Integrand size = 216, antiderivative size = 19 \[ \int \frac {32 x^2+16 x^3+1026 x^4+1280 x^5+576 x^6+112 x^7+8 x^8+\left (-64 x-32 x^2-3076 x^3-4096 x^4-1920 x^5-384 x^6-28 x^7\right ) \log (3)+\left (32+16 x+3074 x^2+4608 x^3+2304 x^4+480 x^5+36 x^6\right ) \log ^2(3)+\left (-1024 x-2048 x^2-1152 x^3-256 x^4-20 x^5\right ) \log ^3(3)+\left (256 x+192 x^2+48 x^3+4 x^4\right ) \log ^4(3)+\left (2+64 x^2+48 x^3+8 x^4+\left (-64 x-64 x^2-12 x^3\right ) \log (3)+\left (16 x+4 x^2\right ) \log ^2(3)\right ) \log (x)}{x} \, dx=\left ((4+x)^2 (x-\log (3))^2+\log (x)\right )^2 \] Output:
((4+x)^2*(-ln(3)+x)^2+ln(x))^2
Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {32 x^2+16 x^3+1026 x^4+1280 x^5+576 x^6+112 x^7+8 x^8+\left (-64 x-32 x^2-3076 x^3-4096 x^4-1920 x^5-384 x^6-28 x^7\right ) \log (3)+\left (32+16 x+3074 x^2+4608 x^3+2304 x^4+480 x^5+36 x^6\right ) \log ^2(3)+\left (-1024 x-2048 x^2-1152 x^3-256 x^4-20 x^5\right ) \log ^3(3)+\left (256 x+192 x^2+48 x^3+4 x^4\right ) \log ^4(3)+\left (2+64 x^2+48 x^3+8 x^4+\left (-64 x-64 x^2-12 x^3\right ) \log (3)+\left (16 x+4 x^2\right ) \log ^2(3)\right ) \log (x)}{x} \, dx=\left ((4+x)^2 (x-\log (3))^2+\log (x)\right )^2 \] Input:
Integrate[(32*x^2 + 16*x^3 + 1026*x^4 + 1280*x^5 + 576*x^6 + 112*x^7 + 8*x ^8 + (-64*x - 32*x^2 - 3076*x^3 - 4096*x^4 - 1920*x^5 - 384*x^6 - 28*x^7)* Log[3] + (32 + 16*x + 3074*x^2 + 4608*x^3 + 2304*x^4 + 480*x^5 + 36*x^6)*L og[3]^2 + (-1024*x - 2048*x^2 - 1152*x^3 - 256*x^4 - 20*x^5)*Log[3]^3 + (2 56*x + 192*x^2 + 48*x^3 + 4*x^4)*Log[3]^4 + (2 + 64*x^2 + 48*x^3 + 8*x^4 + (-64*x - 64*x^2 - 12*x^3)*Log[3] + (16*x + 4*x^2)*Log[3]^2)*Log[x])/x,x]
Output:
((4 + x)^2*(x - Log[3])^2 + Log[x])^2
Leaf count is larger than twice the leaf count of optimal. \(279\) vs. \(2(19)=38\).
Time = 0.64 (sec) , antiderivative size = 279, normalized size of antiderivative = 14.68, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2010, 2009}
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.
\(\displaystyle \int \frac {8 x^8+112 x^7+576 x^6+1280 x^5+1026 x^4+16 x^3+32 x^2+\left (4 x^4+48 x^3+192 x^2+256 x\right ) \log ^4(3)+\left (8 x^4+48 x^3+64 x^2+\left (4 x^2+16 x\right ) \log ^2(3)+\left (-12 x^3-64 x^2-64 x\right ) \log (3)+2\right ) \log (x)+\left (-20 x^5-256 x^4-1152 x^3-2048 x^2-1024 x\right ) \log ^3(3)+\left (36 x^6+480 x^5+2304 x^4+4608 x^3+3074 x^2+16 x+32\right ) \log ^2(3)+\left (-28 x^7-384 x^6-1920 x^5-4096 x^4-3076 x^3-32 x^2-64 x\right ) \log (3)}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {2 (x+4)^2 \left (4 x^4+6 x^3 (4-\log (3))+2 x^2 \left (16+\log ^2(3)-16 \log (3)\right )-8 x (4-\log (3)) \log (3)+1\right ) (x-\log (3))^2}{x}+\frac {2 \left (4 x^4+6 x^3 (4-\log (3))+2 x^2 \left (16+\log ^2(3)-16 \log (3)\right )-8 x (4-\log (3)) \log (3)+1\right ) \log (x)}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^8+4 x^7 (4-\log (3))+2 x^6 \left (48+3 \log ^2(3)-32 \log (3)\right )+4 x^5 (4-\log (3)) \left (16+\log ^2(3)-20 \log (3)\right )-\frac {x^4}{2}+\frac {1}{2} x^4 \left (513+2 \log ^4(3)-128 \log ^3(3)+1152 \log ^2(3)-2048 \log (3)\right )+2 x^4 \log (x)+\frac {16}{3} x^3 \left (1+2 \log ^4(3)-68 \log ^3(3)+288 \log ^2(3)-192 \log (3)\right )+4 x^3 (4-\log (3)) \log (x)-\frac {4}{3} x^3 (4-\log (3))+2 x^2 \left (16+\log ^2(3)-16 \log (3)\right ) \log (x)-x^2 \left (16+\log ^2(3)-16 \log (3)\right )+x^2 \left (16+32 \log ^4(3)-768 \log ^3(3)+1537 \log ^2(3)\right )+16 x \log ^2(3)+\log ^2(x)+32 \log ^2(3) \log (x)-\frac {4}{3} (x+4)^3 \log (3) \left (1-4 \log ^3(3)+16 \log ^2(3)\right )-16 x (4-\log (3)) \log (3) \log (x)+16 x (4-\log (3)) \log (3)\) |
