Integrand size = 409, antiderivative size = 33 \[ \int \frac {10 x-20 x^2+30 x^3-20 x^4-2 x^5+10 x^6-20 x^7+20 x^8-10 x^9+2 x^{10}+\left (5 x-16 x^3+52 x^4-60 x^5+28 x^6-4 x^7\right ) \log (3)+\left (-32 x+48 x^2-18 x^3+2 x^4\right ) \log ^2(3)+\left (-10+10 x-20 x^2+8 x^4-32 x^5+48 x^6-32 x^7+8 x^8+\left (32 x^2-72 x^3+48 x^4-8 x^5\right ) \log (3)\right ) \log (x)+\left (-12 x^3+36 x^4-36 x^5+12 x^6+\left (-16 x+20 x^2-4 x^3\right ) \log (3)\right ) \log ^2(x)+\left (8 x^2-16 x^3+8 x^4\right ) \log ^3(x)+\left (-2 x+2 x^2\right ) \log ^4(x)}{x^5-4 x^6+6 x^7-4 x^8+x^9+\left (8 x^3-18 x^4+12 x^5-2 x^6\right ) \log (3)+\left (16 x-8 x^2+x^3\right ) \log ^2(3)+\left (-4 x^4+12 x^5-12 x^6+4 x^7+\left (-16 x^2+20 x^3-4 x^4\right ) \log (3)\right ) \log (x)+\left (6 x^3-12 x^4+6 x^5+\left (8 x-2 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-4 x^2+4 x^3\right ) \log ^3(x)+x \log ^4(x)} \, dx=-2 x+x^2+\frac {5}{(4-x) \log (3)+\left (x-x^2-\log (x)\right )^2} \]
\[ \int \frac {10 x-20 x^2+30 x^3-20 x^4-2 x^5+10 x^6-20 x^7+20 x^8-10 x^9+2 x^{10}+\left (5 x-16 x^3+52 x^4-60 x^5+28 x^6-4 x^7\right ) \log (3)+\left (-32 x+48 x^2-18 x^3+2 x^4\right ) \log ^2(3)+\left (-10+10 x-20 x^2+8 x^4-32 x^5+48 x^6-32 x^7+8 x^8+\left (32 x^2-72 x^3+48 x^4-8 x^5\right ) \log (3)\right ) \log (x)+\left (-12 x^3+36 x^4-36 x^5+12 x^6+\left (-16 x+20 x^2-4 x^3\right ) \log (3)\right ) \log ^2(x)+\left (8 x^2-16 x^3+8 x^4\right ) \log ^3(x)+\left (-2 x+2 x^2\right ) \log ^4(x)}{x^5-4 x^6+6 x^7-4 x^8+x^9+\left (8 x^3-18 x^4+12 x^5-2 x^6\right ) \log (3)+\left (16 x-8 x^2+x^3\right ) \log ^2(3)+\left (-4 x^4+12 x^5-12 x^6+4 x^7+\left (-16 x^2+20 x^3-4 x^4\right ) \log (3)\right ) \log (x)+\left (6 x^3-12 x^4+6 x^5+\left (8 x-2 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-4 x^2+4 x^3\right ) \log ^3(x)+x \log ^4(x)} \, dx=\int \frac {10 x-20 x^2+30 x^3-20 x^4-2 x^5+10 x^6-20 x^7+20 x^8-10 x^9+2 x^{10}+\left (5 x-16 x^3+52 x^4-60 x^5+28 x^6-4 x^7\right ) \log (3)+\left (-32 x+48 x^2-18 x^3+2 x^4\right ) \log ^2(3)+\left (-10+10 x-20 x^2+8 x^4-32 x^5+48 x^6-32 x^7+8 x^8+\left (32 x^2-72 x^3+48 x^4-8 x^5\right ) \log (3)\right ) \log (x)+\left (-12 x^3+36 x^4-36 x^5+12 x^6+\left (-16 x+20 x^2-4 x^3\right ) \log (3)\right ) \log ^2(x)+\left (8 x^2-16 x^3+8 x^4\right ) \log ^3(x)+\left (-2 x+2 x^2\right ) \log ^4(x)}{x^5-4 x^6+6 x^7-4 x^8+x^9+\left (8 x^3-18 x^4+12 x^5-2 x^6\right ) \log (3)+\left (16 x-8 x^2+x^3\right ) \log ^2(3)+\left (-4 x^4+12 x^5-12 x^6+4 x^7+\left (-16 x^2+20 x^3-4 x^4\right ) \log (3)\right ) \log (x)+\left (6 x^3-12 x^4+6 x^5+\left (8 x-2 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-4 x^2+4 x^3\right ) \log ^3(x)+x \log ^4(x)} \, dx \]
