\(\int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} (12 x^2+3 x^3)+e^x (8+4 x+72 x^2-30 x^3-12 x^4)+(-36+31 x+10 x^2+e^x (-12-11 x-2 x^2)) \log (4 x+x^2)}{(108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} (12 x^3+3 x^4)+e^x (72 x^3-30 x^4-12 x^5)+(12 x-5 x^2-2 x^3+e^x (4 x+x^2)) \log (4 x+x^2)) \log (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x (54 x^4-36 x^5)+(18 x^2+6 e^x x^2-12 x^3) \log (4 x+x^2)+\log ^2(4 x+x^2)}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x (6 x^3-4 x^4)})} \, dx\) [893]

Optimal result

Integrand size = 319, antiderivative size = 31 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\log \left (\log \left (x \left (3-\frac {\log (x (4+x))}{x^2 \left (-3-e^x+2 x\right )}\right )^2\right )\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx \]

[In]

Int[(24 - 4*x + 100*x^2 - 117*x^3 + 12*x^4 + 12*x^5 + E^(2*x)*(12*x^2 + 3*x^3) + E^x*(8 + 4*x + 72*x^2 - 30*x^
3 - 12*x^4) + (-36 + 31*x + 10*x^2 + E^x*(-12 - 11*x - 2*x^2))*Log[4*x + x^2])/((108*x^3 - 117*x^4 + 12*x^5 +
12*x^6 + E^(2*x)*(12*x^3 + 3*x^4) + E^x*(72*x^3 - 30*x^4 - 12*x^5) + (12*x - 5*x^2 - 2*x^3 + E^x*(4*x + x^2))*
Log[4*x + x^2])*Log[(81*x^4 + 9*E^(2*x)*x^4 - 108*x^5 + 36*x^6 + E^x*(54*x^4 - 36*x^5) + (18*x^2 + 6*E^x*x^2 -
12*x^3)*Log[4*x + x^2] + Log[4*x + x^2]^2)/(9*x^3 + E^(2*x)*x^3 - 12*x^4 + 4*x^5 + E^x*(6*x^3 - 4*x^4))]),x]

[Out]