Input:
Int[(32*x^2 + 16*x^3 + 1026*x^4 + 1280*x^5 + 576*x^6 + 112*x^7 + 8*x^8 + ( -64*x - 32*x^2 - 3076*x^3 - 4096*x^4 - 1920*x^5 - 384*x^6 - 28*x^7)*Log[3] + (32 + 16*x + 3074*x^2 + 4608*x^3 + 2304*x^4 + 480*x^5 + 36*x^6)*Log[3]^ 2 + (-1024*x - 2048*x^2 - 1152*x^3 - 256*x^4 - 20*x^5)*Log[3]^3 + (256*x + 192*x^2 + 48*x^3 + 4*x^4)*Log[3]^4 + (2 + 64*x^2 + 48*x^3 + 8*x^4 + (-64* x - 64*x^2 - 12*x^3)*Log[3] + (16*x + 4*x^2)*Log[3]^2)*Log[x])/x,x]
Output:
-1/2*x^4 + x^8 - (4*x^3*(4 - Log[3]))/3 + 4*x^7*(4 - Log[3]) + 16*x*(4 - L og[3])*Log[3] + 16*x*Log[3]^2 + 4*x^5*(4 - Log[3])*(16 - 20*Log[3] + Log[3 ]^2) - x^2*(16 - 16*Log[3] + Log[3]^2) + 2*x^6*(48 - 32*Log[3] + 3*Log[3]^ 2) - (4*(4 + x)^3*Log[3]*(1 + 16*Log[3]^2 - 4*Log[3]^3))/3 + (x^4*(513 - 2 048*Log[3] + 1152*Log[3]^2 - 128*Log[3]^3 + 2*Log[3]^4))/2 + (16*x^3*(1 - 192*Log[3] + 288*Log[3]^2 - 68*Log[3]^3 + 2*Log[3]^4))/3 + x^2*(16 + 1537* Log[3]^2 - 768*Log[3]^3 + 32*Log[3]^4) + 2*x^4*Log[x] + 4*x^3*(4 - Log[3]) *Log[x] - 16*x*(4 - Log[3])*Log[3]*Log[x] + 32*Log[3]^2*Log[x] + 2*x^2*(16 - 16*Log[3] + Log[3]^2)*Log[x] + Log[x]^2
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(207\) vs. \(2(19)=38\).
Time = 38.34 (sec) , antiderivative size = 208, normalized size of antiderivative = 10.95
method | result | size |
norman | \(x^{8}+\ln \left (x \right )^{2}+\left (16-4 \ln \left (3\right )\right ) x^{7}+32 \ln \left (3\right )^{2} \ln \left (x \right )+\left (256 \ln \left (3\right )^{4}-1024 \ln \left (3\right )^{3}\right ) x +\left (6 \ln \left (3\right )^{2}-64 \ln \left (3\right )+96\right ) x^{6}+\left (1536 \ln \left (3\right )^{2}+96 \ln \left (3\right )^{4}-1024 \ln \left (3\right )^{3}\right ) x^{2}+\left (-4 \ln \left (3\right )^{3}+96 \ln \left (3\right )^{2}-384 \ln \left (3\right )+256\right ) x^{5}+\left (-1024 \ln \left (3\right )-384 \ln \left (3\right )^{3}+1536 \ln \left (3\right )^{2}+16 \ln \left (3\right )^{4}\right ) x^{3}+\left (256+\ln \left (3\right )^{4}+576 \ln \left (3\right )^{2}-1024 \ln \left (3\right )-64 \ln \left (3\right )^{3}\right ) x^{4}+\left (16-4 \ln \left (3\right )\right ) x^{3} \ln \left (x \right )+\left (16 \ln \left (3\right )^{2}-64 \ln \left (3\right )\right ) x \ln \left (x \right )+\left (2 \ln \left (3\right )^{2}-32 \ln \left (3\right )+32\right ) x^{2} \ln \left (x \right )+2 x^{4} \ln \left (x \right )\) | \(208\) |
risch | \(-64 x^{6} \ln \left (3\right )-1024 x \ln \left (3\right )^{3}+96 x^{5} \ln \left (3\right )^{2}+256 x \ln \left (3\right )^{4}+1536 x^{3} \ln \left (3\right )^{2}-1024 x^{4} \ln \left (3\right )+1536 x^{2} \ln \left (3\right )^{2}-64 \ln \left (3\right )^{3} x^{4}+576 x^{4} \ln \left (3\right )^{2}+16 \ln \left (3\right )^{4} x^{3}+96 \ln \left (3\right )^{4} x^{2}-384 \ln \left (3\right )^{3} x^{3}-1024 \ln \left (3\right )^{3} x^{2}+\ln \left (3\right )^{4} x^{4}-1024 x^{3} \ln \left (3\right )-384 x^{5} \ln \left (3\right )+x^{8}+16 x^{7}+96 x^{6}+\ln \left (x \right )^{2}+256 x^{4}+256 x^{5}-4 \ln \left (3\right ) x^{7}+6 \ln \left (3\right )^{2} x^{6}+\left (2 x^{2} \ln \left (3\right )^{2}-4 x^{3} \ln \left (3\right )+2 x^{4}+16 x \ln \left (3\right )^{2}-32 x^{2} \ln \left (3\right )+16 x^{3}-64 x \ln \left (3\right )+32 x^{2}\right ) \ln \left (x \right )-4 \ln \left (3\right )^{3} x^{5}+32 \ln \left (3\right )^{2} \ln \left (x \right )\) | \(247\) |