Integrate[(10*x - 20*x^2 + 30*x^3 - 20*x^4 - 2*x^5 + 10*x^6 - 20*x^7 + 20* x^8 - 10*x^9 + 2*x^10 + (5*x - 16*x^3 + 52*x^4 - 60*x^5 + 28*x^6 - 4*x^7)* Log[3] + (-32*x + 48*x^2 - 18*x^3 + 2*x^4)*Log[3]^2 + (-10 + 10*x - 20*x^2 + 8*x^4 - 32*x^5 + 48*x^6 - 32*x^7 + 8*x^8 + (32*x^2 - 72*x^3 + 48*x^4 - 8*x^5)*Log[3])*Log[x] + (-12*x^3 + 36*x^4 - 36*x^5 + 12*x^6 + (-16*x + 20* x^2 - 4*x^3)*Log[3])*Log[x]^2 + (8*x^2 - 16*x^3 + 8*x^4)*Log[x]^3 + (-2*x + 2*x^2)*Log[x]^4)/(x^5 - 4*x^6 + 6*x^7 - 4*x^8 + x^9 + (8*x^3 - 18*x^4 + 12*x^5 - 2*x^6)*Log[3] + (16*x - 8*x^2 + x^3)*Log[3]^2 + (-4*x^4 + 12*x^5 - 12*x^6 + 4*x^7 + (-16*x^2 + 20*x^3 - 4*x^4)*Log[3])*Log[x] + (6*x^3 - 12 *x^4 + 6*x^5 + (8*x - 2*x^2)*Log[3])*Log[x]^2 + (-4*x^2 + 4*x^3)*Log[x]^3 + x*Log[x]^4),x]
Integrate[(10*x - 20*x^2 + 30*x^3 - 20*x^4 - 2*x^5 + 10*x^6 - 20*x^7 + 20* x^8 - 10*x^9 + 2*x^10 + (5*x - 16*x^3 + 52*x^4 - 60*x^5 + 28*x^6 - 4*x^7)* Log[3] + (-32*x + 48*x^2 - 18*x^3 + 2*x^4)*Log[3]^2 + (-10 + 10*x - 20*x^2 + 8*x^4 - 32*x^5 + 48*x^6 - 32*x^7 + 8*x^8 + (32*x^2 - 72*x^3 + 48*x^4 - 8*x^5)*Log[3])*Log[x] + (-12*x^3 + 36*x^4 - 36*x^5 + 12*x^6 + (-16*x + 20* x^2 - 4*x^3)*Log[3])*Log[x]^2 + (8*x^2 - 16*x^3 + 8*x^4)*Log[x]^3 + (-2*x + 2*x^2)*Log[x]^4)/(x^5 - 4*x^6 + 6*x^7 - 4*x^8 + x^9 + (8*x^3 - 18*x^4 + 12*x^5 - 2*x^6)*Log[3] + (16*x - 8*x^2 + x^3)*Log[3]^2 + (-4*x^4 + 12*x^5 - 12*x^6 + 4*x^7 + (-16*x^2 + 20*x^3 - 4*x^4)*Log[3])*Log[x] + (6*x^3 - 12 *x^4 + 6*x^5 + (8*x - 2*x^2)*Log[3])*Log[x]^2 + (-4*x^2 + 4*x^3)*Log[x]^3 + x*Log[x]^4), 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.
| \(\displaystyle \int \frac {2 x^{10}-10 x^9+20 x^8-20 x^7+10 x^6-2 x^5-20 x^4+30 x^3-20 x^2+\left (2 x^2-2 x\right ) \log ^4(x)+\left (8 x^4-16 x^3+8 x^2\right ) \log ^3(x)+\left (2 x^4-18 x^3+48 x^2-32 x\right ) \log ^2(3)+\left (-4 x^7+28 x^6-60 x^5+52 x^4-16 x^3+5 x\right ) \log (3)+\left (12 x^6-36 x^5+36 x^4-12 x^3+\left (-4 x^3+20 x^2-16 x\right ) \log (3)\right ) \log ^2(x)+\left (8 x^8-32 x^7+48 x^6-32 x^5+8 x^4-20 x^2+\left (-8 x^5+48 x^4-72 x^3+32 x^2\right ) \log (3)+10 x-10\right ) \log (x)+10 x}{x^9-4 x^8+6 x^7-4 x^6+x^5+\left (4 x^3-4 x^2\right ) \log ^3(x)+\left (x^3-8 x^2+16 x\right ) \log ^2(3)+\left (-2 x^6+12 x^5-18 x^4+8 x^3\right ) \log (3)+\left (6 x^5-12 x^4+6 x^3+\left (8 x-2 x^2\right ) \log (3)\right ) \log ^2(x)+\left (4 x^7-12 x^6+12 x^5-4 x^4+\left (-4 x^4+20 x^3-16 x^2\right ) \log (3)\right ) \log (x)+x \log ^4(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 x^{10}-10 x^9+20 x^8-20 x^7+10 x^6-2 x^5-20 x^4+30 x^3-20 x^2+\left (2 x^2-2 x\right ) \log ^4(x)+\left (8 x^4-16 x^3+8 x^2\right ) \log ^3(x)+\left (2 x^4-18 x^3+48 x^2-32 x\right ) \log ^2(3)+\left (-4 x^7+28 x^6-60 x^5+52 x^4-16 x^3+5 x\right ) \log (3)+\left (12 x^6-36 x^5+36 