∫ 8x−16x3−16x4+e8(−48x3−48x4)+e12(16x3+16x4)+e4(−8x+48x3+48x4)+(−4+24x2+16x3−8x4+e8(24x2−24x4)+e12(8x3+8x4)+e4(−48x2−24x3+24x4)) log(x)+(−12x+12x3+e4(12x−12x2−24x3)+e8(12x2+12x3)) log2(x)+(2−4x−6x2+e4(6x+6x2)) log3(x)+(1+x) log4(x)+(16−32x+32e4x+(8−16x+16e4x) log(x)) log(2+log(x))____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ −16x4+48e4x4−48e8x4+16e12x4+(24x3−8x4+8e12x4+e8(24x3−24x4)+e4(−48x3+24x4)) log(x)+(−12x2+12x3+12e8x3+e4(12x2−24x3)) log2(x)+(2x−6x2+6e4x2) log3(x)+x log4(x) dx [2277]

3.23.77 \(\int \frac {8 x-16 x^3-16 x^4+e^8 (-48 x^3-48 x^4)+e^{12} (16 x^3+16 x^4)+e^4 (-8 x+48 x^3+48 x^4)+(-4+24 x^2+16 x^3-8 x^4+e^8 (24 x^2-24 x^4)+e^{12} (8 x^3+8 x^4)+e^4 (-48 x^2-24 x^3+24 x^4)) \log (x)+(-12 x+12 x^3+e^4 (12 x-12 x^2-24 x^3)+e^8 (12 x^2+12 x^3)) \log ^2(x)+(2-4 x-6 x^2+e^4 (6 x+6 x^2)) \log ^3(x)+(1+x) \log ^4(x)+(16-32 x+32 e^4 x+(8-16 x+16 e^4 x) \log (x)) \log (2+\log (x))}{-16 x^4+48 e^4 x^4-48 e^8 x^4+16 e^{12} x^4+(24 x^3-8 x^4+8 e^{12} x^4+e^8 (24 x^3-24 x^4)+e^4 (-48 x^3+24 x^4)) \log (x)+(-12 x^2+12 x^3+12 e^8 x^3+e^4 (12 x^2-24 x^3)) \log ^2(x)+(2 x-6 x^2+6 e^4 x^2) \log ^3(x)+x \log ^4(x)} \, dx\) [2277]

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

[Out]

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

________________________________________________________________________________________

Rubi [F]
time = 23.73, 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 x-16 x^3-16 x^4+e^8 \left (-48 x^3-48 x^4\right )+e^{12} \left (16 x^3+16 x^4\right )+e^4 \left (-8 x+48 x^3+48 x^4\right )+\left (-4+24 x^2+16 x^3-8 x^4+e^8 \left (24 x^2-24 x^4\right )+e^{12} \left (8 x^3+8 x^4\right )+e^4 \left (-48 x^2-24 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x+12 x^3+e^4 \left (12 x-12 x^2-24 x^3\right )+e^8 \left (12 x^2+12 x^3\right )\right ) \log ^2(x)+\left (2-4 x-6 x^2+e^4 \left (6 x+6 x^2\right )\right ) \log ^3(x)+(1+x) \log ^4(x)+\left (16-32 x+32 e^4 x+\left (8-16 x+16 e^4 x\right ) \log (x)\right ) \log (2+\log (x))}{-16 x^4+48 e^4 x^4-48 e^8 x^4+16 e^{12} x^4+\left (24 x^3-8 x^4+8 e^{12} x^4+e^8 \left (24 x^3-24 x^4\right )+e^4 \left (-48 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x^2+12 x^3+12 e^8 x^3+e^4 \left (12 x^2-24 x^3\right )\right ) \log ^2(x)+\left (2 x-6 x^2+6 e^4 x^2\right ) \log ^3(x)+x \log ^4(x)} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(8*x - 16*x^3 - 16*x^4 + E^8*(-48*x^3 - 48*x^4) + E^12*(16*x^3 + 16*x^4) + E^4*(-8*x + 48*x^3 + 48*x^4) +

(-4 + 24*x^2 + 16*x^3 - 8*x^4 + E^8*(24*x^2 - 24*x^4) + E^12*(8*x^3 + 8*x^4) + E^4*(-48*x^2 - 24*x^3 + 24*x^4)

)*Log[x] + (-12*x + 12*x^3 + E^4*(12*x - 12*x^2 - 24*x^3) + E^8*(12*x^2 + 12*x^3))*Log[x]^2 + (2 - 4*x - 6*x^2

