Integrand size = 370, antiderivative size = 28 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {\log ^2\left (4 \left (x-\frac {1}{2+x}\right )\right )}{\left (3+e^{4 x}-\log (x)\right )^2} \] Output:
ln(4*x-4/(2+x))^2/(3+exp(4*x)-ln(x))^2
\[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx \] Input:
Integrate[((-30*x - 24*x^2 - 6*x^3 + E^(4*x)*(-10*x - 8*x^2 - 2*x^3) + (10 *x + 8*x^2 + 2*x^3)*Log[x])*Log[(-4 + 8*x + 4*x^2)/(2 + x)] + (4 - 6*x - 8 *x^2 - 2*x^3 + E^(4*x)*(-16*x + 24*x^2 + 32*x^3 + 8*x^4))*Log[(-4 + 8*x + 4*x^2)/(2 + x)]^2)/(54*x - 81*x^2 - 108*x^3 - 27*x^4 + E^(4*x)*(54*x - 81* x^2 - 108*x^3 - 27*x^4) + E^(8*x)*(18*x - 27*x^2 - 36*x^3 - 9*x^4) + E^(12 *x)*(2*x - 3*x^2 - 4*x^3 - x^4) + (-54*x + 81*x^2 + 108*x^3 + 27*x^4 + E^( 8*x)*(-6*x + 9*x^2 + 12*x^3 + 3*x^4) + E^(4*x)*(-36*x + 54*x^2 + 72*x^3 + 18*x^4))*Log[x] + (18*x - 27*x^2 - 36*x^3 - 9*x^4 + E^(4*x)*(6*x - 9*x^2 - 12*x^3 - 3*x^4))*Log[x]^2 + (-2*x + 3*x^2 + 4*x^3 + x^4)*Log[x]^3),x]
Output:
Integrate[((-30*x - 24*x^2 - 6*x^3 + E^(4*x)*(-10*x - 8*x^2 - 2*x^3) + (10 *x + 8*x^2 + 2*x^3)*Log[x])*Log[(-4 + 8*x + 4*x^2)/(2 + x)] + (4 - 6*x - 8 *x^2 - 2*x^3 + E^(4*x)*(-16*x + 24*x^2 + 32*x^3 + 8*x^4))*Log[(-4 + 8*x + 4*x^2)/(2 + x)]^2)/(54*x - 81*x^2 - 108*x^3 - 27*x^4 + E^(4*x)*(54*x - 81* x^2 - 108*x^3 - 27*x^4) + E^(8*x)*(18*x - 27*x^2 - 36*x^3 - 9*x^4) + E^(12 *x)*(2*x - 3*x^2 - 4*x^3 - x^4) + (-54*x + 81*x^2 + 108*x^3 + 27*x^4 + E^( 8*x)*(-6*x + 9*x^2 + 12*x^3 + 3*x^4) + E^(4*x)*(-36*x + 54*x^2 + 72*x^3 + 18*x^4))*Log[x] + (18*x - 27*x^2 - 36*x^3 - 9*x^4 + E^(4*x)*(6*x - 9*x^2 - 12*x^3 - 3*x^4))*Log[x]^2 + (-2*x + 3*x^2 + 4*x^3 + x^4)*Log[x]^3), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-6 x^3-24 x^2+e^{4 x} \left (-2 x^3-8 x^2-10 x\right )+\left (2 x^3+8 x^2+10 x\right ) \log (x)-30 x\right ) \log \left (\frac {4 x^2+8 x-4}{x+2}\right )+\left (-2 x^3-8 x^2+e^{4 x} \left (8 x^4+32 x^3+24 x^2-16 x\right )-6 x+4\right ) \log ^2\left (\frac {4 x^2+8 x-4}{x+2}\right )}{-27 x^4-108 x^3-81 x^2+e^{4 x} \left (-27 x^4-108 x^3-81 x^2+54 x\right )+e^{8 x} \left (-9 x^4-36 x^3-27 x^2+18 x\right )+e^{12 x} \left (-x^4-4 x^3-3 x^2+2 x\right )+\left (x^4+4 x^3+3 x^2-2 x\right ) \log ^3(x)+\left (-9 x^4-36 x^3-27 x^2+e^{4 x} \left (-3 x^4-12 x^3-9 x^2+6 x\right )+18 x\right ) \log ^2(x)+\left (27 x^4+108 x^3+81 x^2+e^{8 x} \left (3 x^4+12 x^3+9 x^2-6 x\right )+e^{4 x} \left (18 x^4+72 x^3+54 x^2-36 x\right )-54 x\right ) \log (x)+54 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right ) \left (-\left (\left (e^{4 x}+3\right ) x \left (x^2+4 x+5\right )\right )+x \left (x^2+4 x+5\right ) \log (x)+\left (4 e^{4 x} x-1\right ) \left (x^3+4 x^2+3 x-2\right ) \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right )\right )}{x \left (-x^3-4 x^2-3 x+2\right ) \left (e^{4 x}-\log (x)+3\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\log \left (-\frac {4 \left (-x^2-2 x+1\right )}{x+2}\right ) \left (\left (3+e^{4 x}\right ) x \left (x^2+4 x+5\right )-x \log (x) \left (x^2+4 x+5\right )-\left (1-4 e^{4 x} x\right ) \left (-x^3-4 x^2-3 x+2\right ) \log \left (-\frac {4 \left (-x^2-2 x+1\right )}{x+2}\right )\right )}{x \left (-x^3-4 x^2-3 x+2\right ) \left (-\log (x)+e^{4 x}+3\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\log \left (-\frac {4 \left (-x^2-2 x+1\right )}{x+2}\right ) \left (\left (3+e^{4 x}\right ) x \left (x^2+4 x+5\right )-x \log (x) \left (x^2+4 x+5\right )-\left (1-4 e^{4 x} x\right ) \left (-x^3-4 x^2-3 x+2\right ) \log \left (-\frac {4 \left (-x^2-2 x+1\right )}{x+2}\right )\right )}{x \left (-x^3-4 x^2-3 x+2\right ) \left (-\log (x)+e^{4 x}+3\right )^3}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle -2 \int \left (\frac {\log \left (-\frac {4 \left (-x^2-2 x+1\right )}{x+2}\right ) \left (\left (3+e^{4 x}\right ) x \left (x^2+4 x+5\right )-x \log (x) \left (x^2+4 x+5\right )-\left (1-4 e^{4 x} x\right ) \left (-x^3-4 x^2-3 x+2\right ) \log \left (-\frac {4 \left (-x^2-2 x+1\right )}{x+2}\right )\right )}{x (x+2) \left (-\log (x)+e^{4 x}+3\right )^3}-\frac {\log \left (-\frac {4 \left (-x^2-2 x+1\right )}{x+2}\right ) \left (\left (3+e^{4 x}\right ) x \left (x^2+4 x+5\right )-x \log (x) \left (x^2+4 x+5\right )-\left (1-4 e^{4 x} x\right ) \left (-x^3-4 x^2-3 x+2\right ) \log \left (-\frac {4 \left (-x^2-2 x+1\right )}{x+2}\right )\right )}{\left (x^2+2 x-1\right ) \left (-\log (x)+e^{4 x}+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right ) \left (\left (3+e^{4 x}\right ) x \left (x^2+4 x+5\right )-x \log (x) \left (x^2+4 x+5\right )-\left (4 e^{4 x} x-1\right ) \left (x^3+4 x^2+3 x-2\right ) \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right )\right )}{x (x+2) \left (-x^2-2 x+1\right ) \left (-\log (x)+e^{4 x}+3\right )^3}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -2 \int \left (\frac {(4 \log (x) x-12 x-1) \log ^2\left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right )}{x \left (-\log (x)+e^{4 x}+3\right )^3}+\frac {\left (4 \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right ) x^3+16 \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right ) x^2-x^2+12 \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right ) x-4 x-8 \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right )-5\right ) \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right )}{(x+2) \left (x^2+2 x-1\right ) \left (-\log (x)+e^{4 x}+3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -2 \int \left (\frac {(4 \log (x) x-12 x-1) \log ^2\left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right )}{x \left (-\log (x)+e^{4 x}+3\right )^3}+\frac {\left (4 \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right ) x^3+16 \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right ) x^2-x^2+12 \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right ) x-4 x-8 \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right )-5\right ) \log \left (\frac {4 \left (x^2+2 x-1\right )}{x+2}\right )}{(x+2) \left (x^2+2 x-1\right ) \left (-\log (x)+e^{4 x}+3\right )^2}\right )dx\) |
Input:
Int[((-30*x - 24*x^2 - 6*x^3 + E^(4*x)*(-10*x - 8*x^2 - 2*x^3) + (10*x + 8 *x^2 + 2*x^3)*Log[x])*Log[(-4 + 8*x + 4*x^2)/(2 + x)] + (4 - 6*x - 8*x^2 - 2*x^3 + E^(4*x)*(-16*x + 24*x^2 + 32*x^3 + 8*x^4))*Log[(-4 + 8*x + 4*x^2) /(2 + x)]^2)/(54*x - 81*x^2 - 108*x^3 - 27*x^4 + E^(4*x)*(54*x - 81*x^2 - 108*x^3 - 27*x^4) + E^(8*x)*(18*x - 27*x^2 - 36*x^3 - 9*x^4) + E^(12*x)*(2 *x - 3*x^2 - 4*x^3 - x^4) + (-54*x + 81*x^2 + 108*x^3 + 27*x^4 + E^(8*x)*( -6*x + 9*x^2 + 12*x^3 + 3*x^4) + E^(4*x)*(-36*x + 54*x^2 + 72*x^3 + 18*x^4 ))*Log[x] + (18*x - 27*x^2 - 36*x^3 - 9*x^4 + E^(4*x)*(6*x - 9*x^2 - 12*x^ 3 - 3*x^4))*Log[x]^2 + (-2*x + 3*x^2 + 4*x^3 + x^4)*Log[x]^3),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 870, normalized size of antiderivative = 31.07
\[\text {Expression too large to display}\]
Input:
int((((8*x^4+32*x^3+24*x^2-16*x)*exp(4*x)-2*x^3-8*x^2-6*x+4)*ln((4*x^2+8*x -4)/(2+x))^2+((2*x^3+8*x^2+10*x)*ln(x)+(-2*x^3-8*x^2-10*x)*exp(4*x)-6*x^3- 24*x^2-30*x)*ln((4*x^2+8*x-4)/(2+x)))/((x^4+4*x^3+3*x^2-2*x)*ln(x)^3+((-3* x^4-12*x^3-9*x^2+6*x)*exp(4*x)-9*x^4-36*x^3-27*x^2+18*x)*ln(x)^2+((3*x^4+1 2*x^3+9*x^2-6*x)*exp(4*x)^2+(18*x^4+72*x^3+54*x^2-36*x)*exp(4*x)+27*x^4+10 8*x^3+81*x^2-54*x)*ln(x)+(-x^4-4*x^3-3*x^2+2*x)*exp(4*x)^3+(-9*x^4-36*x^3- 27*x^2+18*x)*exp(4*x)^2+(-27*x^4-108*x^3-81*x^2+54*x)*exp(4*x)-27*x^4-108* x^3-81*x^2+54*x),x)
Output:
1/(3+exp(4*x)-ln(x))^2*ln(x^2+2*x-1)^2+(-I*Pi*csgn(I/(2+x))*csgn(I*(x^2+2* x-1))*csgn(I/(2+x)*(x^2+2*x-1))+I*Pi*csgn(I/(2+x))*csgn(I/(2+x)*(x^2+2*x-1 ))^2+I*Pi*csgn(I*(x^2+2*x-1))*csgn(I/(2+x)*(x^2+2*x-1))^2-I*Pi*csgn(I/(2+x )*(x^2+2*x-1))^3+4*ln(2)-2*ln(2+x))/(3+exp(4*x)-ln(x))^2*ln(x^2+2*x-1)+1/4 *(-Pi^2*csgn(I/(2+x))^2*csgn(I/(2+x)*(x^2+2*x-1))^4-Pi^2*csgn(I*(x^2+2*x-1 ))^2*csgn(I/(2+x)*(x^2+2*x-1))^4+16*ln(2)^2-Pi^2*csgn(I/(2+x))^2*csgn(I*(x ^2+2*x-1))^2*csgn(I/(2+x)*(x^2+2*x-1))^2+2*Pi^2*csgn(I/(2+x))^2*csgn(I*(x^ 2+2*x-1))*csgn(I/(2+x)*(x^2+2*x-1))^3+2*Pi^2*csgn(I/(2+x))*csgn(I*(x^2+2*x -1))^2*csgn(I/(2+x)*(x^2+2*x-1))^3+2*Pi^2*csgn(I/(2+x))*csgn(I/(2+x)*(x^2+ 