10*Defer[Int][1/((3 + E^x - 2*x)*Log[(-3*x^2*(-3 - E^x + 2*x) + Log[x*(4 + x)])^2/((3 + E^x - 2*x)^2*x^3)]), x
] + Defer[Int][1/(x*Log[(-3*x^2*(-3 - E^x + 2*x) + Log[x*(4 + x)])^2/((3 + E^x - 2*x)^2*x^3)]), x] - 4*Defer[I
nt][x/((3 + E^x - 2*x)*Log[(-3*x^2*(-3 - E^x + 2*x) + Log[x*(4 + x)])^2/((3 + E^x - 2*x)^2*x^3)]), x] - 2*Defe
r[Int][1/(x*(-9*x^2 - 3*E^x*x^2 + 6*x^3 - Log[x*(4 + x)])*Log[(-3*x^2*(-3 - E^x + 2*x) + Log[x*(4 + x)])^2/((3
+ E^x - 2*x)^2*x^3)]), x] + 30*Defer[Int][x^2/((-9*x^2 - 3*E^x*x^2 + 6*x^3 - Log[x*(4 + x)])*Log[(-3*x^2*(-3
- E^x + 2*x) + Log[x*(4 + x)])^2/((3 + E^x - 2*x)^2*x^3)]), x] - 12*Defer[Int][x^3/((-9*x^2 - 3*E^x*x^2 + 6*x^
3 - Log[x*(4 + x)])*Log[(-3*x^2*(-3 - E^x + 2*x) + Log[x*(4 + x)])^2/((3 + E^x - 2*x)^2*x^3)]), x] - 2*Defer[I
nt][1/((4 + x)*(-9*x^2 - 3*E^x*x^2 + 6*x^3 - Log[x*(4 + x)])*Log[(-3*x^2*(-3 - E^x + 2*x) + Log[x*(4 + x)])^2/
((3 + E^x - 2*x)^2*x^3)]), x] + 2*Defer[Int][Log[x*(4 + x)]/((-9*x^2 - 3*E^x*x^2 + 6*x^3 - Log[x*(4 + x)])*Log
[(-3*x^2*(-3 - E^x + 2*x) + Log[x*(4 + x)])^2/((3 + E^x - 2*x)^2*x^3)]), x] + 4*Defer[Int][Log[x*(4 + x)]/(x*(
-9*x^2 - 3*E^x*x^2 + 6*x^3 - Log[x*(4 + x)])*Log[(-3*x^2*(-3 - E^x + 2*x) + Log[x*(4 + x)])^2/((3 + E^x - 2*x)
^2*x^3)]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-24+4 x-100 x^2+117 x^3-12 x^4-12 x^5-3 e^{2 x} x^2 (4+x)-e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+(4+x) \left (9-10 x+e^x (3+2 x)\right ) \log (x (4+x))}{\left (3+e^x-2 x\right ) x (4+x) \left (3 x^2 \left (-3-e^x+2 x\right )-\log (x (4+x))\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx \\ & = \int \left (\frac {1}{x \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )}-\frac {2 (-5+2 x)}{\left (3+e^x-2 x\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )}-\frac {2 \left (4+2 x-60 x^3+9 x^4+6 x^5-8 \log (x (4+x))-6 x \log (x (4+x))-x^2 \log (x (4+x))\right )}{x (4+x) \left (-9 x^2-3 e^x x^2+6 x^3-\log (x (4+x))\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )}\right ) \, dx \\ & = -\left (2 \int \frac {-5+2 x}{\left (3+e^x-2 x\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx\right )-2 \int \frac {4+2 x-60 x^3+9 x^4+6 x^5-8 \log (x (4+x))-6 x \log (x (4+x))-x^2 \log (x (4+x))}{x (4+x) \left (-9 x^2-3 e^x x^2+6 x^3-\log (x (4+x))\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx+\int \frac {1}{x \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx \\ & = -\left (2 \int \left (-\frac {5}{\left (3+e^x-2 x\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )}+\frac {2 x}{\left (3+e^x-2 x\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )}\right ) \, dx\right )-2 \int \left (\frac {4+2 x-60 x^3+9 x^4+6 x^5-8 \log (x (4+x))-6 x \log (x (4+x))-x^2 \log (x (4+x))}{4 x \left (-9 x^2-3 e^x x^2+6 x^3-\log (x (4+x))\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )}-\frac {4+2 x-60 x^3+9 x^4+6 x^5-8 \log (x (4+x))-6 x \log (x (4+x))-x^2 \log (x (4+x))}{4 (4+x) \left (-9 x^2-3 e^x x^2+6 x^3-\log (x (4+x))\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )}\right ) \, dx+\int \frac {1}{x \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx \\ & = -\left (\frac {1}{2} \int \frac {4+2 x-60 x^3+9 x^4+6 x^5-8 \log (x (4+x))-6 x \log (x (4+x))-x^2 \log (x (4+x))}{x \left (-9 x^2-3 e^x x^2+6 x^3-\log (x (4+x))\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx\right )+\frac {1}{2} \int \frac {4+2 x-60 x^3+9 x^4+6 x^5-8 \log (x (4+x))-6 x \log (x (4+x))-x^2 \log (x (4+x))}{(4+x) \left (-9 x^2-3 e^x x^2+6 x^3-\log (x (4+x))\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx-4 \int \frac {x}{\left (3+e^x-2 x\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx+10 \int \frac {1}{\left (3+e^x-2 x\right ) \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx+\int \frac {1}{x \log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\log \left (\log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )\right ) \]

[In]

Integrate[(24 - 4*x + 100*x^2 - 117*x^3 + 12*x^4 + 12*x^5 + E^(2*x)*(12*x^2 + 3*x^3) + E^x*(8 + 4*x + 72*x^2 -
30*x^3 - 12*x^4) + (-36 + 31*x + 10*x^2 + E^x*(-12 - 11*x - 2*x^2))*Log[4*x + x^2])/((108*x^3 - 117*x^4 + 12*
x^5 + 12*x^6 + E^(2*x)*(12*x^3 + 3*x^4) + E^x*(72*x^3 - 30*x^4 - 12*x^5) + (12*x - 5*x^2 - 2*x^3 + E^x*(4*x +
x^2))*Log[4*x + x^2])*Log[(81*x^4 + 9*E^(2*x)*x^4 - 108*x^5 + 36*x^6 + E^x*(54*x^4 - 36*x^5) + (18*x^2 + 6*E^x
*x^2 - 12*x^3)*Log[4*x + x^2] + Log[4*x + x^2]^2)/(9*x^3 + E^(2*x)*x^3 - 12*x^4 + 4*x^5 + E^x*(6*x^3 - 4*x^4))
]),x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.13 (sec) , antiderivative size = 2554, normalized size of antiderivative = 82.39

\[\text {Expression too large to display}\]