x^4-12 x^3+\left (-4 x^3+20 x^2-16 x\right ) \log (3)\right ) \log ^2(x)+\left (8 x^8-32 x^7+48 x^6-32 x^5+8 x^4-20 x^2+\left (-8 x^5+48 x^4-72 x^3+32 x^2\right ) \log (3)+10 x-10\right ) \log (x)+10 x}{x \left (x^4-2 x^3+x^2+2 x^2 \log (x)+\log ^2(x)-2 x \log (x)-x \log (3)+\log (81)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-20 x^4+30 x^3-20 x^2 \left (1+\frac {1}{10} \left (\log ^2(81)-16 \log ^2(3)\right )\right )-20 x^2 \log (x)+10 x \left (1+\frac {1}{10} \left (-32 \log ^2(3)-2 \log ^2(81)+16 \log (3) \log (81)+\log (243)\right )\right )+10 x \log (x)-10 \log (x)}{x \left (x^4-2 x^3+x^2+2 x^2 \log (x)+\log ^2(x)-2 x \log (x)-x \log (3)+\log (81)\right )^2}+2 (x-1)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \left (10-32 \log ^2(3)-2 \log ^2(81)+16 \log (3) \log (81)+\log (243)\right ) \int \frac {1}{\left (x^4-2 x^3+2 \log (x) x^2+x^2-2 \log (x) x-\log (3) x+\log ^2(x)+\log (81)\right )^2}dx-2 \left (10-16 \log ^2(3)+\log ^2(81)\right ) \int \frac {x}{\left (x^4-2 x^3+2 \log (x) x^2+x^2-2 \log (x) x-\log (3) x+\log ^2(x)+\log (81)\right )^2}dx+30 \int \frac {x^2}{\left (x^4-2 x^3+2 \log (x) x^2+x^2-2 \log (x) x-\log (3) x+\log ^2(x)+\log (81)\right )^2}dx-20 \int \frac {x^3}{\left (x^4-2 x^3+2 \log (x) x^2+x^2-2 \log (x) x-\log (3) x+\log ^2(x)+\log (81)\right )^2}dx+10 \int \frac {\log (x)}{\left (x^4-2 x^3+2 \log (x) x^2+x^2-2 \log (x) x-\log (3) x+\log ^2(x)+\log (81)\right )^2}dx-10 \int \frac {\log (x)}{x \left (x^4-2 x^3+2 \log (x) x^2+x^2-2 \log (x) x-\log (3) x+\log ^2(x)+\log (81)\right )^2}dx-20 \int \frac {x \log (x)}{\left (x^4-2 x^3+2 \log (x) x^2+x^2-2 \log (x) x-\log (3) x+\log ^2(x)+\log (81)\right )^2}dx+(x-1)^2\) |
Int[(10*x - 20*x^2 + 30*x^3 - 20*x^4 - 2*x^5 + 10*x^6 - 20*x^7 + 20*x^8 - 10*x^9 + 2*x^10 + (5*x - 16*x^3 + 52*x^4 - 60*x^5 + 28*x^6 - 4*x^7)*Log[3] + (-32*x + 48*x^2 - 18*x^3 + 2*x^4)*Log[3]^2 + (-10 + 10*x - 20*x^2 + 8*x ^4 - 32*x^5 + 48*x^6 - 32*x^7 + 8*x^8 + (32*x^2 - 72*x^3 + 48*x^4 - 8*x^5) *Log[3])*Log[x] + (-12*x^3 + 36*x^4 - 36*x^5 + 12*x^6 + (-16*x + 20*x^2 - 4*x^3)*Log[3])*Log[x]^2 + (8*x^2 - 16*x^3 + 8*x^4)*Log[x]^3 + (-2*x + 2*x^ 2)*Log[x]^4)/(x^5 - 4*x^6 + 6*x^7 - 4*x^8 + x^9 + (8*x^3 - 18*x^4 + 12*x^5 - 2*x^6)*Log[3] + (16*x - 8*x^2 + x^3)*Log[3]^2 + (-4*x^4 + 12*x^5 - 12*x ^6 + 4*x^7 + (-16*x^2 + 20*x^3 - 4*x^4)*Log[3])*Log[x] + (6*x^3 - 12*x^4 + 6*x^5 + (8*x - 2*x^2)*Log[3])*Log[x]^2 + (-4*x^2 + 4*x^3)*Log[x]^3 + x*Lo g[x]^4),x]
3.1.66.3.1 Defintions of rubi rules used
Time = 1.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64
| method | result | size |
| default | \(x^{2}-2 x -\frac {5}{-x^{4}-2 x^{2} \ln \left (x \right )+2 x^{3}+x \ln \left (3\right )-\ln \left (x \right )^{2}+2 x \ln \left (x \right )-x^{2}-4 \ln \left (3\right )}\) | \(54\) |