2*x-1))^5+2*Pi^2*csgn(I*(x^2+2*x-1))*csgn(I/(2+x)*(x^2+2*x-1))^5+8*I*ln(2) *Pi*csgn(I/(2+x))*csgn(I/(2+x)*(x^2+2*x-1))^2-4*I*ln(2+x)*Pi*csgn(I*(x^2+2 *x-1))*csgn(I/(2+x)*(x^2+2*x-1))^2+4*ln(2+x)^2-16*ln(2)*ln(2+x)-Pi^2*csgn( I/(2+x)*(x^2+2*x-1))^6-8*I*ln(2)*Pi*csgn(I/(2+x))*csgn(I*(x^2+2*x-1))*csgn (I/(2+x)*(x^2+2*x-1))+4*I*ln(2+x)*Pi*csgn(I/(2+x)*(x^2+2*x-1))^3-8*I*ln(2) *Pi*csgn(I/(2+x)*(x^2+2*x-1))^3+8*I*ln(2)*Pi*csgn(I*(x^2+2*x-1))*csgn(I/(2 +x)*(x^2+2*x-1))^2-4*Pi^2*csgn(I/(2+x))*csgn(I*(x^2+2*x-1))*csgn(I/(2+x)*( x^2+2*x-1))^4-4*I*ln(2+x)*Pi*csgn(I/(2+x))*csgn(I/(2+x)*(x^2+2*x-1))^2+4*I *ln(2+x)*Pi*csgn(I/(2+x))*csgn(I*(x^2+2*x-1))*csgn(I/(2+x)*(x^2+2*x-1)))/( 3+exp(4*x)-ln(x))^2
Time = 0.10 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=-\frac {\log \left (\frac {4 \, {\left (x^{2} + 2 \, x - 1\right )}}{x + 2}\right )^{2}}{2 \, {\left (e^{\left (4 \, x\right )} + 3\right )} \log \left (x\right ) - \log \left (x\right )^{2} - e^{\left (8 \, x\right )} - 6 \, e^{\left (4 \, x\right )} - 9} \] Input:
integrate((((8*x^4+32*x^3+24*x^2-16*x)*exp(4*x)-2*x^3-8*x^2-6*x+4)*log((4* x^2+8*x-4)/(2+x))^2+((2*x^3+8*x^2+10*x)*log(x)+(-2*x^3-8*x^2-10*x)*exp(4*x )-6*x^3-24*x^2-30*x)*log((4*x^2+8*x-4)/(2+x)))/((x^4+4*x^3+3*x^2-2*x)*log( x)^3+((-3*x^4-12*x^3-9*x^2+6*x)*exp(4*x)-9*x^4-36*x^3-27*x^2+18*x)*log(x)^ 2+((3*x^4+12*x^3+9*x^2-6*x)*exp(4*x)^2+(18*x^4+72*x^3+54*x^2-36*x)*exp(4*x )+27*x^4+108*x^3+81*x^2-54*x)*log(x)+(-x^4-4*x^3-3*x^2+2*x)*exp(4*x)^3+(-9 *x^4-36*x^3-27*x^2+18*x)*exp(4*x)^2+(-27*x^4-108*x^3-81*x^2+54*x)*exp(4*x) -27*x^4-108*x^3-81*x^2+54*x),x, algorithm="fricas")
Output:
-log(4*(x^2 + 2*x - 1)/(x + 2))^2/(2*(e^(4*x) + 3)*log(x) - log(x)^2 - e^( 8*x) - 6*e^(4*x) - 9)
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {\log {\left (\frac {4 x^{2} + 8 x - 4}{x + 2} \right )}^{2}}{\left (6 - 2 \log {\left (x \right )}\right ) e^{4 x} + e^{8 x} + \log {\left (x \right )}^{2} - 6 \log {\left (x \right )} + 9} \] Input:
integrate((((8*x**4+32*x**3+24*x**2-16*x)*exp(4*x)-2*x**3-8*x**2-6*x+4)*ln ((4*x**2+8*x-4)/(2+x))**2+((2*x**3+8*x**2+10*x)*ln(x)+(-2*x**3-8*x**2-10*x )*exp(4*x)-6*x**3-24*x**2-30*x)*ln((4*x**2+8*x-4)/(2+x)))/((x**4+4*x**3+3* x**2-2*x)*ln(x)**3+((-3*x**4-12*x**3-9*x**2+6*x)*exp(4*x)-9*x**4-36*x**3-2 7*x**2+18*x)*ln(x)**2+((3*x**4+12*x**3+9*x**2-6*x)*exp(4*x)**2+(18*x**4+72 *x**3+54*x**2-36*x)*exp(4*x)+27*x**4+108*x**3+81*x**2-54*x)*ln(x)+(-x**4-4 *x**3-3*x**2+2*x)*exp(4*x)**3+(-9*x**4-36*x**3-27*x**2+18*x)*exp(4*x)**2+( -27*x**4-108*x**3-81*x**2+54*x)*exp(4*x)-27*x**4-108*x**3-81*x**2+54*x),x)
Output:
log((4*x**2 + 8*x - 4)/(x + 2))**2/((6 - 2*log(x))*exp(4*x) + exp(8*x) + l og(x)**2 - 6*log(x) + 9)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (27) = 54\).
