Integrand size = 131, antiderivative size = 27 \[ \int \frac {32+60 x+110 x^2+32 x^3+6 x^4+\left (5 x+5 x^2+5 x^3\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )}{-64 x-96 x^2-100 x^3-36 x^4-4 x^5+\left (16 x+24 x^2+25 x^3+9 x^4+x^5\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )} \, dx=3-\frac {5}{4+x}+\log \left (4-\log \left (x^2 \left (1+x+x^2\right )^2\right )\right ) \] Output:
3+ln(4-ln(x^2*(x^2+x+1)^2))-5/(4+x)
Time = 0.18 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {32+60 x+110 x^2+32 x^3+6 x^4+\left (5 x+5 x^2+5 x^3\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )}{-64 x-96 x^2-100 x^3-36 x^4-4 x^5+\left (16 x+24 x^2+25 x^3+9 x^4+x^5\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )} \, dx=-\frac {5}{4+x}+\log \left (4-\log \left (x^2 \left (1+x+x^2\right )^2\right )\right ) \] Input:
Integrate[(32 + 60*x + 110*x^2 + 32*x^3 + 6*x^4 + (5*x + 5*x^2 + 5*x^3)*Lo g[x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6])/(-64*x - 96*x^2 - 100*x^3 - 36*x^4 - 4*x^5 + (16*x + 24*x^2 + 25*x^3 + 9*x^4 + x^5)*Log[x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6]),x]
Output:
-5/(4 + x) + Log[4 - Log[x^2*(1 + x + x^2)^2]]
Time = 1.74 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {7239, 7279, 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.
\(\displaystyle \int \frac {6 x^4+32 x^3+110 x^2+\left (5 x^3+5 x^2+5 x\right ) \log \left (x^6+2 x^5+3 x^4+2 x^3+x^2\right )+60 x+32}{-4 x^5-36 x^4-100 x^3-96 x^2+\left (x^5+9 x^4+25 x^3+24 x^2+16 x\right ) \log \left (x^6+2 x^5+3 x^4+2 x^3+x^2\right )-64 x} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-6 x^4-32 x^3-110 x^2-5 \left (x^2+x+1\right ) x \log \left (x^2 \left (x^2+x+1\right )^2\right )-60 x-32}{x (x+4)^2 \left (x^2+x+1\right ) \left (4-\log \left (x^2 \left (x^2+x+1\right )^2\right )\right )}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {2 \left (3 x^2+2 x+1\right )}{x \left (x^2+x+1\right ) \left (\log \left (x^2 \left (x^2+x+1\right )^2\right )-4\right )}+\frac {5}{(x+4)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (4-\log \left (x^2 \left (x^2+x+1\right )^2\right )\right )-\frac {5}{x+4}\) |
Input:
Int[(32 + 60*x + 110*x^2 + 32*x^3 + 6*x^4 + (5*x + 5*x^2 + 5*x^3)*Log[x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6])/(-64*x - 96*x^2 - 100*x^3 - 36*x^4 - 4*x^5 + (16*x + 24*x^2 + 25*x^3 + 9*x^4 + x^5)*Log[x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6]),x]
Output:
-5/(4 + x) + Log[4 - Log[x^2*(1 + x + x^2)^2]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 1.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30
method | result | size |
norman | \(-\frac {5}{4+x}+\ln \left (\ln \left (x^{6}+2 x^{5}+3 x^{4}+2 x^{3}+x^{2}\right )-4\right )\) | \(35\) |
risch | \(-\frac {5}{4+x}+\ln \left (\ln \left (x^{6}+2 x^{5}+3 x^{4}+2 x^{3}+x^{2}\right )-4\right )\) | \(35\) |
parallelrisch | \(\frac {-10+2 \ln \left (\ln \left (x^{2} \left (x^{4}+2 x^{3}+3 x^{2}+2 x +1\right )\right )-4\right ) x +8 \ln \left (\ln \left (x^{2} \left (x^{4}+2 x^{3}+3 x^{2}+2 x +1\right )\right )-4\right )}{2 x +8}\) | \(67\) |
Input:
int(((5*x^3+5*x^2+5*x)*ln(x^6+2*x^5+3*x^4+2*x^3+x^2)+6*x^4+32*x^3+110*x^2+ 60*x+32)/((x^5+9*x^4+25*x^3+24*x^2+16*x)*ln(x^6+2*x^5+3*x^4+2*x^3+x^2)-4*x ^5-36*x^4-100*x^3-96*x^2-64*x),x,method=_RETURNVERBOSE)
Output:
-5/(4+x)+ln(ln(x^6+2*x^5+3*x^4+2*x^3+x^2)-4)
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {32+60 x+110 x^2+32 x^3+6 x^4+\left (5 x+5 x^2+5 x^3\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )}{-64 x-96 x^2-100 x^3-36 x^4-4 x^5+\left (16 x+24 x^2+25 x^3+9 x^4+x^5\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )} \, dx=\frac {{\left (x + 4\right )} \log \left (\log \left (x^{6} + 2 \, x^{5} + 3 \, x^{4} + 2 \, x^{3} + x^{2}\right ) - 4\right ) - 5}{x + 4} \] Input:
integrate(((5*x^3+5*x^2+5*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)+6*x^4+32*x^3+1 10*x^2+60*x+32)/((x^5+9*x^4+25*x^3+24*x^2+16*x)*log(x^6+2*x^5+3*x^4+2*x^3+ x^2)-4*x^5-36*x^4-100*x^3-96*x^2-64*x),x, algorithm="fricas")