parallelrisch | \(-64 x^{6} \ln \left (3\right )-1024 x \ln \left (3\right )^{3}+96 x^{5} \ln \left (3\right )^{2}+256 x \ln \left (3\right )^{4}+1536 x^{3} \ln \left (3\right )^{2}-32 x^{2} \ln \left (3\right ) \ln \left (x \right )-1024 x^{4} \ln \left (3\right )+2 x^{4} \ln \left (x \right )+1536 x^{2} \ln \left (3\right )^{2}-64 \ln \left (3\right )^{3} x^{4}+576 x^{4} \ln \left (3\right )^{2}+16 \ln \left (3\right )^{4} x^{3}+96 \ln \left (3\right )^{4} x^{2}-384 \ln \left (3\right )^{3} x^{3}-1024 \ln \left (3\right )^{3} x^{2}+\ln \left (3\right )^{4} x^{4}-1024 x^{3} \ln \left (3\right )-384 x^{5} \ln \left (3\right )+x^{8}+16 x^{7}+16 x^{3} \ln \left (x \right )+32 x^{2} \ln \left (x \right )+96 x^{6}+\ln \left (x \right )^{2}+256 x^{4}+256 x^{5}-64 x \ln \left (3\right ) \ln \left (x \right )+2 \ln \left (3\right )^{2} \ln \left (x \right ) x^{2}-4 \ln \left (3\right ) x^{7}+6 \ln \left (3\right )^{2} x^{6}-4 \ln \left (3\right )^{3} x^{5}-4 \ln \left (3\right ) \ln \left (x \right ) x^{3}+32 \ln \left (3\right )^{2} \ln \left (x \right )+16 \ln \left (x \right ) \ln \left (3\right )^{2} x\) | \(259\) |
default | \(-64 x^{6} \ln \left (3\right )-1024 x \ln \left (3\right )^{3}+96 x^{5} \ln \left (3\right )^{2}+256 x \ln \left (3\right )^{4}+1536 x^{3} \ln \left (3\right )^{2}-1024 x^{4} \ln \left (3\right )+2 x^{4} \ln \left (x \right )+1537 x^{2} \ln \left (3\right )^{2}-16 x^{2} \ln \left (3\right )-64 \ln \left (3\right )^{3} x^{4}+576 x^{4} \ln \left (3\right )^{2}+16 \ln \left (3\right )^{4} x^{3}+96 \ln \left (3\right )^{4} x^{2}-384 \ln \left (3\right )^{3} x^{3}-1024 \ln \left (3\right )^{3} x^{2}+\ln \left (3\right )^{4} x^{4}-\frac {3076 x^{3} \ln \left (3\right )}{3}-384 x^{5} \ln \left (3\right )+x^{8}+16 x^{7}+16 x^{3} \ln \left (x \right )-64 x \ln \left (3\right )+32 x^{2} \ln \left (x \right )+16 x \ln \left (3\right )^{2}+96 x^{6}+\ln \left (x \right )^{2}+256 x^{4}+256 x^{5}+4 \ln \left (3\right )^{2} \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+16 \ln \left (3\right )^{2} \left (x \ln \left (x \right )-x \right )-64 \ln \left (3\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )-64 \ln \left (3\right ) \left (x \ln \left (x \right )-x \right )-12 \ln \left (3\right ) \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )-4 \ln \left (3\right ) x^{7}+6 \ln \left (3\right )^{2} x^{6}-4 \ln \left (3\right )^{3} x^{5}+32 \ln \left (3\right )^{2} \ln \left (x \right )\) | \(312\) |
parts | \(-64 x^{6} \ln \left (3\right )-1024 x \ln \left (3\right )^{3}+96 x^{5} \ln \left (3\right )^{2}+256 x \ln \left (3\right )^{4}+1536 x^{3} \ln \left (3\right )^{2}-1024 x^{4} \ln \left (3\right )+2 x^{4} \ln \left (x \right )+1537 x^{2} \ln \left (3\right )^{2}-16 x^{2} \ln \left (3\right )-64 \ln \left (3\right )^{3} x^{4}+576 x^{4} \ln \left (3\right )^{2}+16 \ln \left (3\right )^{4} x^{3}+96 \ln \left (3\right )^{4} x^{2}-384 \ln \left (3\right )^{3} x^{3}-1024 \ln \left (3\right )^{3} x^{2}+\ln \left (3\right )^{4} x^{4}-\frac {3076 x^{3} \ln \left (3\right )}{3}-384 x^{5} \ln \left (3\right )+x^{8}+16 x^{7}+16 x^{3} \ln \left (x \right )-64 x \ln \left (3\right )+32 x^{2} \ln \left (x \right )+16 x \ln \left (3\right )^{2}+96 x^{6}+\ln \left (x \right )^{2}+256 x^{4}+256 x^{5}+4 \ln \left (3\right )^{2} \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+16 \ln \left (3\right )^{2} \left (x \ln \left (x \right )-x \right )-64 \ln \left (3\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )-64 \ln \left (3\right ) \left (x \ln \left (x \right )-x \right )-12 \ln \left (3\right ) \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )-4 \ln \left (3\right ) x^{7}+6 \ln \left (3\right )^{2} x^{6}-4 \ln \left (3\right )^{3} x^{5}+32 \ln \left (3\right )^{2} \ln \left (x \right )\) | \(312\) |