+ E^4*(6*x + 6*x^2))*Log[x]^3 + (1 + x)*Log[x]^4 + (16 - 32*x + 32*E^4*x + (8 - 16*x + 16*E^4*x)*Log[x])*Log[

2 + Log[x]])/(-16*x^4 + 48*E^4*x^4 - 48*E^8*x^4 + 16*E^12*x^4 + (24*x^3 - 8*x^4 + 8*E^12*x^4 + E^8*(24*x^3 - 2

4*x^4) + E^4*(-48*x^3 + 24*x^4))*Log[x] + (-12*x^2 + 12*x^3 + 12*E^8*x^3 + E^4*(12*x^2 - 24*x^3))*Log[x]^2 + (

2*x - 6*x^2 + 6*E^4*x^2)*Log[x]^3 + x*Log[x]^4),x]

[Out]

x + Log[x] + 32*E^4*Defer[Int][(2*(1 - E^4)*x - Log[x])^(-3), x] - 8*(2 + E^4)*Defer[Int][(2*(1 - E^4)*x - Log

[x])^(-3), x] + 8*(1 - E^4)*Defer[Int][x/(2*(1 - E^4)*x - Log[x])^3, x] + 24*E^4*(1 - E^4)*Defer[Int][x/(2*(1

- E^4)*x - Log[x])^3, x] - 24*(1 - E^4)^2*Defer[Int][x/(2*(1 - E^4)*x - Log[x])^3, x] + 8*(1 - E^4)^2*Defer[In

t][x^2/(2*(1 - E^4)*x - Log[x])^3, x] + 8*E^4*(1 - E^4)^2*Defer[Int][x^2/(2*(1 - E^4)*x - Log[x])^3, x] - 8*(1

- E^4)^2*(2 + E^4)*Defer[Int][x^2/(2*(1 - E^4)*x - Log[x])^3, x] - 8*(1 - E^4)^2*(1 + 3*E^4)*Defer[Int][x^2/(

2*(1 - E^4)*x - Log[x])^3, x] + 24*(1 - E^4)*(1 - E^8)*Defer[Int][x^2/(2*(1 - E^4)*x - Log[x])^3, x] + 24*(1 -

E^4)^3*Defer[Int][x^3/(2*(1 - E^4)*x - Log[x])^3, x] + 8*E^4*Defer[Int][1/((-1 - (1 - E^4)*x)*(2*(1 - E^4)*x

- Log[x])^3), x] + 8*Defer[Int][1/((1 + (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])^3), x] - 16*E^4*Defer[Int][1/((1

+ (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])^3), x] - 20*(1 - E^4)*Defer[Int][1/((1 + (1 - E^4)*x)*(2*(1 - E^4)*x

- Log[x])^3), x] - 4*(1 + E^4)*Defer[Int][1/((1 + (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])^3), x] + 8*(2 + E^4)*D

efer[Int][1/((1 + (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])^3), x] + 8*Defer[Int][(2*(1 - E^4)*x - Log[x])^(-2), x

] + 4*(4 - 5*E^4)*Defer[Int][(2*(1 - E^4)*x - Log[x])^(-2), x] - 4*(1 - 2*E^4)*Defer[Int][(2*(1 - E^4)*x - Log

[x])^(-2), x] - 12*(2 - E^4)*Defer[Int][(2*(1 - E^4)*x - Log[x])^(-2), x] - 4*(1 + E^4)*Defer[Int][(2*(1 - E^4

)*x - Log[x])^(-2), x] + 4*(2 + E^4)*Defer[Int][(2*(1 - E^4)*x - Log[x])^(-2), x] - 2*Defer[Int][1/(x*(2*(1 -

E^4)*x - Log[x])^2), x] - 12*E^4*(1 - E^4)*Defer[Int][x/(2*(1 - E^4)*x - Log[x])^2, x] - 4*(1 - 6*E^4)*(1 - E^

4)*Defer[Int][x/(2*(1 - E^4)*x - Log[x])^2, x] + 4*(1 - 3*E^4)*(1 - E^4)*Defer[Int][x/(2*(1 - E^4)*x - Log[x])

^2, x] + 8*(2 + E^4)*Defer[Int][1/((-1 - (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])^2), x] + 4*Defer[Int][1/((1 + (

1 - E^4)*x)^2*(2*(1 - E^4)*x - Log[x])^2), x] - 12*E^4*Defer[Int][1/((1 + (1 - E^4)*x)^2*(2*(1 - E^4)*x - Log[

x])^2), x] - 10*(1 - E^4)*Defer[Int][1/((1 + (1 - E^4)*x)^2*(2*(1 - E^4)*x - Log[x])^2), x] - 2*(1 + E^4)*Defe

r[Int][1/((1 + (1 - E^4)*x)^2*(2*(1 - E^4)*x - Log[x])^2), x] + 4*(2 + E^4)*Defer[Int][1/((1 + (1 - E^4)*x)^2*