| risch | \(x^{2}-2 x -\frac {5}{-x^{4}-2 x^{2} \ln \left (x \right )+2 x^{3}+x \ln \left (3\right )-\ln \left (x \right )^{2}+2 x \ln \left (x \right )-x^{2}-4 \ln \left (3\right )}\) | \(54\) |
| parallelrisch | \(\frac {-5+x^{3} \ln \left (3\right )+2 x \ln \left (x \right )^{2}+8 x \ln \left (3\right )-2 x^{4} \ln \left (x \right )-6 x^{2} \ln \left (3\right )-x^{2} \ln \left (x \right )^{2}-x^{6}+4 x^{5}-5 x^{4}+2 x^{3}+6 x^{3} \ln \left (x \right )-4 x^{2} \ln \left (x \right )}{-x^{4}-2 x^{2} \ln \left (x \right )+2 x^{3}+x \ln \left (3\right )-\ln \left (x \right )^{2}+2 x \ln \left (x \right )-x^{2}-4 \ln \left (3\right )}\) | \(123\) |
int(((2*x^2-2*x)*ln(x)^4+(8*x^4-16*x^3+8*x^2)*ln(x)^3+((-4*x^3+20*x^2-16*x )*ln(3)+12*x^6-36*x^5+36*x^4-12*x^3)*ln(x)^2+((-8*x^5+48*x^4-72*x^3+32*x^2 )*ln(3)+8*x^8-32*x^7+48*x^6-32*x^5+8*x^4-20*x^2+10*x-10)*ln(x)+(2*x^4-18*x ^3+48*x^2-32*x)*ln(3)^2+(-4*x^7+28*x^6-60*x^5+52*x^4-16*x^3+5*x)*ln(3)+2*x ^10-10*x^9+20*x^8-20*x^7+10*x^6-2*x^5-20*x^4+30*x^3-20*x^2+10*x)/(x*ln(x)^ 4+(4*x^3-4*x^2)*ln(x)^3+((-2*x^2+8*x)*ln(3)+6*x^5-12*x^4+6*x^3)*ln(x)^2+(( -4*x^4+20*x^3-16*x^2)*ln(3)+4*x^7-12*x^6+12*x^5-4*x^4)*ln(x)+(x^3-8*x^2+16 *x)*ln(3)^2+(-2*x^6+12*x^5-18*x^4+8*x^3)*ln(3)+x^9-4*x^8+6*x^7-4*x^6+x^5), x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (30) = 60\).
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.12 \[ \int \frac {10 x-20 x^2+30 x^3-20 x^4-2 x^5+10 x^6-20 x^7+20 x^8-10 x^9+2 x^{10}+\left (5 x-16 x^3+52 x^4-60 x^5+28 x^6-4 x^7\right ) \log (3)+\left (-32 x+48 x^2-18 x^3+2 x^4\right ) \log ^2(3)+\left (-10+10 x-20 x^2+8 x^4-32 x^5+48 x^6-32 x^7+8 x^8+\left (32 x^2-72 x^3+48 x^4-8 x^5\right ) \log (3)\right ) \log (x)+\left (-12 x^3+36 x^4-36 x^5+12 x^6+\left (-16 x+20 x^2-4 x^3\right ) \log (3)\right ) \log ^2(x)+\left (8 x^2-16 x^3+8 x^4\right ) \log ^3(x)+\left (-2 x+2 x^2\right ) \log ^4(x)}{x^5-4 x^6+6 x^7-4 x^8+x^9+\left (8 x^3-18 x^4+12 x^5-2 x^6\right ) \log (3)+\left (16 x-8 x^2+x^3\right ) \log ^2(3)+\left (-4 x^4+12 x^5-12 x^6+4 x^7+\left (-16 x^2+20 x^3-4 x^4\right ) \log (3)\right ) \log (x)+\left (6 x^3-12 x^4+6 x^5+\left (8 x-2 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-4 x^2+4 x^3\right ) \log ^3(x)+x \log ^4(x)} \, dx=\frac {x^{6} - 4 \, x^{5} + 5 \, x^{4} - 2 \, x^{3} + {\left (x^{2} - 2 \, x\right )} \log \left (x\right )^{2} - {\left (x^{3} - 6 \, x^{2} + 8 \, x\right )} \log \left (3\right ) + 2 \, {\left (x^{4} - 3 \, x^{3} + 2 \, x^{2}\right )} \log \left (x\right ) + 5}{x^{4} - 2 \, x^{3} + x^{2} - {\left (x - 4\right )} \log \left (3\right ) + 2 \, {\left (x^{2} - x\right )} \log \left (x\right ) + \log \left (x\right )^{2}} \]