Input:
int((((4*x^2+16*x)*ln(3)^2+(-12*x^3-64*x^2-64*x)*ln(3)+8*x^4+48*x^3+64*x^2 +2)*ln(x)+(4*x^4+48*x^3+192*x^2+256*x)*ln(3)^4+(-20*x^5-256*x^4-1152*x^3-2 048*x^2-1024*x)*ln(3)^3+(36*x^6+480*x^5+2304*x^4+4608*x^3+3074*x^2+16*x+32 )*ln(3)^2+(-28*x^7-384*x^6-1920*x^5-4096*x^4-3076*x^3-32*x^2-64*x)*ln(3)+8 *x^8+112*x^7+576*x^6+1280*x^5+1026*x^4+16*x^3+32*x^2)/x,x,method=_RETURNVE RBOSE)
Output:
x^8+ln(x)^2+(16-4*ln(3))*x^7+32*ln(3)^2*ln(x)+(256*ln(3)^4-1024*ln(3)^3)*x +(6*ln(3)^2-64*ln(3)+96)*x^6+(1536*ln(3)^2+96*ln(3)^4-1024*ln(3)^3)*x^2+(- 4*ln(3)^3+96*ln(3)^2-384*ln(3)+256)*x^5+(-1024*ln(3)-384*ln(3)^3+1536*ln(3 )^2+16*ln(3)^4)*x^3+(256+ln(3)^4+576*ln(3)^2-1024*ln(3)-64*ln(3)^3)*x^4+(1 6-4*ln(3))*x^3*ln(x)+(16*ln(3)^2-64*ln(3))*x*ln(x)+(2*ln(3)^2-32*ln(3)+32) *x^2*ln(x)+2*x^4*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 183, normalized size of antiderivative = 9.63 \[ \int \frac {32 x^2+16 x^3+1026 x^4+1280 x^5+576 x^6+112 x^7+8 x^8+\left (-64 x-32 x^2-3076 x^3-4096 x^4-1920 x^5-384 x^6-28 x^7\right ) \log (3)+\left (32+16 x+3074 x^2+4608 x^3+2304 x^4+480 x^5+36 x^6\right ) \log ^2(3)+\left (-1024 x-2048 x^2-1152 x^3-256 x^4-20 x^5\right ) \log ^3(3)+\left (256 x+192 x^2+48 x^3+4 x^4\right ) \log ^4(3)+\left (2+64 x^2+48 x^3+8 x^4+\left (-64 x-64 x^2-12 x^3\right ) \log (3)+\left (16 x+4 x^2\right ) \log ^2(3)\right ) \log (x)}{x} \, dx=x^{8} + 16 \, x^{7} + 96 \, x^{6} + 256 \, x^{5} + {\left (x^{4} + 16 \, x^{3} + 96 \, x^{2} + 256 \, x\right )} \log \left (3\right )^{4} + 256 \, x^{4} - 4 \, {\left (x^{5} + 16 \, x^{4} + 96 \, x^{3} + 256 \, x^{2} + 256 \, x\right )} \log \left (3\right )^{3} + 6 \, {\left (x^{6} + 16 \, x^{5} + 96 \, x^{4} + 256 \, x^{3} + 256 \, x^{2}\right )} \log \left (3\right )^{2} - 4 \, {\left (x^{7} + 16 \, x^{6} + 96 \, x^{5} + 256 \, x^{4} + 256 \, x^{3}\right )} \log \left (3\right ) + 2 \, {\left (x^{4} + 8 \, x^{3} + {\left (x^{2} + 8 \, x + 16\right )} \log \left (3\right )^{2} + 16 \, x^{2} - 2 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} \log \left (3\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2} \] Input:
integrate((((4*x^2+16*x)*log(3)^2+(-12*x^3-64*x^2-64*x)*log(3)+8*x^4+48*x^ 3+64*x^2+2)*log(x)+(4*x^4+48*x^3+192*x^2+256*x)*log(3)^4+(-20*x^5-256*x^4- 1152*x^3-2048*x^2-1024*x)*log(3)^3+(36*x^6+480*x^5+2304*x^4+4608*x^3+3074* x^2+16*x+32)*log(3)^2+(-28*x^7-384*x^6-1920*x^5-4096*x^4-3076*x^3-32*x^2-6 4*x)*log(3)+8*x^8+112*x^7+576*x^6+1280*x^5+1026*x^4+16*x^3+32*x^2)/x,x, al gorithm="fricas")
Output:
x^8 + 16*x^7 + 96*x^6 + 256*x^5 + (x^4 + 16*x^3 + 96*x^2 + 256*x)*log(3)^4 + 256*x^4 - 4*(x^5 + 16*x^4 + 96*x^3 + 256*x^2 + 256*x)*log(3)^3 + 6*(x^6 + 16*x^5 + 96*x^4 + 256*x^3 + 256*x^2)*log(3)^2 - 4*(x^7 + 16*x^6 + 96*x^ 5 + 256*x^4 + 256*x^3)*log(3) + 2*(x^4 + 8*x^3 + (x^2 + 8*x + 16)*log(3)^2 + 16*x^2 - 2*(x^3 + 8*x^2 + 16*x)*log(3))*log(x) + log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (15) = 30\).