Time = 0.89 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.07 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=-\frac {4 \, \log \left (2\right )^{2} + 2 \, {\left (2 \, \log \left (2\right ) - \log \left (x + 2\right )\right )} \log \left (x^{2} + 2 \, x - 1\right ) + \log \left (x^{2} + 2 \, x - 1\right )^{2} - 4 \, \log \left (2\right ) \log \left (x + 2\right ) + \log \left (x + 2\right )^{2}}{2 \, {\left (\log \left (x\right ) - 3\right )} e^{\left (4 \, x\right )} - \log \left (x\right )^{2} - e^{\left (8 \, x\right )} + 6 \, \log \left (x\right ) - 9} \] Input:
integrate((((8*x^4+32*x^3+24*x^2-16*x)*exp(4*x)-2*x^3-8*x^2-6*x+4)*log((4* x^2+8*x-4)/(2+x))^2+((2*x^3+8*x^2+10*x)*log(x)+(-2*x^3-8*x^2-10*x)*exp(4*x )-6*x^3-24*x^2-30*x)*log((4*x^2+8*x-4)/(2+x)))/((x^4+4*x^3+3*x^2-2*x)*log( x)^3+((-3*x^4-12*x^3-9*x^2+6*x)*exp(4*x)-9*x^4-36*x^3-27*x^2+18*x)*log(x)^ 2+((3*x^4+12*x^3+9*x^2-6*x)*exp(4*x)^2+(18*x^4+72*x^3+54*x^2-36*x)*exp(4*x )+27*x^4+108*x^3+81*x^2-54*x)*log(x)+(-x^4-4*x^3-3*x^2+2*x)*exp(4*x)^3+(-9 *x^4-36*x^3-27*x^2+18*x)*exp(4*x)^2+(-27*x^4-108*x^3-81*x^2+54*x)*exp(4*x) -27*x^4-108*x^3-81*x^2+54*x),x, algorithm="maxima")
Output:
-(4*log(2)^2 + 2*(2*log(2) - log(x + 2))*log(x^2 + 2*x - 1) + log(x^2 + 2* x - 1)^2 - 4*log(2)*log(x + 2) + log(x + 2)^2)/(2*(log(x) - 3)*e^(4*x) - l og(x)^2 - e^(8*x) + 6*log(x) - 9)
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (27) = 54\).