[In]

int((((-2*x^2-11*x-12)*exp(x)+10*x^2+31*x-36)*ln(x^2+4*x)+(3*x^3+12*x^2)*exp(x)^2+(-12*x^4-30*x^3+72*x^2+4*x+8
)*exp(x)+12*x^5+12*x^4-117*x^3+100*x^2-4*x+24)/(((x^2+4*x)*exp(x)-2*x^3-5*x^2+12*x)*ln(x^2+4*x)+(3*x^4+12*x^3)
*exp(x)^2+(-12*x^5-30*x^4+72*x^3)*exp(x)+12*x^6+12*x^5-117*x^4+108*x^3)/ln((ln(x^2+4*x)^2+(6*exp(x)*x^2-12*x^3
+18*x^2)*ln(x^2+4*x)+9*exp(x)^2*x^4+(-36*x^5+54*x^4)*exp(x)+36*x^6-108*x^5+81*x^4)/(exp(x)^2*x^3+(-4*x^4+6*x^3
)*exp(x)+4*x^5-12*x^4+9*x^3)),x)

[Out]

ln(ln(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*
x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))+1/4*I*(2*Pi+Pi*csgn(I*x^3)*c
sgn(I*x)*csgn(I*x^2)-Pi*csgn(I/(-x+1/2*exp(x)+3/2)^2)*csgn(I*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*cs
gn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^2*exp(x)+18*I
*x^2+2*I*ln(x)+2*I*ln(4+x))^2)*csgn(I/(-x+1/2*exp(x)+3/2)^2*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*csg
n(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^2*exp(x)+18*I*
x^2+2*I*ln(x)+2*I*ln(4+x))^2)-Pi*csgn(I/x^3)*csgn(I/(-x+1/2*exp(x)+3/2)^2*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(
4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^
2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)*csgn(I/x^3*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x
)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^2*exp(x)+18*I*x^2+2
*I*ln(x)+2*I*ln(4+x))^2/(-x+1/2*exp(x)+3/2)^2)+Pi*csgn(I*(-x+1/2*exp(x)+3/2))^2*csgn(I*(-x+1/2*exp(x)+3/2)^2)-
Pi*csgn(I*Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-I*Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-I*Pi*csgn(I*(4+x)*x)^3+I
*Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+6*exp(x)*x^2-12*x^3+18*x^2+2*ln(x)+2*ln(4+x))^2*csgn(I*(Pi*csgn(I*x)*csgn(
I*(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^
3-12*I*x^3+6*I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*
x^2)^2+Pi*csgn(I*x^2)^3-2*Pi*csgn(I/x^3*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*
x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln
(4+x))^2/(-x+1/2*exp(x)+3/2)^2)^2-Pi*csgn(I/(-x+1/2*exp(x)+3/2)^2*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-
Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^2*exp(x)
+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)^3-Pi*csgn(I*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(
I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^2*exp(x)+18*I*x^2+2*I*ln(x
)+2*I*ln(4+x))^2)^3+Pi*csgn(I*(-x+1/2*exp(x)+3/2)^2)^3+Pi*csgn(I/x^3*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*
x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^2*exp
(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2/(-x+1/2*exp(x)+3/2)^2)^3+Pi*csgn(I*x^3)^3+2*Pi*csgn(I*(-x+1/2*exp(x)+3/2
))*csgn(I*(-x+1/2*exp(x)+3/2)^2)^2+Pi*csgn(I/(-x+1/2*exp(x)+3/2)^2)*csgn(I/(-x+1/2*exp(x)+3/2)^2*(Pi*csgn(I*x)
*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+
x)*x)^3-12*I*x^3+6*I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)^2+Pi*csgn(I*(Pi*csgn(I*x)*csgn(I*(4+x))*csg
n(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6
*I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)*csgn(I/(-x+1/2*exp(x)+3/2)^2*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn
(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*
I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)^2-Pi*csgn(I*x^3)^2*csgn(I*x)-Pi*csgn(I*x^3)^2*csgn(I*x^2)-2*Pi
*csgn(I*Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-I*Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-I*Pi*csgn(I*(4+x)*x)^3+I*P
i*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+6*exp(x)*x^2-12*x^3+18*x^2+2*ln(x)+2*ln(4+x))*csgn(I*(Pi*csgn(I*x)*csgn(I*(4
+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12
*I*x^3+6*I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)^2+Pi*csgn(I/x^3)*csgn(I/x^3*(Pi*csgn(I*x)*csgn(I*(4+x
))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I
*x^3+6*I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2/(-x+1/2*exp(x)+3/2)^2)^2+Pi*csgn(I/(-x+1/2*exp(x)+3/2)^2
*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2
+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)*csgn(I/x^3*(Pi*csgn(I*x)*csgn
(I*(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)
^3-12*I*x^3+6*I*x^2*exp(x)+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2/(-x+1/2*exp(x)+3/2)^2)^2+6*I*ln(x)+8*I*ln(2)+4*I*
ln(x-1/2*exp(x)-3/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (29) = 58\).

Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.94 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\log \left (\log \left (\frac {36 \, x^{6} - 108 \, x^{5} + 9 \, x^{4} e^{\left (2 \, x\right )} + 81 \, x^{4} - 18 \, {\left (2 \, x^{5} - 3 \, x^{4}\right )} e^{x} - 6 \, {\left (2 \, x^{3} - x^{2} e^{x} - 3 \, x^{2}\right )} \log \left (x^{2} + 4 \, x\right ) + \log \left (x^{2} + 4 \, x\right )^{2}}{4 \, x^{5} - 12 \, x^{4} + x^{3} e^{\left (2 \, x\right )} + 9 \, x^{3} - 2 \, {\left (2 \, x^{4} - 3 \, x^{3}\right )} e^{x}}\right )\right ) \]

[In]

integrate((((-2*x^2-11*x-12)*exp(x)+10*x^2+31*x-36)*log(x^2+4*x)+(3*x^3+12*x^2)*exp(x)^2+(-12*x^4-30*x^3+72*x^
2+4*x+8)*exp(x)+12*x^5+12*x^4-117*x^3+100*x^2-4*x+24)/(((x^2+4*x)*exp(x)-2*x^3-5*x^2+12*x)*log(x^2+4*x)+(3*x^4
+12*x^3)*exp(x)^2+(-12*x^5-30*x^4+72*x^3)*exp(x)+12*x^6+12*x^5-117*x^4+108*x^3)/log((log(x^2+4*x)^2+(6*exp(x)*
x^2-12*x^3+18*x^2)*log(x^2+4*x)+9*exp(x)^2*x^4+(-36*x^5+54*x^4)*exp(x)+36*x^6-108*x^5+81*x^4)/(exp(x)^2*x^3+(-
4*x^4+6*x^3)*exp(x)+4*x^5-12*x^4+9*x^3)),x, algorithm="fricas")

[Out]

log(log((36*x^6 - 108*x^5 + 9*x^4*e^(2*x) + 81*x^4 - 18*(2*x^5 - 3*x^4)*e^x - 6*(2*x^3 - x^2*e^x - 3*x^2)*log(
x^2 + 4*x) + log(x^2 + 4*x)^2)/(4*x^5 - 12*x^4 + x^3*e^(2*x) + 9*x^3 - 2*(2*x^4 - 3*x^3)*e^x)))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (26) = 52\).

Time = 116.33 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.74 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\log {\left (\log {\left (\frac {36 x^{6} - 108 x^{5} + 9 x^{4} e^{2 x} + 81 x^{4} + \left (- 36 x^{5} + 54 x^{4}\right ) e^{x} + \left (- 12 x^{3} + 6 x^{2} e^{x} + 18 x^{2}\right ) \log {\left (x^{2} + 4 x \right )} + \log {\left (x^{2} + 4 x \right )}^{2}}{4 x^{5} - 12 x^{4} + x^{3} e^{2 x} + 9 x^{3} + \left (- 4 x^{4} + 6 x^{3}\right ) e^{x}} \right )} \right )} \]

[In]

integrate((((-2*x**2-11*x-12)*exp(x)+10*x**2+31*x-36)*ln(x**2+4*x)+(3*x**3+12*x**2)*exp(x)**2+(-12*x**4-30*x**
3+72*x**2+4*x+8)*exp(x)+12*x**5+12*x**4-117*x**3+100*x**2-4*x+24)/(((x**2+4*x)*exp(x)-2*x**3-5*x**2+12*x)*ln(x
**2+4*x)+(3*x**4+12*x**3)*exp(x)**2+(-12*x**5-30*x**4+72*x**3)*exp(x)+12*x**6+12*x**5-117*x**4+108*x**3)/ln((l
n(x**2+4*x)**2+(6*exp(x)*x**2-12*x**3+18*x**2)*ln(x**2+4*x)+9*exp(x)**2*x**4+(-36*x**5+54*x**4)*exp(x)+36*x**6
-108*x**5+81*x**4)/(exp(x)**2*x**3+(-4*x**4+6*x**3)*exp(x)+4*x**5-12*x**4+9*x**3)),x)