Time = 0.26 (sec) , antiderivative size = 226, normalized size of antiderivative = 11.89 \[ \int \frac {32 x^2+16 x^3+1026 x^4+1280 x^5+576 x^6+112 x^7+8 x^8+\left (-64 x-32 x^2-3076 x^3-4096 x^4-1920 x^5-384 x^6-28 x^7\right ) \log (3)+\left (32+16 x+3074 x^2+4608 x^3+2304 x^4+480 x^5+36 x^6\right ) \log ^2(3)+\left (-1024 x-2048 x^2-1152 x^3-256 x^4-20 x^5\right ) \log ^3(3)+\left (256 x+192 x^2+48 x^3+4 x^4\right ) \log ^4(3)+\left (2+64 x^2+48 x^3+8 x^4+\left (-64 x-64 x^2-12 x^3\right ) \log (3)+\left (16 x+4 x^2\right ) \log ^2(3)\right ) \log (x)}{x} \, dx=x^{8} + x^{7} \cdot \left (16 - 4 \log {\left (3 \right )}\right ) + x^{6} \left (- 64 \log {\left (3 \right )} + 6 \log {\left (3 \right )}^{2} + 96\right ) + x^{5} \left (- 384 \log {\left (3 \right )} - 4 \log {\left (3 \right )}^{3} + 96 \log {\left (3 \right )}^{2} + 256\right ) + x^{4} \left (- 1024 \log {\left (3 \right )} - 64 \log {\left (3 \right )}^{3} + \log {\left (3 \right )}^{4} + 256 + 576 \log {\left (3 \right )}^{2}\right ) + x^{3} \left (- 1024 \log {\left (3 \right )} - 384 \log {\left (3 \right )}^{3} + 16 \log {\left (3 \right )}^{4} + 1536 \log {\left (3 \right )}^{2}\right ) + x^{2} \left (- 1024 \log {\left (3 \right )}^{3} + 96 \log {\left (3 \right )}^{4} + 1536 \log {\left (3 \right )}^{2}\right ) + x \left (- 1024 \log {\left (3 \right )}^{3} + 256 \log {\left (3 \right )}^{4}\right ) + \left (2 x^{4} - 4 x^{3} \log {\left (3 \right )} + 16 x^{3} - 32 x^{2} \log {\left (3 \right )} + 2 x^{2} \log {\left (3 \right )}^{2} + 32 x^{2} - 64 x \log {\left (3 \right )} + 16 x \log {\left (3 \right )}^{2}\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 32 \log {\left (3 \right )}^{2} \log {\left (x \right )} \] Input:
integrate((((4*x**2+16*x)*ln(3)**2+(-12*x**3-64*x**2-64*x)*ln(3)+8*x**4+48 *x**3+64*x**2+2)*ln(x)+(4*x**4+48*x**3+192*x**2+256*x)*ln(3)**4+(-20*x**5- 256*x**4-1152*x**3-2048*x**2-1024*x)*ln(3)**3+(36*x**6+480*x**5+2304*x**4+ 4608*x**3+3074*x**2+16*x+32)*ln(3)**2+(-28*x**7-384*x**6-1920*x**5-4096*x* *4-3076*x**3-32*x**2-64*x)*ln(3)+8*x**8+112*x**7+576*x**6+1280*x**5+1026*x **4+16*x**3+32*x**2)/x,x)
Output:
x**8 + x**7*(16 - 4*log(3)) + x**6*(-64*log(3) + 6*log(3)**2 + 96) + x**5* (-384*log(3) - 4*log(3)**3 + 96*log(3)**2 + 256) + x**4*(-1024*log(3) - 64 *log(3)**3 + log(3)**4 + 256 + 576*log(3)**2) + x**3*(-1024*log(3) - 384*l og(3)**3 + 16*log(3)**4 + 1536*log(3)**2) + x**2*(-1024*log(3)**3 + 96*log (3)**4 + 1536*log(3)**2) + x*(-1024*log(3)**3 + 256*log(3)**4) + (2*x**4 - 4*x**3*log(3) + 16*x**3 - 32*x**2*log(3) + 2*x**2*log(3)**2 + 32*x**2 - 6 4*x*log(3) + 16*x*log(3)**2)*log(x) + log(x)**2 + 32*log(3)**2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (19) = 38\).