integrate(((2*x^2-2*x)*log(x)^4+(8*x^4-16*x^3+8*x^2)*log(x)^3+((-4*x^3+20* x^2-16*x)*log(3)+12*x^6-36*x^5+36*x^4-12*x^3)*log(x)^2+((-8*x^5+48*x^4-72* x^3+32*x^2)*log(3)+8*x^8-32*x^7+48*x^6-32*x^5+8*x^4-20*x^2+10*x-10)*log(x) +(2*x^4-18*x^3+48*x^2-32*x)*log(3)^2+(-4*x^7+28*x^6-60*x^5+52*x^4-16*x^3+5 *x)*log(3)+2*x^10-10*x^9+20*x^8-20*x^7+10*x^6-2*x^5-20*x^4+30*x^3-20*x^2+1 0*x)/(x*log(x)^4+(4*x^3-4*x^2)*log(x)^3+((-2*x^2+8*x)*log(3)+6*x^5-12*x^4+ 6*x^3)*log(x)^2+((-4*x^4+20*x^3-16*x^2)*log(3)+4*x^7-12*x^6+12*x^5-4*x^4)* log(x)+(x^3-8*x^2+16*x)*log(3)^2+(-2*x^6+12*x^5-18*x^4+8*x^3)*log(3)+x^9-4 *x^8+6*x^7-4*x^6+x^5),x, algorithm=\
(x^6 - 4*x^5 + 5*x^4 - 2*x^3 + (x^2 - 2*x)*log(x)^2 - (x^3 - 6*x^2 + 8*x)* log(3) + 2*(x^4 - 3*x^3 + 2*x^2)*log(x) + 5)/(x^4 - 2*x^3 + x^2 - (x - 4)* log(3) + 2*(x^2 - x)*log(x) + log(x)^2)
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {10 x-20 x^2+30 x^3-20 x^4-2 x^5+10 x^6-20 x^7+20 x^8-10 x^9+2 x^{10}+\left (5 x-16 x^3+52 x^4-60 x^5+28 x^6-4 x^7\right ) \log (3)+\left (-32 x+48 x^2-18 x^3+2 x^4\right ) \log ^2(3)+\left (-10+10 x-20 x^2+8 x^4-32 x^5+48 x^6-32 x^7+8 x^8+\left (32 x^2-72 x^3+48 x^4-8 x^5\right ) \log (3)\right ) \log (x)+\left (-12 x^3+36 x^4-36 x^5+12 x^6+\left (-16 x+20 x^2-4 x^3\right ) \log (3)\right ) \log ^2(x)+\left (8 x^2-16 x^3+8 x^4\right ) \log ^3(x)+\left (-2 x+2 x^2\right ) \log ^4(x)}{x^5-4 x^6+6 x^7-4 x^8+x^9+\left (8 x^3-18 x^4+12 x^5-2 x^6\right ) \log (3)+\left (16 x-8 x^2+x^3\right ) \log ^2(3)+\left (-4 x^4+12 x^5-12 x^6+4 x^7+\left (-16 x^2+20 x^3-4 x^4\right ) \log (3)\right ) \log (x)+\left (6 x^3-12 x^4+6 x^5+\left (8 x-2 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-4 x^2+4 x^3\right ) \log ^3(x)+x \log ^4(x)} \, dx=x^{2} - 2 x + \frac {5}{x^{4} - 2 x^{3} + x^{2} - x \log {\left (3 \right )} + \left (2 x^{2} - 2 x\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 4 \log {\left (3 \right )}} \]
integrate(((2*x**2-2*x)*ln(x)**4+(8*x**4-16*x**3+8*x**2)*ln(x)**3+((-4*x** 3+20*x**2-16*x)*ln(3)+12*x**6-36*x**5+36*x**4-12*x**3)*ln(x)**2+((-8*x**5+ 48*x**4-72*x**3+32*x**2)*ln(3)+8*x**8-32*x**7+48*x**6-32*x**5+8*x**4-20*x* *2+10*x-10)*ln(x)+(2*x**4-18*x**3+48*x**2-32*x)*ln(3)**2+(-4*x**7+28*x**6- 60*x**5+52*x**4-16*x**3+5*x)*ln(3)+2*x**10-10*x**9+20*x**8-20*x**7+10*x**6 -2*x**5-20*x**4+30*x**3-20*x**2+10*x)/(x*ln(x)**4+(4*x**3-4*x**2)*ln(x)**3 +((-2*x**2+8*x)*ln(3)+6*x**5-12*x**4+6*x**3)*ln(x)**2+((-4*x**4+20*x**3-16 *x**2)*ln(3)+4*x**7-12*x**6+12*x**5-4*x**4)*ln(x)+(x**3-8*x**2+16*x)*ln(3) **2+(-2*x**6+12*x**5-18*x**4+8*x**3)*ln(3)+x**9-4*x**8+6*x**7-4*x**6+x**5) ,x)
x**2 - 2*x + 5/(x**4 - 2*x**3 + x**2 - x*log(3) + (2*x**2 - 2*x)*log(x) + log(x)**2 + 4*log(3))
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (30) = 60\).