(2*(1 - E^4)*x - Log[x])^2), x] - 12*Defer[Int][1/((1 + (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])^2), x] + 18*(1 -

E^4)*Defer[Int][1/((1 + (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])^2), x] - 4*(4 - E^4)*Defer[Int][1/((1 + (1 - E^

4)*x)*(2*(1 - E^4)*x - Log[x])^2), x] + 12*(2 + E^4)*Defer[Int][1/((1 + (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])^

2), x] + 4*(1 + 2*E^4)*Defer[Int][1/((1 + (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])^2), x] + 2*E^4*Defer[Int][1/((

-1 - (1 - E^4)*x)^3*(2*(1 - E^4)*x - Log[x])), x] - 2*(2 + E^4)*Defer[Int][1/((-1 - (1 - E^4)*x)^3*(2*(1 - E^4

)*x - Log[x])), x] + 2*(1 - E^4)*Defer[Int][1/((-1 - (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])), x] - 2*(2 + E^4)*

Defer[Int][1/((-1 - (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])), x] + 2*Defer[Int][1/((1 + (1 - E^4)*x)^3*(2*(1 - E

^4)*x - Log[x])), x] - 4*E^4*Defer[Int][1/((1 + (1 - E^4)*x)^3*(2*(1 - E^4)*x - Log[x])), x] - 5*(1 - E^4)*Def

er[Int][1/((1 + (1 - E^4)*x)^3*(2*(1 - E^4)*x - Log[x])), x] - (1 + E^4)*Defer[Int][1/((1 + (1 - E^4)*x)^3*(2*

(1 - E^4)*x - Log[x])), x] - 6*Defer[Int][1/((1 + (1 - E^4)*x)^2*(2*(1 - E^4)*x - Log[x])), x] + 9*(1 - E^4)*D

efer[Int][1/((1 + (1 - E^4)*x)^2*(2*(1 - E^4)*x - Log[x])), x] - 2*(4 - E^4)*Defer[Int][1/((1 + (1 - E^4)*x)^2

*(2*(1 - E^4)*x - Log[x])), x] + 2*(2 + E^4)*Defer[Int][1/((1 + (1 - E^4)*x)^2*(2*(1 - E^4)*x - Log[x])), x] +

2*(1 + 2*E^4)*Defer[Int][1/((1 + (1 - E^4)*x)^2*(2*(1 - E^4)*x - Log[x])), x] + 3*(1 - E^4)*Defer[Int][1/((1

+ (1 - E^4)*x)*(2*(1 - E^4)*x - Log[x])), x] - 2*(2 + E^4)*Defer[Int][1/((1 + (1 - E^4)*x)*(2*(1 - E^4)*x - Lo

g[x])), x] + 2*Defer[Int][(-1 - x)/(x*(1 + (1 - E^4)*x)^3*(2 + Log[x])), x] + 6*(1 - E^4)^2*Defer[Int][x/((1 +

(1 - E^4)*x)^3*(2 + Log[x])), x] + Defer[Int][1/(x*(-1 + (-1 + E^4)*x)^3*(2 + Log[x])), x] - 2*(1 - E^4)^2*(2

+ E^4)*Defer[Int][x^2/((-1 + (-1 + E^4)*x)^3*(2 + Log[x])), x] + 2*(1 - E^4)^3*Defer[Int][x^3/((-1 + (-1 + E^

4)*x)^3*(2 + Log[x])), x] + 2*Defer[Int][(1 - (2 - 3*E^4)*x - 3*(1 - E^4)*x^2)/(x*(1 + (1 - E^4)*x)^3*(2 + Log

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

^4)*Defer[Int][(-1 + 2*(1 - E^4)^2*x^2 + 2*(1 - E^4)^2*x^3)/((1 + (1 - E^4)*x)^3*(2 + Log[x])), x] + 8*Defer[I

nt][(-2*(1 - E^4)*x + Log[x])^(-3), x] - 24*(1 - E^4)*Defer[Int][(-2*(1 - E^4)*x + Log[x])^(-3), x] + 56*E^4*(