Time = 0.04 (sec) , antiderivative size = 310, normalized size of antiderivative = 16.32 \[ \int \frac {32 x^2+16 x^3+1026 x^4+1280 x^5+576 x^6+112 x^7+8 x^8+\left (-64 x-32 x^2-3076 x^3-4096 x^4-1920 x^5-384 x^6-28 x^7\right ) \log (3)+\left (32+16 x+3074 x^2+4608 x^3+2304 x^4+480 x^5+36 x^6\right ) \log ^2(3)+\left (-1024 x-2048 x^2-1152 x^3-256 x^4-20 x^5\right ) \log ^3(3)+\left (256 x+192 x^2+48 x^3+4 x^4\right ) \log ^4(3)+\left (2+64 x^2+48 x^3+8 x^4+\left (-64 x-64 x^2-12 x^3\right ) \log (3)+\left (16 x+4 x^2\right ) \log ^2(3)\right ) \log (x)}{x} \, dx=x^{8} - 4 \, x^{7} \log \left (3\right ) + 6 \, x^{6} \log \left (3\right )^{2} - 4 \, x^{5} \log \left (3\right )^{3} + x^{4} \log \left (3\right )^{4} + 16 \, x^{7} - 64 \, x^{6} \log \left (3\right ) + 96 \, x^{5} \log \left (3\right )^{2} - 64 \, x^{4} \log \left (3\right )^{3} + 16 \, x^{3} \log \left (3\right )^{4} + 96 \, x^{6} - 384 \, x^{5} \log \left (3\right ) + 576 \, x^{4} \log \left (3\right )^{2} - 384 \, x^{3} \log \left (3\right )^{3} + 96 \, x^{2} \log \left (3\right )^{4} + 256 \, x^{5} - 1024 \, x^{4} \log \left (3\right ) + 1536 \, x^{3} \log \left (3\right )^{2} - 1024 \, x^{2} \log \left (3\right )^{3} + 256 \, x \log \left (3\right )^{4} + 2 \, x^{4} \log \left (x\right ) + 256 \, x^{4} - \frac {3076}{3} \, x^{3} \log \left (3\right ) + 1537 \, x^{2} \log \left (3\right )^{2} - 1024 \, x \log \left (3\right )^{3} + 16 \, x^{3} \log \left (x\right ) - 16 \, x^{2} \log \left (3\right ) + {\left (2 \, x^{2} \log \left (x\right ) - x^{2}\right )} \log \left (3\right )^{2} + 16 \, {\left (x \log \left (x\right ) - x\right )} \log \left (3\right )^{2} + 16 \, x \log \left (3\right )^{2} + 32 \, x^{2} \log \left (x\right ) + 32 \, \log \left (3\right )^{2} \log \left (x\right ) - \frac {4}{3} \, {\left (3 \, x^{3} \log \left (x\right ) - x^{3}\right )} \log \left (3\right ) - 16 \, {\left (2 \, x^{2} \log \left (x\right ) - x^{2}\right )} \log \left (3\right ) - 64 \, {\left (x \log \left (x\right ) - x\right )} \log \left (3\right ) - 64 \, x \log \left (3\right ) + \log \left (x\right )^{2} \] Input:
integrate((((4*x^2+16*x)*log(3)^2+(-12*x^3-64*x^2-64*x)*log(3)+8*x^4+48*x^ 3+64*x^2+2)*log(x)+(4*x^4+48*x^3+192*x^2+256*x)*log(3)^4+(-20*x^5-256*x^4- 1152*x^3-2048*x^2-1024*x)*log(3)^3+(36*x^6+480*x^5+2304*x^4+4608*x^3+3074* x^2+16*x+32)*log(3)^2+(-28*x^7-384*x^6-1920*x^5-4096*x^4-3076*x^3-32*x^2-6 4*x)*log(3)+8*x^8+112*x^7+576*x^6+1280*x^5+1026*x^4+16*x^3+32*x^2)/x,x, al gorithm="maxima")
Output:
x^8 - 4*x^7*log(3) + 6*x^6*log(3)^2 - 4*x^5*log(3)^3 + x^4*log(3)^4 + 16*x ^7 - 64*x^6*log(3) + 96*x^5*log(3)^2 - 64*x^4*log(3)^3 + 16*x^3*log(3)^4 + 96*x^6 - 384*x^5*log(3) + 576*x^4*log(3)^2 - 384*x^3*log(3)^3 + 96*x^2*lo g(3)^4 + 256*x^5 - 1024*x^4*log(3) + 1536*x^3*log(3)^2 - 1024*x^2*log(3)^3 + 256*x*log(3)^4 + 2*x^4*log(x) + 256*x^4 - 3076/3*x^3*log(3) + 1537*x^2* log(3)^2 - 1024*x*log(3)^3 + 16*x^3*log(x) - 16*x^2*log(3) + (2*x^2*log(x) - x^2)*log(3)^2 + 16*(x*log(x) - x)*log(3)^2 + 16*x*log(3)^2 + 32*x^2*log (x) + 32*log(3)^2*log(x) - 4/3*(3*x^3*log(x) - x^3)*log(3) - 16*(2*x^2*log (x) - x^2)*log(3) - 64*(x*log(x) - x)*log(3) - 64*x*log(3) + log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 196, normalized size of antiderivative = 10.32 \[ \int \frac {32 x^2+16 x^3+1026 x^4+1280 x^5+576 x^6+112 x^7+8 x^8+\left (-64 x-32 x^2-3076 x^3-4096 x^4-1920 x^5-384 x^6-28 x^7\right ) \log (3)+\left (32+16 x+3074 x^2+4608 x^3+2304 x^4+480 x^5+36 x^6\right ) \log ^2(3)+\left (-1024 x-2048 x^2-1152 x^3-256 x^4-20 x^5\right ) \log ^3(3)+\left (256 x+192 x^2+48 x^3+4 x^4\right ) \log ^4(3)+\left (2+64 x^2+48 x^3+8 x^4+\left (-64 x-64 x^2-12 x^3\right ) \log (3)+\left (16 x+4 x^2\right ) \log ^2(3)\right ) \log (x)}{x} \, dx=x^{8} - 4 \, x^{7} {\left (\log \left (3\right ) - 4\right )} + 2 \, {\left (3 \, \log \left (3\right )^{2} - 32 \, \log \left (3\right ) + 48\right )} x^{6} - 4 \, {\left (\log \left (3\right )^{3} - 24 \, \log \left (3\right )^{2} + 96 \, \log \left (3\right ) - 64\right )} x^{5} + {\left (\log \left (3\right )^{4} - 64 \, \log \left (3\right )^{3} + 576 \, \log \left (3\right )^{2} - 1024 \, \log \left (3\right ) + 256\right )} x^{4} + 16 \, {\left (\log \left (3\right )^{4} - 24 \, \log \left (3\right )^{3} + 96 \, \log \left (3\right )^{2} - 64 \, \log \left (3\right )\right )} x^{3} + 32 \, {\left (3 \, \log \left (3\right )^{4} - 32 \, \log \left (3\right )^{3} + 48 \, \log \left (3\right )^{2}\right )} x^{2} + 32 \, \log \left (3\right )^{2} \log \left (x\right ) + 256 \, {\left (\log \left (3\right )^{4} - 4 \, \log \left (3\right )^{3}\right )} x + 2 \, {\left (x^{4} - 2 \, x^{3} {\left (\log \left (3\right ) - 4\right )} + {\left (\log \left (3\right )^{2} - 16 \, \log \left (3\right ) + 16\right )} x^{2} + 8 \, {\left (\log \left (3\right )^{2} - 4 \, \log \left (3\right )\right )} x\right )} \log \left (x\right ) + \log \left (x\right )^{2} \] Input:
integrate((((4*x^2+16*x)*log(3)^2+(-12*x^3-64*x^2-64*x)*log(3)+8*x^4+48*x^ 3+64*x^2+2)*log(x)+(4*x^4+48*x^3+192*x^2+256*x)*log(3)^4+(-20*x^5-256*x^4- 1152*x^3-2048*x^2-1024*x)*log(3)^3+(36*x^6+480*x^5+2304*x^4+4608*x^3+3074* x^2+16*x+32)*log(3)^2+(-28*x^7-384*x^6-1920*x^5-4096*x^4-3076*x^3-32*x^2-6 4*x)*log(3)+8*x^8+112*x^7+576*x^6+1280*x^5+1026*x^4+16*x^3+32*x^2)/x,x, al gorithm="giac")
Output:
x^8 - 4*x^7*(log(3) - 4) + 2*(3*log(3)^2 - 32*log(3) + 48)*x^6 - 4*(log(3) ^3 - 24*log(3)^2 + 96*log(3) - 64)*x^5 + (log(3)^4 - 64*log(3)^3 + 576*log (3)^2 - 1024*log(3) + 256)*x^4 + 16*(log(3)^4 - 24*log(3)^3 + 96*log(3)^2 - 64*log(3))*x^3 + 32*(3*log(3)^4 - 32*log(3)^3 + 48*log(3)^2)*x^2 + 32*lo g(3)^2*log(x) + 256*(log(3)^4 - 4*log(3)^3)*x + 2*(x^4 - 2*x^3*(log(3) - 4 ) + (log(3)^2 - 16*log(3) + 16)*x^2 + 8*(log(3)^2 - 4*log(3))*x)*log(x) + log(x)^2
Time = 2.72 (sec) , antiderivative size = 184, normalized size of antiderivative = 9.68 \[ \int \frac {32 x^2+16 x^3+1026 x^4+1280 x^5+576 x^6+112 x^7+8 x^8+\left (-64 x-32 x^2-3076 x^3-4096 x^4-1920 x^5-384 x^6-28 x^7\right ) \log (3)+\left (32+16 x+3074 x^2+4608 x^3+2304 x^4+480 x^5+36 x^6\right ) \log ^2(3)+\left (-1024 x-2048 x^2-1152 x^3-256 x^4-20 x^5\right ) \log ^3(3)+\left (256 x+192 x^2+48 x^3+4 x^4\right ) \log ^4(3)+\left (2+64 x^2+48 x^3+8 x^4+\left (-64 x-64 x^2-12 x^3\right ) \log (3)+\left (16 x+4 x^2\right ) \log ^2(3)\right ) \log (x)}{x} \, dx=32\,{\ln \left (3\right )}^2\,\ln \left (x\right )-x^7\,\left (\ln \left (81\right )-16\right )+{\ln \left (x\right )}^2+x^6\,\left (6\,{\ln \left (3\right )}^2-64\,\ln \left (3\right )+96\right )-x^3\,\left (\ln \left (x\right )\,\left (\ln \left (81\right )-16\right )-16\,\ln \left (3\right )\,\left (96\,\ln \left (3\right )-24\,{\ln \left (3\right )}^2+{\ln \left (3\right )}^3-64\right )\right )+x\,\left (256\,{\ln \left (3\right )}^3\,\left (\ln \left (3\right )-4\right )+16\,\ln \left (3\right )\,\ln \left (x\right )\,\left (\ln \left (3\right )-4\right )\right )+x^2\,\left (32\,{\ln \left (3\right )}^2\,\left (3\,{\ln \left (3\right )}^2-32\,\ln \left (3\right )+48\right )+\ln \left (x\right )\,\left (2\,{\ln \left (3\right )}^2-32\,\ln \left (3\right )+32\right )\right )+x^4\,\left (2\,\ln \left (x\right )-1024\,\ln \left (3\right )+576\,{\ln \left (3\right )}^2-64\,{\ln \left (3\right )}^3+{\ln \left (3\right )}^4+256\right )+x^8-4\,x^5\,\left (\ln \left (3\right )-4\right )\,\left ({\ln \left (3\right )}^2-20\,\ln \left (3\right )+16\right ) \] Input:
