\(\int \frac {e^{\frac {2 (-e^6+x+4 e^3 x-4 x^2+(2+e^6-x-4 e^3 x+4 x^2+(e^6 x-x^2-4 e^3 x^2+4 x^3) \log (4)) \log (x))}{-1+(1+x \log (4)) \log (x)}} (-4-2 x-8 e^3 x+16 x^2+(4 x+16 e^3 x-32 x^2+(4 x^2+16 e^3 x^2-32 x^3) \log (4)) \log (x)+(-2 x-8 e^3 x+16 x^2+(-4 x-4 x^2-16 e^3 x^2+32 x^3) \log (4)+(-2 x^3-8 e^3 x^3+16 x^4) \log ^2(4)) \log ^2(x))}{x+(-2 x-2 x^2 \log (4)) \log (x)+(x+2 x^2 \log (4)+x^3 \log ^2(4)) \log ^2(x)} \, dx\) [4209]

Optimal result

Integrand size = 250, antiderivative size = 33 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=e^{2 \left (e^3-2 x\right )^2-2 x+\frac {4}{1+x \log (4)-\frac {1}{\log (x)}}} \]

[Out]

exp((exp(3)-2*x)^2-x+2/(1-1/ln(x)+2*x*ln(2)))^2

Rubi [F]

\[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=\int \frac {\exp \left (\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \left (-4+\left (-2-8 e^3\right ) x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx \\ & = \int \frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \left (-4+\left (-2-8 e^3\right ) x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x (1-\log (x)-x \log (4) \log (x))^2} \, dx \\ & = \int \left (\frac {2 \exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \left (-1-4 e^3+8 x^3 \log ^2(4)+x^2 \log (4) \left (16-\log (4)-4 e^3 \log (4)\right )+x \left (8-8 e^3 \log (4)-\log (16)\right )-\log (16)\right )}{(1+x \log (4))^2}-\frac {4 \exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \left (1+x^2 \log ^2(4)+x \log (64)\right )}{x (1+x \log (4))^2 (-1+\log (x)+x \log (4) \log (x))^2}-\frac {8 \exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \log (4)}{(1+x \log (4))^2 (-1+\log (x)+x \log (4) \log (x))}\right ) \, dx \\ & = 2 \int \frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \left (-1-4 e^3+8 x^3 \log ^2(4)+x^2 \log (4) \left (16-\log (4)-4 e^3 \log (4)\right )+x \left (8-8 e^3 \log (4)-\log (16)\right )-\log (16)\right )}{(1+x \log (4))^2} \, dx-4 \int \frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \left (1+x^2 \log ^2(4)+x \log (64)\right )}{x (1+x \log (4))^2 (-1+\log (x)+x \log (4) \log (x))^2} \, dx-(8 \log (4)) \int \frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right )}{(1+x \log (4))^2 (-1+\log (x)+x \log (4) \log (x))} \, dx \\ & = 2 \int \left (\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \left (-1-4 e^3\right )+8 \exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) x-\frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \log (16)}{(1+x \log (4))^2}\right ) \, dx-4 \int \left (\frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right )}{x (-1+\log (x)+x \log (4) \log (x))^2}+\frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \log (4)}{(1+x \log (4))^2 (-1+\log (x)+x \log (4) \log (x))^2}\right ) \, dx-(8 \log (4)) \int \frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right )}{(1+x \log (4))^2 (-1+\log (x)+x \log (4) \log (x))} \, dx \\ & = -\left (4 \int \frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right )}{x (-1+\log (x)+x \log (4) \log (x))^2} \, dx\right )+16 \int \exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) x \, dx-\left (2 \left (1+4 e^3\right )\right ) \int \exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right ) \, dx-(4 \log (4)) \int \frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right )}{(1+x \log (4))^2 (-1+\log (x)+x \log (4) \log (x))^2} \, dx-(8 \log (4)) \int \frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right )}{(1+x \log (4))^2 (-1+\log (x)+x \log (4) \log (x))} \, dx-(2 \log (16)) \int \frac {\exp \left (\frac {2 \left (-e^6+\left (1+4 e^3\right ) x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}\right )}{(1+x \log (4))^2} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=\int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(32)=64\).