Time = 0.33 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.18 \[ \int \frac {10 x-20 x^2+30 x^3-20 x^4-2 x^5+10 x^6-20 x^7+20 x^8-10 x^9+2 x^{10}+\left (5 x-16 x^3+52 x^4-60 x^5+28 x^6-4 x^7\right ) \log (3)+\left (-32 x+48 x^2-18 x^3+2 x^4\right ) \log ^2(3)+\left (-10+10 x-20 x^2+8 x^4-32 x^5+48 x^6-32 x^7+8 x^8+\left (32 x^2-72 x^3+48 x^4-8 x^5\right ) \log (3)\right ) \log (x)+\left (-12 x^3+36 x^4-36 x^5+12 x^6+\left (-16 x+20 x^2-4 x^3\right ) \log (3)\right ) \log ^2(x)+\left (8 x^2-16 x^3+8 x^4\right ) \log ^3(x)+\left (-2 x+2 x^2\right ) \log ^4(x)}{x^5-4 x^6+6 x^7-4 x^8+x^9+\left (8 x^3-18 x^4+12 x^5-2 x^6\right ) \log (3)+\left (16 x-8 x^2+x^3\right ) \log ^2(3)+\left (-4 x^4+12 x^5-12 x^6+4 x^7+\left (-16 x^2+20 x^3-4 x^4\right ) \log (3)\right ) \log (x)+\left (6 x^3-12 x^4+6 x^5+\left (8 x-2 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-4 x^2+4 x^3\right ) \log ^3(x)+x \log ^4(x)} \, dx=\frac {x^{6} - 4 \, x^{5} + 5 \, x^{4} - x^{3} {\left (\log \left (3\right ) + 2\right )} + 6 \, x^{2} \log \left (3\right ) + {\left (x^{2} - 2 \, x\right )} \log \left (x\right )^{2} - 8 \, x \log \left (3\right ) + 2 \, {\left (x^{4} - 3 \, x^{3} + 2 \, x^{2}\right )} \log \left (x\right ) + 5}{x^{4} - 2 \, x^{3} + x^{2} - x \log \left (3\right ) + 2 \, {\left (x^{2} - x\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 4 \, \log \left (3\right )} \]
integrate(((2*x^2-2*x)*log(x)^4+(8*x^4-16*x^3+8*x^2)*log(x)^3+((-4*x^3+20* x^2-16*x)*log(3)+12*x^6-36*x^5+36*x^4-12*x^3)*log(x)^2+((-8*x^5+48*x^4-72* x^3+32*x^2)*log(3)+8*x^8-32*x^7+48*x^6-32*x^5+8*x^4-20*x^2+10*x-10)*log(x) +(2*x^4-18*x^3+48*x^2-32*x)*log(3)^2+(-4*x^7+28*x^6-60*x^5+52*x^4-16*x^3+5 *x)*log(3)+2*x^10-10*x^9+20*x^8-20*x^7+10*x^6-2*x^5-20*x^4+30*x^3-20*x^2+1 0*x)/(x*log(x)^4+(4*x^3-4*x^2)*log(x)^3+((-2*x^2+8*x)*log(3)+6*x^5-12*x^4+ 6*x^3)*log(x)^2+((-4*x^4+20*x^3-16*x^2)*log(3)+4*x^7-12*x^6+12*x^5-4*x^4)* log(x)+(x^3-8*x^2+16*x)*log(3)^2+(-2*x^6+12*x^5-18*x^4+8*x^3)*log(3)+x^9-4 *x^8+6*x^7-4*x^6+x^5),x, algorithm=\
(x^6 - 4*x^5 + 5*x^4 - x^3*(log(3) + 2) + 6*x^2*log(3) + (x^2 - 2*x)*log(x )^2 - 8*x*log(3) + 2*(x^4 - 3*x^3 + 2*x^2)*log(x) + 5)/(x^4 - 2*x^3 + x^2 - x*log(3) + 2*(x^2 - x)*log(x) + log(x)^2 + 4*log(3))