int((log(3)^4*(256*x + 192*x^2 + 48*x^3 + 4*x^4) - log(3)^3*(1024*x + 2048 *x^2 + 1152*x^3 + 256*x^4 + 20*x^5) - log(3)*(64*x + 32*x^2 + 3076*x^3 + 4 096*x^4 + 1920*x^5 + 384*x^6 + 28*x^7) + log(x)*(log(3)^2*(16*x + 4*x^2) - log(3)*(64*x + 64*x^2 + 12*x^3) + 64*x^2 + 48*x^3 + 8*x^4 + 2) + log(3)^2 *(16*x + 3074*x^2 + 4608*x^3 + 2304*x^4 + 480*x^5 + 36*x^6 + 32) + 32*x^2 + 16*x^3 + 1026*x^4 + 1280*x^5 + 576*x^6 + 112*x^7 + 8*x^8)/x,x)
Output:
32*log(3)^2*log(x) - x^7*(log(81) - 16) + log(x)^2 + x^6*(6*log(3)^2 - 64* log(3) + 96) - x^3*(log(x)*(log(81) - 16) - 16*log(3)*(96*log(3) - 24*log( 3)^2 + log(3)^3 - 64)) + x*(256*log(3)^3*(log(3) - 4) + 16*log(3)*log(x)*( log(3) - 4)) + x^2*(32*log(3)^2*(3*log(3)^2 - 32*log(3) + 48) + log(x)*(2* log(3)^2 - 32*log(3) + 32)) + x^4*(2*log(x) - 1024*log(3) + 576*log(3)^2 - 64*log(3)^3 + log(3)^4 + 256) + x^8 - 4*x^5*(log(3) - 4)*(log(3)^2 - 20*l og(3) + 16)
Time = 0.21 (sec) , antiderivative size = 258, normalized size of antiderivative = 13.58 \[ \int \frac {32 x^2+16 x^3+1026 x^4+1280 x^5+576 x^6+112 x^7+8 x^8+\left (-64 x-32 x^2-3076 x^3-4096 x^4-1920 x^5-384 x^6-28 x^7\right ) \log (3)+\left (32+16 x+3074 x^2+4608 x^3+2304 x^4+480 x^5+36 x^6\right ) \log ^2(3)+\left (-1024 x-2048 x^2-1152 x^3-256 x^4-20 x^5\right ) \log ^3(3)+\left (256 x+192 x^2+48 x^3+4 x^4\right ) \log ^4(3)+\left (2+64 x^2+48 x^3+8 x^4+\left (-64 x-64 x^2-12 x^3\right ) \log (3)+\left (16 x+4 x^2\right ) \log ^2(3)\right ) \log (x)}{x} \, dx=-384 \,\mathrm {log}\left (3\right ) x^{5}-1024 \,\mathrm {log}\left (3\right ) x^{4}+2 \,\mathrm {log}\left (x \right ) x^{4}-1024 \,\mathrm {log}\left (3\right ) x^{3}+2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right )^{2} x^{2}+16 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right )^{2} x -4 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x^{3}-32 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x^{2}-64 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x +256 x^{5}+96 \mathrm {log}\left (3\right )^{2} x^{5}+16 x^{7}+1536 \mathrm {log}\left (3\right )^{2} x^{2}+x^{8}+96 x^{6}+32 \,\mathrm {log}\left (x \right ) x^{2}+\mathrm {log}\left (3\right )^{4} x^{4}+32 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right )^{2}+16 \mathrm {log}\left (3\right )^{4} x^{3}+96 \mathrm {log}\left (3\right )^{4} x^{2}+256 \mathrm {log}\left (3\right )^{4} x -4 \mathrm {log}\left (3\right )^{3} x^{5}-64 \mathrm {log}\left (3\right )^{3} x^{4}-384 \mathrm {log}\left (3\right )^{3} x^{3}-1024 \mathrm {log}\left (3\right )^{3} x^{2}-1024 \mathrm {log}\left (3\right )^{3} x +6 \mathrm {log}\left (3\right )^{2} x^{6}+576 \mathrm {log}\left (3\right )^{2} x^{4}-4 \,\mathrm {log}\left (3\right ) x^{7}+16 \,\mathrm {log}\left (x \right ) x^{3}+256 x^{4}-64 \,\mathrm {log}\left (3\right ) x^{6}+\mathrm {log}\left (x \right )^{2}+1536 \mathrm {log}\left (3\right )^{2} x^{3} \] Input:
int((((4*x^2+16*x)*log(3)^2+(-12*x^3-64*x^2-64*x)*log(3)+8*x^4+48*x^3+64*x ^2+2)*log(x)+(4*x^4+48*x^3+192*x^2+256*x)*log(3)^4+(-20*x^5-256*x^4-1152*x ^3-2048*x^2-1024*x)*log(3)^3+(36*x^6+480*x^5+2304*x^4+4608*x^3+3074*x^2+16 *x+32)*log(3)^2+(-28*x^7-384*x^6-1920*x^5-4096*x^4-3076*x^3-32*x^2-64*x)*l og(3)+8*x^8+112*x^7+576*x^6+1280*x^5+1026*x^4+16*x^3+32*x^2)/x,x)
Output:
log(x)**2 + 2*log(x)*log(3)**2*x**2 + 16*log(x)*log(3)**2*x + 32*log(x)*lo g(3)**2 - 4*log(x)*log(3)*x**3 - 32*log(x)*log(3)*x**2 - 64*log(x)*log(3)* x + 2*log(x)*x**4 + 16*log(x)*x**3 + 32*log(x)*x**2 + log(3)**4*x**4 + 16* log(3)**4*x**3 + 96*log(3)**4*x**2 + 256*log(3)**4*x - 4*log(3)**3*x**5 - 64*log(3)**3*x**4 - 384*log(3)**3*x**3 - 1024*log(3)**3*x**2 - 1024*log(3) **3*x + 6*log(3)**2*x**6 + 96*log(3)**2*x**5 + 576*log(3)**2*x**4 + 1536*l og(3)**2*x**3 + 1536*log(3)**2*x**2 - 4*log(3)*x**7 - 64*log(3)*x**6 - 384 *log(3)*x**5 - 1024*log(3)*x**4 - 1024*log(3)*x**3 + x**8 + 16*x**7 + 96*x **6 + 256*x**5 + 256*x**4