Time = 36.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61

method	result	size
parallelrisch	\({\mathrm e}^{\frac {2 \left (2 \left (x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{3}+4 x^{3}-x^{2}\right ) \ln \left (2\right )+{\mathrm e}^{6}-4 x \,{\mathrm e}^{3}+4 x^{2}-x +2\right ) \ln \left (x \right )-2 \,{\mathrm e}^{6}+8 x \,{\mathrm e}^{3}-8 x^{2}+2 x}{2 x \ln \left (2\right ) \ln \left (x \right )+\ln \left (x \right )-1}}\)	\(86\)
risch	\({\mathrm e}^{-\frac {2 \left (8 \ln \left (x \right ) {\mathrm e}^{3} \ln \left (2\right ) x^{2}-8 \ln \left (2\right ) \ln \left (x \right ) x^{3}-2 \ln \left (x \right ) {\mathrm e}^{6} \ln \left (2\right ) x +2 x^{2} \ln \left (2\right ) \ln \left (x \right )+4 x \,{\mathrm e}^{3} \ln \left (x \right )-4 x^{2} \ln \left (x \right )-\ln \left (x \right ) {\mathrm e}^{6}+x \ln \left (x \right )-4 x \,{\mathrm e}^{3}+4 x^{2}-2 \ln \left (x \right )+{\mathrm e}^{6}-x \right )}{2 x \ln \left (2\right ) \ln \left (x \right )+\ln \left (x \right )-1}}\)	\(99\)

[In]

int(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*ln(2)^2+2*(-16*x^2*exp(3)+32*x^3-4*x^2-4*x)*ln(2)-8*x*exp(3)+16*x^2-2*x)*
ln(x)^2+(2*(16*x^2*exp(3)-32*x^3+4*x^2)*ln(2)+16*x*exp(3)-32*x^2+4*x)*ln(x)-8*x*exp(3)+16*x^2-2*x-4)*exp(((2*(
x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*ln(2)+exp(3)^2-4*x*exp(3)+4*x^2-x+2)*ln(x)-exp(3)^2+4*x*exp(3)-4*x^2+x)/((2
*x*ln(2)+1)*ln(x)-1))^2/((4*x^3*ln(2)^2+4*x^2*ln(2)+x)*ln(x)^2+(-4*x^2*ln(2)-2*x)*ln(x)+x),x,method=_RETURNVER
BOSE)

[Out]

exp(((2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*ln(2)+exp(3)^2-4*x*exp(3)+4*x^2-x+2)*ln(x)-exp(3)^2+4*x*exp(3)-4*x
^2+x)/(2*x*ln(2)*ln(x)+ln(x)-1))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (34) = 68\).

Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=e^{\left (-\frac {2 \, {\left (4 \, x^{2} - 4 \, x e^{3} - {\left (4 \, x^{2} - 4 \, x e^{3} + 2 \, {\left (4 \, x^{3} - 4 \, x^{2} e^{3} - x^{2} + x e^{6}\right )} \log \left (2\right ) - x + e^{6} + 2\right )} \log \left (x\right ) - x + e^{6}\right )}}{{\left (2 \, x \log \left (2\right ) + 1\right )} \log \left (x\right ) - 1}\right )} \]

[In]

integrate(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*log(2)^2+2*(-16*x^2*exp(3)+32*x^3-4*x^2-4*x)*log(2)-8*x*exp(3)+16*x
^2-2*x)*log(x)^2+(2*(16*x^2*exp(3)-32*x^3+4*x^2)*log(2)+16*x*exp(3)-32*x^2+4*x)*log(x)-8*x*exp(3)+16*x^2-2*x-4
)*exp(((2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*log(2)+exp(3)^2-4*x*exp(3)+4*x^2-x+2)*log(x)-exp(3)^2+4*x*exp(3)
-4*x^2+x)/((2*x*log(2)+1)*log(x)-1))^2/((4*x^3*log(2)^2+4*x^2*log(2)+x)*log(x)^2+(-4*x^2*log(2)-2*x)*log(x)+x)
,x, algorithm="fricas")

[Out]

e^(-2*(4*x^2 - 4*x*e^3 - (4*x^2 - 4*x*e^3 + 2*(4*x^3 - 4*x^2*e^3 - x^2 + x*e^6)*log(2) - x + e^6 + 2)*log(x) -
x + e^6)/((2*x*log(2) + 1)*log(x) - 1))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (29) = 58\).

Time = 0.84 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.64 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=e^{\frac {2 \left (- 4 x^{2} + x + 4 x e^{3} + \left (4 x^{2} - 4 x e^{3} - x + \left (8 x^{3} - 8 x^{2} e^{3} - 2 x^{2} + 2 x e^{6}\right ) \log {\left (2 \right )} + 2 + e^{6}\right ) \log {\left (x \right )} - e^{6}\right )}{\left (2 x \log {\left (2 \right )} + 1\right ) \log {\left (x \right )} - 1}} \]