Output:
((x + 4)*log(log(x^6 + 2*x^5 + 3*x^4 + 2*x^3 + x^2) - 4) - 5)/(x + 4)
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {32+60 x+110 x^2+32 x^3+6 x^4+\left (5 x+5 x^2+5 x^3\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )}{-64 x-96 x^2-100 x^3-36 x^4-4 x^5+\left (16 x+24 x^2+25 x^3+9 x^4+x^5\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )} \, dx=\log {\left (\log {\left (x^{6} + 2 x^{5} + 3 x^{4} + 2 x^{3} + x^{2} \right )} - 4 \right )} - \frac {5}{x + 4} \] Input:
integrate(((5*x**3+5*x**2+5*x)*ln(x**6+2*x**5+3*x**4+2*x**3+x**2)+6*x**4+3 2*x**3+110*x**2+60*x+32)/((x**5+9*x**4+25*x**3+24*x**2+16*x)*ln(x**6+2*x** 5+3*x**4+2*x**3+x**2)-4*x**5-36*x**4-100*x**3-96*x**2-64*x),x)
Output:
log(log(x**6 + 2*x**5 + 3*x**4 + 2*x**3 + x**2) - 4) - 5/(x + 4)
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {32+60 x+110 x^2+32 x^3+6 x^4+\left (5 x+5 x^2+5 x^3\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )}{-64 x-96 x^2-100 x^3-36 x^4-4 x^5+\left (16 x+24 x^2+25 x^3+9 x^4+x^5\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )} \, dx=-\frac {5}{x + 4} + \log \left (\log \left (x^{2} + x + 1\right ) + \log \left (x\right ) - 2\right ) \] Input:
integrate(((5*x^3+5*x^2+5*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)+6*x^4+32*x^3+1 10*x^2+60*x+32)/((x^5+9*x^4+25*x^3+24*x^2+16*x)*log(x^6+2*x^5+3*x^4+2*x^3+ x^2)-4*x^5-36*x^4-100*x^3-96*x^2-64*x),x, algorithm="maxima")
Output:
-5/(x + 4) + log(log(x^2 + x + 1) + log(x) - 2)
Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {32+60 x+110 x^2+32 x^3+6 x^4+\left (5 x+5 x^2+5 x^3\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )}{-64 x-96 x^2-100 x^3-36 x^4-4 x^5+\left (16 x+24 x^2+25 x^3+9 x^4+x^5\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )} \, dx=-\frac {5}{x + 4} + \log \left (\log \left (x^{6} + 2 \, x^{5} + 3 \, x^{4} + 2 \, x^{3} + x^{2}\right ) - 4\right ) \] Input:
integrate(((5*x^3+5*x^2+5*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)+6*x^4+32*x^3+1 10*x^2+60*x+32)/((x^5+9*x^4+25*x^3+24*x^2+16*x)*log(x^6+2*x^5+3*x^4+2*x^3+ x^2)-4*x^5-36*x^4-100*x^3-96*x^2-64*x),x, algorithm="giac")
Output:
-5/(x + 4) + log(log(x^6 + 2*x^5 + 3*x^4 + 2*x^3 + x^2) - 4)
Time = 0.38 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {32+60 x+110 x^2+32 x^3+6 x^4+\left (5 x+5 x^2+5 x^3\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )}{-64 x-96 x^2-100 x^3-36 x^4-4 x^5+\left (16 x+24 x^2+25 x^3+9 x^4+x^5\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )} \, dx=\ln \left (\ln \left (x^6+2\,x^5+3\,x^4+2\,x^3+x^2\right )-4\right )-\frac {5}{x+4} \] Input:
int(-(60*x + log(x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6)*(5*x + 5*x^2 + 5*x^3) + 110*x^2 + 32*x^3 + 6*x^4 + 32)/(64*x - log(x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6)*(16*x + 24*x^2 + 25*x^3 + 9*x^4 + x^5) + 96*x^2 + 100*x^3 + 36*x^4 + 4*x^5),x)
Output:
log(log(x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6) - 4) - 5/(x + 4)
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.56 \[ \int \frac {32+60 x+110 x^2+32 x^3+6 x^4+\left (5 x+5 x^2+5 x^3\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )}{-64 x-96 x^2-100 x^3-36 x^4-4 x^5+\left (16 x+24 x^2+25 x^3+9 x^4+x^5\right ) \log \left (x^2+2 x^3+3 x^4+2 x^5+x^6\right )} \, dx=\frac {4 \,\mathrm {log}\left (\mathrm {log}\left (x^{6}+2 x^{5}+3 x^{4}+2 x^{3}+x^{2}\right )-4\right ) x +16 \,\mathrm {log}\left (\mathrm {log}\left (x^{6}+2 x^{5}+3 x^{4}+2 x^{3}+x^{2}\right )-4\right )+5 x}{4 x +16} \] Input:
int(((5*x^3+5*x^2+5*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)+6*x^4+32*x^3+110*x^2 +60*x+32)/((x^5+9*x^4+25*x^3+24*x^2+16*x)*log(x^6+2*x^5+3*x^4+2*x^3+x^2)-4 *x^5-36*x^4-100*x^3-96*x^2-64*x),x)
Output:
(4*log(log(x**6 + 2*x**5 + 3*x**4 + 2*x**3 + x**2) - 4)*x + 16*log(log(x** 6 + 2*x**5 + 3*x**4 + 2*x**3 + x**2) - 4) + 5*x)/(4*(x + 4))