1 - E^4)*Defer[Int][x/(-2*(1 - E^4)*x + Log[x])^3, x] - 8*(1 - E^4)*(2 + E^4)*Defer[Int][x/(-2*(1 - E^4)*x + L

og[x])^3, x] + 8*(1 - E^4)^2*Defer[Int][x^2/(-2*(1 - E^4)*x + Log[x])^3, x] + 24*(1 - E^4)^3*Defer[Int][x^3/(-

2*(1 - E^4)*x + Log[x])^3, x] + Defer[Int][1/(x*(-2*(1 - E^4)*x + Log[x])), x] + 16*(1 - E^4)*Defer[Int][Log[2

+ Log[x]]/(2*(1 - E^4)*x - Log[x])^3, x] + 8*Defer[Int][Log[2 + Log[x]]/(x*(-2*(1 - E^4)*x + Log[x])^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 x-16 x^3-16 x^4+e^8 \left (-48 x^3-48 x^4\right )+e^{12} \left (16 x^3+16 x^4\right )+e^4 \left (-8 x+48 x^3+48 x^4\right )+\left (-4+24 x^2+16 x^3-8 x^4+e^8 \left (24 x^2-24 x^4\right )+e^{12} \left (8 x^3+8 x^4\right )+e^4 \left (-48 x^2-24 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x+12 x^3+e^4 \left (12 x-12 x^2-24 x^3\right )+e^8 \left (12 x^2+12 x^3\right )\right ) \log ^2(x)+\left (2-4 x-6 x^2+e^4 \left (6 x+6 x^2\right )\right ) \log ^3(x)+(1+x) \log ^4(x)+\left (16-32 x+32 e^4 x+\left (8-16 x+16 e^4 x\right ) \log (x)\right ) \log (2+\log (x))}{-48 e^8 x^4+16 e^{12} x^4+\left (-16+48 e^4\right ) x^4+\left (24 x^3-8 x^4+8 e^{12} x^4+e^8 \left (24 x^3-24 x^4\right )+e^4 \left (-48 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x^2+12 x^3+12 e^8 x^3+e^4 \left (12 x^2-24 x^3\right )\right ) \log ^2(x)+\left (2 x-6 x^2+6 e^4 x^2\right ) \log ^3(x)+x \log ^4(x)} \, dx\\ &=\int \frac {8 x-16 x^3-16 x^4+e^8 \left (-48 x^3-48 x^4\right )+e^{12} \left (16 x^3+16 x^4\right )+e^4 \left (-8 x+48 x^3+48 x^4\right )+\left (-4+24 x^2+16 x^3-8 x^4+e^8 \left (24 x^2-24 x^4\right )+e^{12} \left (8 x^3+8 x^4\right )+e^4 \left (-48 x^2-24 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x+12 x^3+e^4 \left (12 x-12 x^2-24 x^3\right )+e^8 \left (12 x^2+12 x^3\right )\right ) \log ^2(x)+\left (2-4 x-6 x^2+e^4 \left (6 x+6 x^2\right )\right ) \log ^3(x)+(1+x) \log ^4(x)+\left (16-32 x+32 e^4 x+\left (8-16 x+16 e^4 x\right ) \log (x)\right ) \log (2+\log (x))}{\left (-16+48 e^4\right ) x^4+\left (-48 e^8+16 e^{12}\right ) x^4+\left (24 x^3-8 x^4+8 e^{12} x^4+e^8 \left (24 x^3-24 x^4\right )+e^4 \left (-48 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x^2+12 x^3+12 e^8 x^3+e^4 \left (12 x^2-24 x^3\right )\right ) \log ^2(x)+\left (2 x-6 x^2+6 e^4 x^2\right ) \log ^3(x)+x \log ^4(x)} \, dx\\ &=\int \frac {8 x-16 x^3-16 x^4+e^8 \left (-48 x^3-48 x^4\right )+e^{12} \left (16 x^3+16 x^4\right )+e^4 \left (-8 x+48 x^3+48 x^4\right )+\left (-4+24 x^2+16 x^3-8 x^4+e^8 \left (24 x^2-24 x^4\right )+e^{12} \left (8 x^3+8 x^4\right )+e^4 \left (-48 x^2-24 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x+12 x^3+e^4 \left (12 x-12 x^2-24 x^3\right )+e^8 \left (12 x^2+12 x^3\right )\right ) \log ^2(x)+\left (2-4 x-6 x^2+e^4 \left (6 x+6 x^2\right )\right ) \log ^3(x)+(1+x) \log ^4(x)+\left (16-32 x+32 e^4 x+\left (8-16 x+16 e^4 x\right ) \log (x)\right ) \log (2+\log (x))}{\left (-16+48 e^4-48 e^8+16 e^{12}\right ) x^4+\left (24 x^3-8 x^4+8 e^{12} x^4+e^8 \left (24 x^3-24 x^4\right )+e^4 \left (-48 x^3+24 x^4\right )\right ) \log (x)+\left (-12 x^2+12 x^3+12 e^8 x^3+e^4 \left (12 x^2-24 x^3\right )\right ) \log ^2(x)+\left (2 x-6 x^2+6 e^4 x^2\right ) \log ^3(x)+x \log ^4(x)} \, dx\\ &=\int \frac {12 \left (-1+e^4\right ) x \left (1-x^2+e^4 x (1+x)\right ) \log ^2(x)+\left (2+\left (-4+6 e^4\right ) x+6 \left (-1+e^4\right ) x^2\right ) \log ^3(x)+(1+x) \log ^4(x)+4 \log (x) \left (-1+6 \left (-1+e^4\right )^2 x^2+2 \left (2-3 e^4+e^{12}\right ) x^3+2 \left (-1+e^4\right )^3 x^4+\left (2+4 \left (-1+e^4\right ) x\right ) \log (2+\log (x))\right )+8 \left (\left (-1+e^4\right ) x \left (-1+2 \left (-1+e^4\right )^2 x^2+2 \left (-1+e^4\right )^2 x^3\right )+\left (2+4 \left (-1+e^4\right ) x\right ) \log (2+\log (x))\right )}{x (2+\log (x)) \left (2 \left (-1+e^4\right ) x+\log (x)\right )^3} \, dx\\ &=\int \left (\frac {8 (1-e) (1+e) \left (1+e^2\right ) \left (-1+2 \left (1-e^4\right )^2 x^2+2 \left (1-e^4\right )^2 x^3\right )}{\left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))}+\frac {4 \log (x)}{x \left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))}+\frac {8 \left (-2-e^4\right ) \left (1-e^4\right )^2 x^2 \log (x)}{\left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))}+\frac {8 \left (1-e^4\right )^3 x^3 \log (x)}{\left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))}+\frac {12 (1-e) (1+e) \left (1+e^2\right ) (1+x) \left (1-\left (1-e^4\right ) x\right ) \log ^2(x)}{\left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))}+\frac {2 (1+x) \left (-1+3 \left (1-e^4\right ) x\right ) \log ^3(x)}{x \left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))}+\frac {(-1-x) \log ^4(x)}{x \left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))}+\frac {24 \left (1-e^4\right )^2 x \log (x)}{(2+\log (x)) \left (-2 \left (1-e^4\right ) x+\log (x)\right )^3}+\frac {8 \left (-1+2 \left (1-e^4\right ) x\right ) \log (2+\log (x))}{x \left (2 \left (1-e^4\right ) x-\log (x)\right )^3}\right ) \, dx\\ &=2 \int \frac {(1+x) \left (-1+3 \left (1-e^4\right ) x\right ) \log ^3(x)}{x \left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))} \, dx+4 \int \frac {\log (x)}{x \left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))} \, dx+8 \int \frac {\left (-1+2 \left (1-e^4\right ) x\right ) \log (2+\log (x))}{x \left (2 \left (1-e^4\right ) x-\log (x)\right )^3} \, dx+\left (8 \left (1-e^4\right )\right ) \int \frac {-1+2 \left (1-e^4\right )^2 x^2+2 \left (1-e^4\right )^2 x^3}{\left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))} \, dx+\left (12 \left (1-e^4\right )\right ) \int \frac {(1+x) \left (1-\left (1-e^4\right ) x\right ) \log ^2(x)}{\left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))} \, dx+\left (24 \left (1-e^4\right )^2\right ) \int \frac {x \log (x)}{(2+\log (x)) \left (-2 \left (1-e^4\right ) x+\log (x)\right )^3} \, dx+\left (8 \left (1-e^4\right )^3\right ) \int \frac {x^3 \log (x)}{\left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))} \, dx-\left (8 \left (1-e^4\right )^2 \left (2+e^4\right )\right ) \int \frac {x^2 \log (x)}{\left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))} \, dx+\int \frac {(-1-x) \log ^4(x)}{x \left (2 \left (1-e^4\right ) x-\log (x)\right )^3 (2+\log (x))} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.22, size = 25, normalized size = 0.89 \begin {gather*} x+\log (x)-\frac {4 \log (2+\log (x))}{\left (-2 x+2 e^4 x+\log (x)\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(8*x - 16*x^3 - 16*x^4 + E^8*(-48*x^3 - 48*x^4) + E^12*(16*x^3 + 16*x^4) + E^4*(-8*x + 48*x^3 + 48*x