Time = 0.58 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {10 x-20 x^2+30 x^3-20 x^4-2 x^5+10 x^6-20 x^7+20 x^8-10 x^9+2 x^{10}+\left (5 x-16 x^3+52 x^4-60 x^5+28 x^6-4 x^7\right ) \log (3)+\left (-32 x+48 x^2-18 x^3+2 x^4\right ) \log ^2(3)+\left (-10+10 x-20 x^2+8 x^4-32 x^5+48 x^6-32 x^7+8 x^8+\left (32 x^2-72 x^3+48 x^4-8 x^5\right ) \log (3)\right ) \log (x)+\left (-12 x^3+36 x^4-36 x^5+12 x^6+\left (-16 x+20 x^2-4 x^3\right ) \log (3)\right ) \log ^2(x)+\left (8 x^2-16 x^3+8 x^4\right ) \log ^3(x)+\left (-2 x+2 x^2\right ) \log ^4(x)}{x^5-4 x^6+6 x^7-4 x^8+x^9+\left (8 x^3-18 x^4+12 x^5-2 x^6\right ) \log (3)+\left (16 x-8 x^2+x^3\right ) \log ^2(3)+\left (-4 x^4+12 x^5-12 x^6+4 x^7+\left (-16 x^2+20 x^3-4 x^4\right ) \log (3)\right ) \log (x)+\left (6 x^3-12 x^4+6 x^5+\left (8 x-2 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-4 x^2+4 x^3\right ) \log ^3(x)+x \log ^4(x)} \, dx=x^{2} - 2 \, x + \frac {5}{x^{4} - 2 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + x^{2} - x \log \left (3\right ) - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + 4 \, \log \left (3\right )} \]
integrate(((2*x^2-2*x)*log(x)^4+(8*x^4-16*x^3+8*x^2)*log(x)^3+((-4*x^3+20* x^2-16*x)*log(3)+12*x^6-36*x^5+36*x^4-12*x^3)*log(x)^2+((-8*x^5+48*x^4-72* x^3+32*x^2)*log(3)+8*x^8-32*x^7+48*x^6-32*x^5+8*x^4-20*x^2+10*x-10)*log(x) +(2*x^4-18*x^3+48*x^2-32*x)*log(3)^2+(-4*x^7+28*x^6-60*x^5+52*x^4-16*x^3+5 *x)*log(3)+2*x^10-10*x^9+20*x^8-20*x^7+10*x^6-2*x^5-20*x^4+30*x^3-20*x^2+1 0*x)/(x*log(x)^4+(4*x^3-4*x^2)*log(x)^3+((-2*x^2+8*x)*log(3)+6*x^5-12*x^4+ 6*x^3)*log(x)^2+((-4*x^4+20*x^3-16*x^2)*log(3)+4*x^7-12*x^6+12*x^5-4*x^4)* log(x)+(x^3-8*x^2+16*x)*log(3)^2+(-2*x^6+12*x^5-18*x^4+8*x^3)*log(3)+x^9-4 *x^8+6*x^7-4*x^6+x^5),x, algorithm=\
x^2 - 2*x + 5/(x^4 - 2*x^3 + 2*x^2*log(x) + x^2 - x*log(3) - 2*x*log(x) + log(x)^2 + 4*log(3))
Timed out. \[ \int \frac {10 x-20 x^2+30 x^3-20 x^4-2 x^5+10 x^6-20 x^7+20 x^8-10 x^9+2 x^{10}+\left (5 x-16 x^3+52 x^4-60 x^5+28 x^6-4 x^7\right ) \log (3)+\left (-32 x+48 x^2-18 x^3+2 x^4\right ) \log ^2(3)+\left (-10+10 x-20 x^2+8 x^4-32 x^5+48 x^6-32 x^7+8 x^8+\left (32 x^2-72 x^3+48 x^4-8 x^5\right ) \log (3)\right ) \log (x)+\left (-12 x^3+36 x^4-36 x^5+12 x^6+\left (-16 x+20 x^2-4 x^3\right ) \log (3)\right ) \log ^2(x)+\left (8 x^2-16 x^3+8 x^4\right ) \log ^3(x)+\left (-2 x+2 x^2\right ) \log ^4(x)}{x^5-4 x^6+6 x^7-4 x^8+x^9+\left (8 x^3-18 x^4+12 x^5-2 x^6\right ) \log (3)+\left (16 x-8 x^2+x^3\right ) \log ^2(3)+\left (-4 x^4+12 x^5-12 x^6+4 x^7+\left (-16 x^2+20 x^3-4 x^4\right ) \log (3)\right ) \log (x)+\left (6 x^3-12 x^4+6 x^5+\left (8 x-2 x^2\right ) \log (3)\right ) \log ^2(x)+\left (-4 x^2+4 x^3\right ) \log ^3(x)+x \log ^4(x)} \, dx=\int \frac {10\,x-{\ln \left (3\right )}^2\,\left (-2\,x^4+18\,x^3-48\,x^2+32\,x\right )-{\ln \left (x\right )}^4\,\left (2\,x-2\,x^2\right )+\ln \left (3\right )\,\left (-4\,x^7+28\,x^6-60\,x^5+52\,x^4-16\,x^3+5\,x\right )-{\ln \left (x\right )}^2\,\left (\ln \left (3\right )\,\left (4\,x^3-20\,x^2+16\,x\right )+12\,x^3-36\,x^4+36\,x^5-12\,x^6\right )+{\ln \left (x\right )}^3\,\left (8\,x^4-16\,x^3+8\,x^2\right )+\ln \left (x\right )\,\left (10\,x+\ln \left (3\right )\,\left (-8\,x^5+48\,x^4-72\,x^3+32\,x^2\right )-20\,x^2+8\,x^4-32\,x^5+48\,x^6-32\,x^7+8\,x^8-10\right )-20\,x^2+30\,x^3-20\,x^4-2\,x^5+10\,x^6-20\,x^7+20\,x^8-10\,x^9+2\,x^{10}}{\ln \left (3\right )\,\left (-2\,x^6+12\,x^5-18\,x^4+8\,x^3\right )+x\,{\ln \left (x\right )}^4-\ln \left (x\right )\,\left (\ln \left (3\right )\,\left (4\,x^4-20\,x^3+16\,x^2\right )+4\,x^4-12\,x^5+12\,x^6-4\,x^7\right )-{\ln \left (x\right )}^3\,\left (4\,x^2-4\,x^3\right )+{\ln \left (3\right )}^2\,\left (x^3-8\,x^2+16\,x\right )+x^5-4\,x^6+6\,x^7-4\,x^8+x^9+{\ln \left (x\right )}^2\,\left (\ln \left (3\right )\,\left (8\,x-2\,x^2\right )+6\,x^3-12\,x^4+6\,x^5\right )} \,d x \]
int((10*x - log(3)^2*(32*x - 48*x^2 + 18*x^3 - 2*x^4) - log(x)^4*(2*x - 2* x^2) + log(3)*(5*x - 16*x^3 + 52*x^4 - 60*x^5 + 28*x^6 - 4*x^7) - log(x)^2 *(log(3)*(16*x - 20*x^2 + 4*x^3) + 12*x^3 - 36*x^4 + 36*x^5 - 12*x^6) + lo g(x)^3*(8*x^2 - 16*x^3 + 8*x^4) + log(x)*(10*x + log(3)*(32*x^2 - 72*x^3 + 48*x^4 - 8*x^5) - 20*x^2 + 8*x^4 - 32*x^5 + 48*x^6 - 32*x^7 + 8*x^8 - 10) - 20*x^2 + 30*x^3 - 20*x^4 - 2*x^5 + 10*x^6 - 20*x^7 + 20*x^8 - 10*x^9 + 2*x^10)/(log(3)*(8*x^3 - 18*x^4 + 12*x^5 - 2*x^6) + x*log(x)^4 - log(x)*(l og(3)*(16*x^2 - 20*x^3 + 4*x^4) + 4*x^4 - 12*x^5 + 12*x^6 - 4*x^7) - log(x )^3*(4*x^2 - 4*x^3) + log(3)^2*(16*x - 8*x^2 + x^3) + x^5 - 4*x^6 + 6*x^7 - 4*x^8 + x^9 + log(x)^2*(log(3)*(8*x - 2*x^2) + 6*x^3 - 12*x^4 + 6*x^5)), x)
int((10*x - log(3)^2*(32*x - 48*x^2 + 18*x^3 - 2*x^4) - log(x)^4*(2*x - 2* x^2) + log(3)*(5*x - 16*x^3 + 52*x^4 - 60*x^5 + 28*x^6 - 4*x^7) - log(x)^2 *(log(3)*(16*x - 20*x^2 + 4*x^3) + 12*x^3 - 36*x^4 + 36*x^5 - 12*x^6) + lo g(x)^3*(8*x^2 - 16*x^3 + 8*x^4) + log(x)*(10*x + log(3)*(32*x^2 - 72*x^3 + 48*x^4 - 8*x^5) - 20*x^2 + 8*x^4 - 32*x^5 + 48*x^6 - 32*x^7 + 8*x^8 - 10) - 20*x^2 + 30*x^3 - 20*x^4 - 2*x^5 + 10*x^6 - 20*x^7 + 20*x^8 - 10*x^9 + 2*x^10)/(log(3)*(8*x^3 - 18*x^4 + 12*x^5 - 2*x^6) + x*log(x)^4 - log(x)*(l og(3)*(16*x^2 - 20*x^3 + 4*x^4) + 4*x^4 - 12*x^5 + 12*x^6 - 4*x^7) - log(x )^3*(4*x^2 - 4*x^3) + log(3)^2*(16*x - 8*x^2 + x^3) + x^5 - 4*x^6 + 6*x^7 - 4*x^8 + x^9 + log(x)^2*(log(3)*(8*x - 2*x^2) + 6*x^3 - 12*x^4 + 6*x^5)), x)