[In]

integrate(((4*(-8*x**3*exp(3)+16*x**4-2*x**3)*ln(2)**2+2*(-16*x**2*exp(3)+32*x**3-4*x**2-4*x)*ln(2)-8*x*exp(3)
+16*x**2-2*x)*ln(x)**2+(2*(16*x**2*exp(3)-32*x**3+4*x**2)*ln(2)+16*x*exp(3)-32*x**2+4*x)*ln(x)-8*x*exp(3)+16*x
**2-2*x-4)*exp(((2*(x*exp(3)**2-4*x**2*exp(3)+4*x**3-x**2)*ln(2)+exp(3)**2-4*x*exp(3)+4*x**2-x+2)*ln(x)-exp(3)
**2+4*x*exp(3)-4*x**2+x)/((2*x*ln(2)+1)*ln(x)-1))**2/((4*x**3*ln(2)**2+4*x**2*ln(2)+x)*ln(x)**2+(-4*x**2*ln(2)
-2*x)*ln(x)+x),x)

[Out]

exp(2*(-4*x**2 + x + 4*x*exp(3) + (4*x**2 - 4*x*exp(3) - x + (8*x**3 - 8*x**2*exp(3) - 2*x**2 + 2*x*exp(6))*lo
g(2) + 2 + exp(6))*log(x) - exp(6))/((2*x*log(2) + 1)*log(x) - 1))

Maxima [A] (verification not implemented)

none

Time = 1.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=e^{\left (8 \, x^{2} - 8 \, x e^{3} - 2 \, x + \frac {4 \, \log \left (x\right )}{{\left (2 \, x \log \left (2\right ) + 1\right )} \log \left (x\right ) - 1} + 2 \, e^{6}\right )} \]

[In]

integrate(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*log(2)^2+2*(-16*x^2*exp(3)+32*x^3-4*x^2-4*x)*log(2)-8*x*exp(3)+16*x
^2-2*x)*log(x)^2+(2*(16*x^2*exp(3)-32*x^3+4*x^2)*log(2)+16*x*exp(3)-32*x^2+4*x)*log(x)-8*x*exp(3)+16*x^2-2*x-4
)*exp(((2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*log(2)+exp(3)^2-4*x*exp(3)+4*x^2-x+2)*log(x)-exp(3)^2+4*x*exp(3)
-4*x^2+x)/((2*x*log(2)+1)*log(x)-1))^2/((4*x^3*log(2)^2+4*x^2*log(2)+x)*log(x)^2+(-4*x^2*log(2)-2*x)*log(x)+x)
,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*log(2)^2+2*(-16*x^2*exp(3)+32*x^3-4*x^2-4*x)*log(2)-8*x*exp(3)+16*x
^2-2*x)*log(x)^2+(2*(16*x^2*exp(3)-32*x^3+4*x^2)*log(2)+16*x*exp(3)-32*x^2+4*x)*log(x)-8*x*exp(3)+16*x^2-2*x-4
)*exp(((2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*log(2)+exp(3)^2-4*x*exp(3)+4*x^2-x+2)*log(x)-exp(3)^2+4*x*exp(3)
-4*x^2+x)/((2*x*log(2)+1)*log(x)-1))^2/((4*x^3*log(2)^2+4*x^2*log(2)+x)*log(x)^2+(-4*x^2*log(2)-2*x)*log(x)+x)
,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{4194304,[0,20,16,0,1]%%%}+%%%{46137344,[0,19,15,0,1]%%%}
+%%%{229638

Mupad [B] (verification not implemented)

Time = 12.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.18 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=x^{\frac {2\,\left ({\mathrm {e}}^6-x-4\,x\,{\mathrm {e}}^3-2\,x^2\,\ln \left (2\right )+8\,x^3\,\ln \left (2\right )+4\,x^2-8\,x^2\,{\mathrm {e}}^3\,\ln \left (2\right )+2\,x\,{\mathrm {e}}^6\,\ln \left (2\right )+2\right )}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}}\,{\mathrm {e}}^{\frac {2\,x}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}}\,{\mathrm {e}}^{-\frac {8\,x^2}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}}\,{\mathrm {e}}^{\frac {8\,x\,{\mathrm {e}}^3}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^6}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}} \]

[In]

int(-(exp((2*(x - exp(6) + 4*x*exp(3) + log(x)*(exp(6) - x - 4*x*exp(3) + 2*log(2)*(x*exp(6) - 4*x^2*exp(3) -
x^2 + 4*x^3) + 4*x^2 + 2) - 4*x^2))/(log(x)*(2*x*log(2) + 1) - 1))*(2*x - log(x)*(4*x + 2*log(2)*(16*x^2*exp(3
) + 4*x^2 - 32*x^3) + 16*x*exp(3) - 32*x^2) + 8*x*exp(3) - 16*x^2 + log(x)^2*(2*x + 2*log(2)*(4*x + 16*x^2*exp
(3) + 4*x^2 - 32*x^3) + 8*x*exp(3) + 4*log(2)^2*(8*x^3*exp(3) + 2*x^3 - 16*x^4) - 16*x^2) + 4))/(x - log(x)*(2
*x + 4*x^2*log(2)) + log(x)^2*(x + 4*x^3*log(2)^2 + 4*x^2*log(2))),x)

[Out]

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