^4) + (-4 + 24*x^2 + 16*x^3 - 8*x^4 + E^8*(24*x^2 - 24*x^4) + E^12*(8*x^3 + 8*x^4) + E^4*(-48*x^2 - 24*x^3 + 2

4*x^4))*Log[x] + (-12*x + 12*x^3 + E^4*(12*x - 12*x^2 - 24*x^3) + E^8*(12*x^2 + 12*x^3))*Log[x]^2 + (2 - 4*x -

6*x^2 + E^4*(6*x + 6*x^2))*Log[x]^3 + (1 + x)*Log[x]^4 + (16 - 32*x + 32*E^4*x + (8 - 16*x + 16*E^4*x)*Log[x]

)*Log[2 + Log[x]])/(-16*x^4 + 48*E^4*x^4 - 48*E^8*x^4 + 16*E^12*x^4 + (24*x^3 - 8*x^4 + 8*E^12*x^4 + E^8*(24*x

^3 - 24*x^4) + E^4*(-48*x^3 + 24*x^4))*Log[x] + (-12*x^2 + 12*x^3 + 12*E^8*x^3 + E^4*(12*x^2 - 24*x^3))*Log[x]

^2 + (2*x - 6*x^2 + 6*E^4*x^2)*Log[x]^3 + x*Log[x]^4),x]

[Out]

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

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Maple [A]
time = 20.85, size = 25, normalized size = 0.89


method	result	size

risch	\(-\frac {4 \ln \left (\ln \left (x \right )+2\right )}{\left (2 x \,{\mathrm e}^{4}+\ln \left (x \right )-2 x \right )^{2}}+x +\ln \left (x \right )\)	\(25\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((((16*x*exp(4)-16*x+8)*ln(x)+32*x*exp(4)-32*x+16)*ln(ln(x)+2)+(x+1)*ln(x)^4+((6*x^2+6*x)*exp(4)-6*x^2-4*x+

2)*ln(x)^3+((12*x^3+12*x^2)*exp(4)^2+(-24*x^3-12*x^2+12*x)*exp(4)+12*x^3-12*x)*ln(x)^2+((8*x^4+8*x^3)*exp(4)^3

+(-24*x^4+24*x^2)*exp(4)^2+(24*x^4-24*x^3-48*x^2)*exp(4)-8*x^4+16*x^3+24*x^2-4)*ln(x)+(16*x^4+16*x^3)*exp(4)^3

+(-48*x^4-48*x^3)*exp(4)^2+(48*x^4+48*x^3-8*x)*exp(4)-16*x^4-16*x^3+8*x)/(x*ln(x)^4+(6*x^2*exp(4)-6*x^2+2*x)*l

n(x)^3+(12*x^3*exp(4)^2+(-24*x^3+12*x^2)*exp(4)+12*x^3-12*x^2)*ln(x)^2+(8*x^4*exp(4)^3+(-24*x^4+24*x^3)*exp(4)

^2+(24*x^4-48*x^3)*exp(4)-8*x^4+24*x^3)*ln(x)+16*x^4*exp(4)^3-48*x^4*exp(4)^2+48*x^4*exp(4)-16*x^4),x,method=_

RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (24) = 48\).
time = 0.39, size = 81, normalized size = 2.89 \begin {gather*} \frac {4 \, x^{3} {\left (e^{8} - 2 \, e^{4} + 1\right )} + 4 \, x^{2} {\left (e^{8} - e^{4}\right )} \log \left (x\right ) + x {\left (4 \, e^{4} - 3\right )} \log \left (x\right )^{2} + \log \left (x\right )^{3} - 4 \, \log \left (\log \left (x\right ) + 2\right )}{4 \, x^{2} {\left (e^{8} - 2 \, e^{4} + 1\right )} + 4 \, x {\left (e^{4} - 1\right )} \log \left (x\right ) + \log \left (x\right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x*exp(4)-16*x+8)*log(x)+32*x*exp(4)-32*x+16)*log(log(x)+2)+(1+x)*log(x)^4+((6*x^2+6*x)*exp(4)-

6*x^2-4*x+2)*log(x)^3+((12*x^3+12*x^2)*exp(4)^2+(-24*x^3-12*x^2+12*x)*exp(4)+12*x^3-12*x)*log(x)^2+((8*x^4+8*x