Time = 0.86 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.61 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=-\frac {\log \left (4 \, x^{2} + 8 \, x - 4\right )^{2} - 2 \, \log \left (4 \, x^{2} + 8 \, x - 4\right ) \log \left (x + 2\right ) + \log \left (x + 2\right )^{2}}{2 \, e^{\left (4 \, x\right )} \log \left (x\right ) - \log \left (x\right )^{2} - e^{\left (8 \, x\right )} - 6 \, e^{\left (4 \, x\right )} + 6 \, \log \left (x\right ) - 9} \] Input:
integrate((((8*x^4+32*x^3+24*x^2-16*x)*exp(4*x)-2*x^3-8*x^2-6*x+4)*log((4* x^2+8*x-4)/(2+x))^2+((2*x^3+8*x^2+10*x)*log(x)+(-2*x^3-8*x^2-10*x)*exp(4*x )-6*x^3-24*x^2-30*x)*log((4*x^2+8*x-4)/(2+x)))/((x^4+4*x^3+3*x^2-2*x)*log( x)^3+((-3*x^4-12*x^3-9*x^2+6*x)*exp(4*x)-9*x^4-36*x^3-27*x^2+18*x)*log(x)^ 2+((3*x^4+12*x^3+9*x^2-6*x)*exp(4*x)^2+(18*x^4+72*x^3+54*x^2-36*x)*exp(4*x )+27*x^4+108*x^3+81*x^2-54*x)*log(x)+(-x^4-4*x^3-3*x^2+2*x)*exp(4*x)^3+(-9 *x^4-36*x^3-27*x^2+18*x)*exp(4*x)^2+(-27*x^4-108*x^3-81*x^2+54*x)*exp(4*x) -27*x^4-108*x^3-81*x^2+54*x),x, algorithm="giac")
Output:
-(log(4*x^2 + 8*x - 4)^2 - 2*log(4*x^2 + 8*x - 4)*log(x + 2) + log(x + 2)^ 2)/(2*e^(4*x)*log(x) - log(x)^2 - e^(8*x) - 6*e^(4*x) + 6*log(x) - 9)
Time = 3.81 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {{\ln \left (\frac {4\,x^2+8\,x-4}{x+2}\right )}^2}{{\ln \left (x\right )}^2+\left (-2\,{\mathrm {e}}^{4\,x}-6\right )\,\ln \left (x\right )+6\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{8\,x}+9} \] Input:
int((log((8*x + 4*x^2 - 4)/(x + 2))^2*(6*x - exp(4*x)*(24*x^2 - 16*x + 32* x^3 + 8*x^4) + 8*x^2 + 2*x^3 - 4) + log((8*x + 4*x^2 - 4)/(x + 2))*(30*x + exp(4*x)*(10*x + 8*x^2 + 2*x^3) + 24*x^2 + 6*x^3 - log(x)*(10*x + 8*x^2 + 2*x^3)))/(exp(12*x)*(3*x^2 - 2*x + 4*x^3 + x^4) - log(x)*(exp(8*x)*(9*x^2 - 6*x + 12*x^3 + 3*x^4) - 54*x + exp(4*x)*(54*x^2 - 36*x + 72*x^3 + 18*x^ 4) + 81*x^2 + 108*x^3 + 27*x^4) - 54*x - log(x)^3*(3*x^2 - 2*x + 4*x^3 + x ^4) + exp(8*x)*(27*x^2 - 18*x + 36*x^3 + 9*x^4) + exp(4*x)*(81*x^2 - 54*x + 108*x^3 + 27*x^4) + 81*x^2 + 108*x^3 + 27*x^4 + log(x)^2*(exp(4*x)*(9*x^ 2 - 6*x + 12*x^3 + 3*x^4) - 18*x + 27*x^2 + 36*x^3 + 9*x^4)),x)
Output:
log((8*x + 4*x^2 - 4)/(x + 2))^2/(6*exp(4*x) + exp(8*x) - log(x)*(2*exp(4* x) + 6) + log(x)^2 + 9)
