43, size = 60, normalized size = 2.50 \begin {gather*} \frac {x}{\log {\left (\frac {x^{4} + x^{3} + \left (2 x^{3} + 6 x^{2} + 2 x^{2} e^{6}\right ) e^{x} + \left (x^{2} + 6 x + \left (2 x + 6\right ) e^{6} + 9 + e^{12}\right ) e^{2 x}}{x^{2}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((exp(6)**2+(2*x+6)*exp(6)+x**2+6*x+9)*exp(x)**2+(2*x**2*exp(6)+2*x**3+6*x**2)*exp(x)+x**4+x**3)*ln
(((exp(6)**2+(2*x+6)*exp(6)+x**2+6*x+9)*exp(x)**2+(2*x**2*exp(6)+2*x**3+6*x**2)*exp(x)+x**4+x**3)/x**2)+((-2*x
+2)*exp(6)**2+(-4*x**2-10*x+12)*exp(6)-2*x**3-12*x**2-12*x+18)*exp(x)**2+(-2*x**3*exp(6)-2*x**4-8*x**3)*exp(x)
-2*x**4-x**3)/((exp(6)**2+(2*x+6)*exp(6)+x**2+6*x+9)*exp(x)**2+(2*x**2*exp(6)+2*x**3+6*x**2)*exp(x)+x**4+x**3)
/ln(((exp(6)**2+(2*x+6)*exp(6)+x**2+6*x+9)*exp(x)**2+(2*x**2*exp(6)+2*x**3+6*x**2)*exp(x)+x**4+x**3)/x**2)**2,
x)

[Out]

x/log((x**4 + x**3 + (2*x**3 + 6*x**2 + 2*x**2*exp(6))*exp(x) + (x**2 + 6*x + (2*x + 6)*exp(6) + 9 + exp(12))*
exp(2*x))/x**2)

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