^3)*exp(4)^3+(-24*x^4+24*x^2)*exp(4)^2+(24*x^4-24*x^3-48*x^2)*exp(4)-8*x^4+16*x^3+24*x^2-4)*log(x)+(16*x^4+16*

x^3)*exp(4)^3+(-48*x^4-48*x^3)*exp(4)^2+(48*x^4+48*x^3-8*x)*exp(4)-16*x^4-16*x^3+8*x)/(x*log(x)^4+(6*x^2*exp(4

)-6*x^2+2*x)*log(x)^3+(12*x^3*exp(4)^2+(-24*x^3+12*x^2)*exp(4)+12*x^3-12*x^2)*log(x)^2+(8*x^4*exp(4)^3+(-24*x^

4+24*x^3)*exp(4)^2+(24*x^4-48*x^3)*exp(4)-8*x^4+24*x^3)*log(x)+16*x^4*exp(4)^3-48*x^4*exp(4)^2+48*x^4*exp(4)-1

6*x^4),x, algorithm="maxima")

[Out]

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

4*x^2*(e^8 - 2*e^4 + 1) + 4*x*(e^4 - 1)*log(x) + log(x)^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (24) = 48\).
time = 0.37, size = 102, normalized size = 3.64 \begin {gather*} \frac {4 \, x^{3} e^{8} - 8 \, x^{3} e^{4} + 4 \, x^{3} + {\left (4 \, x e^{4} - 3 \, x\right )} \log \left (x\right )^{2} + \log \left (x\right )^{3} + 4 \, {\left (x^{2} e^{8} - x^{2} e^{4}\right )} \log \left (x\right ) - 4 \, \log \left (\log \left (x\right ) + 2\right )}{4 \, x^{2} e^{8} - 8 \, x^{2} e^{4} + 4 \, x^{2} + 4 \, {\left (x e^{4} - x\right )} \log \left (x\right ) + \log \left (x\right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x*exp(4)-16*x+8)*log(x)+32*x*exp(4)-32*x+16)*log(log(x)+2)+(1+x)*log(x)^4+((6*x^2+6*x)*exp(4)-

6*x^2-4*x+2)*log(x)^3+((12*x^3+12*x^2)*exp(4)^2+(-24*x^3-12*x^2+12*x)*exp(4)+12*x^3-12*x)*log(x)^2+((8*x^4+8*x

^3)*exp(4)^3+(-24*x^4+24*x^2)*exp(4)^2+(24*x^4-24*x^3-48*x^2)*exp(4)-8*x^4+16*x^3+24*x^2-4)*log(x)+(16*x^4+16*

x^3)*exp(4)^3+(-48*x^4-48*x^3)*exp(4)^2+(48*x^4+48*x^3-8*x)*exp(4)-16*x^4-16*x^3+8*x)/(x*log(x)^4+(6*x^2*exp(4

)-6*x^2+2*x)*log(x)^3+(12*x^3*exp(4)^2+(-24*x^3+12*x^2)*exp(4)+12*x^3-12*x^2)*log(x)^2+(8*x^4*exp(4)^3+(-24*x^

4+24*x^3)*exp(4)^2+(24*x^4-48*x^3)*exp(4)-8*x^4+24*x^3)*log(x)+16*x^4*exp(4)^3-48*x^4*exp(4)^2+48*x^4*exp(4)-1

6*x^4),x, algorithm="fricas")

[Out]

(4*x^3*e^8 - 8*x^3*e^4 + 4*x^3 + (4*x*e^4 - 3*x)*log(x)^2 + log(x)^3 + 4*(x^2*e^8 - x^2*e^4)*log(x) - 4*log(lo

g(x) + 2))/(4*x^2*e^8 - 8*x^2*e^4 + 4*x^2 + 4*(x*e^4 - x)*log(x) + log(x)^2)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x*exp(4)-16*x+8)*ln(x)+32*x*exp(4)-32*x+16)*ln(ln(x)+2)+(1+x)*ln(x)**4+((6*x**2+6*x)*exp(4)-6*

x**2-4*x+2)*ln(x)**3+((12*x**3+12*x**2)*exp(4)**2+(-24*x**3-12*x**2+12*x)*exp(4)+12*x**3-12*x)*ln(x)**2+((8*x*

*4+8*x**3)*exp(4)**3+(-24*x**4+24*x**2)*exp(4)**2+(24*x**4-24*x**3-48*x**2)*exp(4)-8*x**4+16*x**3+24*x**2-4)*l

n(x)+(16*x**4+16*x**3)*exp(4)**3+(-48*x**4-48*x**3)*exp(4)**2+(48*x**4+48*x**3-8*x)*exp(4)-16*x**4-16*x**3+8*x