[Out]

log(log((36*x**6 - 108*x**5 + 9*x**4*exp(2*x) + 81*x**4 + (-36*x**5 + 54*x**4)*exp(x) + (-12*x**3 + 6*x**2*exp
(x) + 18*x**2)*log(x**2 + 4*x) + log(x**2 + 4*x)**2)/(4*x**5 - 12*x**4 + x**3*exp(2*x) + 9*x**3 + (-4*x**4 + 6
*x**3)*exp(x))))

Maxima [A] (verification not implemented)

none

Time = 57.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\log \left (\log \left (-6 \, x^{3} + 3 \, x^{2} e^{x} + 9 \, x^{2} + \log \left (x + 4\right ) + \log \left (x\right )\right ) - \frac {3}{2} \, \log \left (x\right ) - \log \left (-2 \, x + e^{x} + 3\right )\right ) \]

[In]

integrate((((-2*x^2-11*x-12)*exp(x)+10*x^2+31*x-36)*log(x^2+4*x)+(3*x^3+12*x^2)*exp(x)^2+(-12*x^4-30*x^3+72*x^
2+4*x+8)*exp(x)+12*x^5+12*x^4-117*x^3+100*x^2-4*x+24)/(((x^2+4*x)*exp(x)-2*x^3-5*x^2+12*x)*log(x^2+4*x)+(3*x^4
+12*x^3)*exp(x)^2+(-12*x^5-30*x^4+72*x^3)*exp(x)+12*x^6+12*x^5-117*x^4+108*x^3)/log((log(x^2+4*x)^2+(6*exp(x)*
x^2-12*x^3+18*x^2)*log(x^2+4*x)+9*exp(x)^2*x^4+(-36*x^5+54*x^4)*exp(x)+36*x^6-108*x^5+81*x^4)/(exp(x)^2*x^3+(-
4*x^4+6*x^3)*exp(x)+4*x^5-12*x^4+9*x^3)),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\int { \frac {12 \, x^{5} + 12 \, x^{4} - 117 \, x^{3} + 100 \, x^{2} + 3 \, {\left (x^{3} + 4 \, x^{2}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (6 \, x^{4} + 15 \, x^{3} - 36 \, x^{2} - 2 \, x - 4\right )} e^{x} + {\left (10 \, x^{2} - {\left (2 \, x^{2} + 11 \, x + 12\right )} e^{x} + 31 \, x - 36\right )} \log \left (x^{2} + 4 \, x\right ) - 4 \, x + 24}{{\left (12 \, x^{6} + 12 \, x^{5} - 117 \, x^{4} + 108 \, x^{3} + 3 \, {\left (x^{4} + 4 \, x^{3}\right )} e^{\left (2 \, x\right )} - 6 \, {\left (2 \, x^{5} + 5 \, x^{4} - 12 \, x^{3}\right )} e^{x} - {\left (2 \, x^{3} + 5 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x} - 12 \, x\right )} \log \left (x^{2} + 4 \, x\right )\right )} \log \left (\frac {36 \, x^{6} - 108 \, x^{5} + 9 \, x^{4} e^{\left (2 \, x\right )} + 81 \, x^{4} - 18 \, {\left (2 \, x^{5} - 3 \, x^{4}\right )} e^{x} - 6 \, {\left (2 \, x^{3} - x^{2} e^{x} - 3 \, x^{2}\right )} \log \left (x^{2} + 4 \, x\right ) + \log \left (x^{2} + 4 \, x\right )^{2}}{4 \, x^{5} - 12 \, x^{4} + x^{3} e^{\left (2 \, x\right )} + 9 \, x^{3} - 2 \, {\left (2 \, x^{4} - 3 \, x^{3}\right )} e^{x}}\right )} \,d x } \]