\[ \int \frac {\left (-30 x-24 x^2-6 x^3+e^{4 x} \left (-10 x-8 x^2-2 x^3\right )+\left (10 x+8 x^2+2 x^3\right ) \log (x)\right ) \log \left (\frac {-4+8 x+4 x^2}{2+x}\right )+\left (4-6 x-8 x^2-2 x^3+e^{4 x} \left (-16 x+24 x^2+32 x^3+8 x^4\right )\right ) \log ^2\left (\frac {-4+8 x+4 x^2}{2+x}\right )}{54 x-81 x^2-108 x^3-27 x^4+e^{4 x} \left (54 x-81 x^2-108 x^3-27 x^4\right )+e^{8 x} \left (18 x-27 x^2-36 x^3-9 x^4\right )+e^{12 x} \left (2 x-3 x^2-4 x^3-x^4\right )+\left (-54 x+81 x^2+108 x^3+27 x^4+e^{8 x} \left (-6 x+9 x^2+12 x^3+3 x^4\right )+e^{4 x} \left (-36 x+54 x^2+72 x^3+18 x^4\right )\right ) \log (x)+\left (18 x-27 x^2-36 x^3-9 x^4+e^{4 x} \left (6 x-9 x^2-12 x^3-3 x^4\right )\right ) \log ^2(x)+\left (-2 x+3 x^2+4 x^3+x^4\right ) \log ^3(x)} \, dx=\text {too large to display} \] Input:
int((((8*x^4+32*x^3+24*x^2-16*x)*exp(4*x)-2*x^3-8*x^2-6*x+4)*log((4*x^2+8* x-4)/(2+x))^2+((2*x^3+8*x^2+10*x)*log(x)+(-2*x^3-8*x^2-10*x)*exp(4*x)-6*x^ 3-24*x^2-30*x)*log((4*x^2+8*x-4)/(2+x)))/((x^4+4*x^3+3*x^2-2*x)*log(x)^3+( (-3*x^4-12*x^3-9*x^2+6*x)*exp(4*x)-9*x^4-36*x^3-27*x^2+18*x)*log(x)^2+((3* x^4+12*x^3+9*x^2-6*x)*exp(4*x)^2+(18*x^4+72*x^3+54*x^2-36*x)*exp(4*x)+27*x ^4+108*x^3+81*x^2-54*x)*log(x)+(-x^4-4*x^3-3*x^2+2*x)*exp(4*x)^3+(-9*x^4-3 6*x^3-27*x^2+18*x)*exp(4*x)^2+(-27*x^4-108*x^3-81*x^2+54*x)*exp(4*x)-27*x^ 4-108*x^3-81*x^2+54*x),x)
Output:
2*( - 2*int(log((4*x**2 + 8*x - 4)/(x + 2))**2/(e**(12*x)*x**4 + 4*e**(12* x)*x**3 + 3*e**(12*x)*x**2 - 2*e**(12*x)*x - 3*e**(8*x)*log(x)*x**4 - 12*e **(8*x)*log(x)*x**3 - 9*e**(8*x)*log(x)*x**2 + 6*e**(8*x)*log(x)*x + 9*e** (8*x)*x**4 + 36*e**(8*x)*x**3 + 27*e**(8*x)*x**2 - 18*e**(8*x)*x + 3*e**(4 *x)*log(x)**2*x**4 + 12*e**(4*x)*log(x)**2*x**3 + 9*e**(4*x)*log(x)**2*x** 2 - 6*e**(4*x)*log(x)**2*x - 18*e**(4*x)*log(x)*x**4 - 72*e**(4*x)*log(x)* x**3 - 54*e**(4*x)*log(x)*x**2 + 36*e**(4*x)*log(x)*x + 27*e**(4*x)*x**4 + 108*e**(4*x)*x**3 + 81*e**(4*x)*x**2 - 54*e**(4*x)*x - log(x)**3*x**4 - 4 *log(x)**3*x**3 - 3*log(x)**3*x**2 + 2*log(x)**3*x + 9*log(x)**2*x**4 + 36 *log(x)**2*x**3 + 27*log(x)**2*x**2 - 18*log(x)**2*x - 27*log(x)*x**4 - 10 8*log(x)*x**3 - 81*log(x)*x**2 + 54*log(x)*x + 27*x**4 + 108*x**3 + 81*x** 2 - 54*x),x) + 3*int(log((4*x**2 + 8*x - 4)/(x + 2))**2/(e**(12*x)*x**3 + 4*e**(12*x)*x**2 + 3*e**(12*x)*x - 2*e**(12*x) - 3*e**(8*x)*log(x)*x**3 - 12*e**(8*x)*log(x)*x**2 - 9*e**(8*x)*log(x)*x + 6*e**(8*x)*log(x) + 9*e**( 8*x)*x**3 + 36*e**(8*x)*x**2 + 27*e**(8*x)*x - 18*e**(8*x) + 3*e**(4*x)*lo g(x)**2*x**3 + 12*e**(4*x)*log(x)**2*x**2 + 9*e**(4*x)*log(x)**2*x - 6*e** (4*x)*log(x)**2 - 18*e**(4*x)*log(x)*x**3 - 72*e**(4*x)*log(x)*x**2 - 54*e **(4*x)*log(x)*x + 36*e**(4*x)*log(x) + 27*e**(4*x)*x**3 + 108*e**(4*x)*x* *2 + 81*e**(4*x)*x - 54*e**(4*x) - log(x)**3*x**3 - 4*log(x)**3*x**2 - 3*l og(x)**3*x + 2*log(x)**3 + 9*log(x)**2*x**3 + 36*log(x)**2*x**2 + 27*lo...