)/(x*ln(x)**4+(6*x**2*exp(4)-6*x**2+2*x)*ln(x)**3+(12*x**3*exp(4)**2+(-24*x**3+12*x**2)*exp(4)+12*x**3-12*x**2

)*ln(x)**2+(8*x**4*exp(4)**3+(-24*x**4+24*x**3)*exp(4)**2+(24*x**4-48*x**3)*exp(4)-8*x**4+24*x**3)*ln(x)+16*x*

*4*exp(4)**3-48*x**4*exp(4)**2+48*x**4*exp(4)-16*x**4),x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

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Giac [F(-1)] Timed out
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((((16*x*exp(4)-16*x+8)*log(x)+32*x*exp(4)-32*x+16)*log(log(x)+2)+(1+x)*log(x)^4+((6*x^2+6*x)*exp(4)-

6*x^2-4*x+2)*log(x)^3+((12*x^3+12*x^2)*exp(4)^2+(-24*x^3-12*x^2+12*x)*exp(4)+12*x^3-12*x)*log(x)^2+((8*x^4+8*x

^3)*exp(4)^3+(-24*x^4+24*x^2)*exp(4)^2+(24*x^4-24*x^3-48*x^2)*exp(4)-8*x^4+16*x^3+24*x^2-4)*log(x)+(16*x^4+16*

x^3)*exp(4)^3+(-48*x^4-48*x^3)*exp(4)^2+(48*x^4+48*x^3-8*x)*exp(4)-16*x^4-16*x^3+8*x)/(x*log(x)^4+(6*x^2*exp(4

)-6*x^2+2*x)*log(x)^3+(12*x^3*exp(4)^2+(-24*x^3+12*x^2)*exp(4)+12*x^3-12*x^2)*log(x)^2+(8*x^4*exp(4)^3+(-24*x^

4+24*x^3)*exp(4)^2+(24*x^4-48*x^3)*exp(4)-8*x^4+24*x^3)*log(x)+16*x^4*exp(4)^3-48*x^4*exp(4)^2+48*x^4*exp(4)-1

6*x^4),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 2.24, size = 79, normalized size = 2.82 \begin {gather*} \frac {{\ln \left (x\right )}^3-3\,x\,{\ln \left (x\right )}^2-4\,\ln \left (\ln \left (x\right )+2\right )-8\,x^3\,{\mathrm {e}}^4+4\,x^3\,{\mathrm {e}}^8+4\,x^3+4\,x\,{\mathrm {e}}^4\,{\ln \left (x\right )}^2-4\,x^2\,{\mathrm {e}}^4\,\ln \left (x\right )+4\,x^2\,{\mathrm {e}}^8\,\ln \left (x\right )}{{\left (\ln \left (x\right )-2\,x+2\,x\,{\mathrm {e}}^4\right )}^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((8*x - log(x)^2*(12*x + exp(4)*(12*x^2 - 12*x + 24*x^3) - exp(8)*(12*x^2 + 12*x^3) - 12*x^3) + log(x)*(exp

(12)*(8*x^3 + 8*x^4) + exp(8)*(24*x^2 - 24*x^4) - exp(4)*(48*x^2 + 24*x^3 - 24*x^4) + 24*x^2 + 16*x^3 - 8*x^4

- 4) + log(log(x) + 2)*(32*x*exp(4) - 32*x + log(x)*(16*x*exp(4) - 16*x + 8) + 16) - log(x)^3*(4*x - exp(4)*(6

*x + 6*x^2) + 6*x^2 - 2) + exp(4)*(48*x^3 - 8*x + 48*x^4) + log(x)^4*(x + 1) + exp(12)*(16*x^3 + 16*x^4) - exp

(8)*(48*x^3 + 48*x^4) - 16*x^3 - 16*x^4)/(x*log(x)^4 + 48*x^4*exp(4) - 48*x^4*exp(8) + 16*x^4*exp(12) + log(x)

^2*(exp(4)*(12*x^2 - 24*x^3) + 12*x^3*exp(8) - 12*x^2 + 12*x^3) + log(x)^3*(2*x + 6*x^2*exp(4) - 6*x^2) + log(

x)*(exp(8)*(24*x^3 - 24*x^4) - exp(4)*(48*x^3 - 24*x^4) + 8*x^4*exp(12) + 24*x^3 - 8*x^4) - 16*x^4),x)

[Out]

(log(x)^3 - 3*x*log(x)^2 - 4*log(log(x) + 2) - 8*x^3*exp(4) + 4*x^3*exp(8) + 4*x^3 + 4*x*exp(4)*log(x)^2 - 4*x

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

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