[In]

integrate((((-2*x^2-11*x-12)*exp(x)+10*x^2+31*x-36)*log(x^2+4*x)+(3*x^3+12*x^2)*exp(x)^2+(-12*x^4-30*x^3+72*x^
2+4*x+8)*exp(x)+12*x^5+12*x^4-117*x^3+100*x^2-4*x+24)/(((x^2+4*x)*exp(x)-2*x^3-5*x^2+12*x)*log(x^2+4*x)+(3*x^4
+12*x^3)*exp(x)^2+(-12*x^5-30*x^4+72*x^3)*exp(x)+12*x^6+12*x^5-117*x^4+108*x^3)/log((log(x^2+4*x)^2+(6*exp(x)*
x^2-12*x^3+18*x^2)*log(x^2+4*x)+9*exp(x)^2*x^4+(-36*x^5+54*x^4)*exp(x)+36*x^6-108*x^5+81*x^4)/(exp(x)^2*x^3+(-
4*x^4+6*x^3)*exp(x)+4*x^5-12*x^4+9*x^3)),x, algorithm="giac")

[Out]

integrate((12*x^5 + 12*x^4 - 117*x^3 + 100*x^2 + 3*(x^3 + 4*x^2)*e^(2*x) - 2*(6*x^4 + 15*x^3 - 36*x^2 - 2*x -
4)*e^x + (10*x^2 - (2*x^2 + 11*x + 12)*e^x + 31*x - 36)*log(x^2 + 4*x) - 4*x + 24)/((12*x^6 + 12*x^5 - 117*x^4
+ 108*x^3 + 3*(x^4 + 4*x^3)*e^(2*x) - 6*(2*x^5 + 5*x^4 - 12*x^3)*e^x - (2*x^3 + 5*x^2 - (x^2 + 4*x)*e^x - 12*
x)*log(x^2 + 4*x))*log((36*x^6 - 108*x^5 + 9*x^4*e^(2*x) + 81*x^4 - 18*(2*x^5 - 3*x^4)*e^x - 6*(2*x^3 - x^2*e^
x - 3*x^2)*log(x^2 + 4*x) + log(x^2 + 4*x)^2)/(4*x^5 - 12*x^4 + x^3*e^(2*x) + 9*x^3 - 2*(2*x^4 - 3*x^3)*e^x)))
, x)

Mupad [B] (verification not implemented)

Time = 10.96 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.84 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\ln \left (\ln \left (\frac {{\mathrm {e}}^x\,\left (54\,x^4-36\,x^5\right )+9\,x^4\,{\mathrm {e}}^{2\,x}+\ln \left (x^2+4\,x\right )\,\left (6\,x^2\,{\mathrm {e}}^x+18\,x^2-12\,x^3\right )+81\,x^4-108\,x^5+36\,x^6+{\ln \left (x^2+4\,x\right )}^2}{{\mathrm {e}}^x\,\left (6\,x^3-4\,x^4\right )+x^3\,{\mathrm {e}}^{2\,x}+9\,x^3-12\,x^4+4\,x^5}\right )\right ) \]

[In]

int((exp(x)*(4*x + 72*x^2 - 30*x^3 - 12*x^4 + 8) - 4*x + exp(2*x)*(12*x^2 + 3*x^3) + 100*x^2 - 117*x^3 + 12*x^
4 + 12*x^5 + log(4*x + x^2)*(31*x - exp(x)*(11*x + 2*x^2 + 12) + 10*x^2 - 36) + 24)/(log((exp(x)*(54*x^4 - 36*
x^5) + 9*x^4*exp(2*x) + log(4*x + x^2)*(6*x^2*exp(x) + 18*x^2 - 12*x^3) + 81*x^4 - 108*x^5 + 36*x^6 + log(4*x
+ x^2)^2)/(exp(x)*(6*x^3 - 4*x^4) + x^3*exp(2*x) + 9*x^3 - 12*x^4 + 4*x^5))*(exp(2*x)*(12*x^3 + 3*x^4) - exp(x
)*(30*x^4 - 72*x^3 + 12*x^5) + log(4*x + x^2)*(12*x + exp(x)*(4*x + x^2) - 5*x^2 - 2*x^3) + 108*x^3 - 117*x^4
+ 12*x^5 + 12*x^6)),x)

[Out]

log(log((exp(x)*(54*x^4 - 36*x^5) + 9*x^4*exp(2*x) + log(4*x + x^2)*(6*x^2*exp(x) + 18*x^2 - 12*x^3) + 81*x^4
- 108*x^5 + 36*x^6 + log(4*x + x^2)^2)/(exp(x)*(6*x^3 - 4*x^4) + x^3*exp(2*x) + 9*x^3 - 12*x^4 + 4*x^5)))