Integrand size = 83, antiderivative size = 31 \[ \int \frac {-7+6 x+6 x^2-6 x^3+x^4+\left (6 x+28 x^2-4 x^3-2 x^4\right ) \log \left (\frac {2+10 x+2 x^2}{x}\right )}{\left (x+5 x^2+x^3\right ) \log ^2\left (\frac {2+10 x+2 x^2}{x}\right )} \, dx=\frac {5+(3-x) (-4+x)-x}{\log \left (\frac {2 (1+x+x (4+x))}{x}\right )} \]
Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-7+6 x+6 x^2-6 x^3+x^4+\left (6 x+28 x^2-4 x^3-2 x^4\right ) \log \left (\frac {2+10 x+2 x^2}{x}\right )}{\left (x+5 x^2+x^3\right ) \log ^2\left (\frac {2+10 x+2 x^2}{x}\right )} \, dx=\frac {-7+6 x-x^2}{\log \left (2 \left (5+\frac {1}{x}+x\right )\right )} \]
Integrate[(-7 + 6*x + 6*x^2 - 6*x^3 + x^4 + (6*x + 28*x^2 - 4*x^3 - 2*x^4) *Log[(2 + 10*x + 2*x^2)/x])/((x + 5*x^2 + x^3)*Log[(2 + 10*x + 2*x^2)/x]^2 ),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.
\(\displaystyle \int \frac {x^4-6 x^3+6 x^2+\left (-2 x^4-4 x^3+28 x^2+6 x\right ) \log \left (\frac {2 x^2+10 x+2}{x}\right )+6 x-7}{\left (x^3+5 x^2+x\right ) \log ^2\left (\frac {2 x^2+10 x+2}{x}\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^4-6 x^3+6 x^2+\left (-2 x^4-4 x^3+28 x^2+6 x\right ) \log \left (\frac {2 x^2+10 x+2}{x}\right )+6 x-7}{x \left (x^2+5 x+1\right ) \log ^2\left (\frac {2 x^2+10 x+2}{x}\right )}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {x^4-6 x^3+6 x^2+6 x-7}{x \left (x^2+5 x+1\right ) \log ^2\left (2 \left (x+\frac {1}{x}+5\right )\right )}-\frac {2 (x-3)}{\log \left (2 \left (x+\frac {1}{x}+5\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -11 \int \frac {1}{\log ^2\left (2 \left (x+5+\frac {1}{x}\right )\right )}dx-\frac {104 \int \frac {1}{\left (-2 x+\sqrt {21}-5\right ) \log ^2\left (2 \left (x+5+\frac {1}{x}\right )\right )}dx}{\sqrt {21}}-7 \int \frac {1}{x \log ^2\left (2 \left (x+5+\frac {1}{x}\right )\right )}dx+\int \frac {x}{\log ^2\left (2 \left (x+5+\frac {1}{x}\right )\right )}dx+\frac {67}{21} \left (21-5 \sqrt {21}\right ) \int \frac {1}{\left (2 x-\sqrt {21}+5\right ) \log ^2\left (2 \left (x+5+\frac {1}{x}\right )\right )}dx+\frac {67}{21} \left (21+5 \sqrt {21}\right ) \int \frac {1}{\left (2 x+\sqrt {21}+5\right ) \log ^2\left (2 \left (x+5+\frac {1}{x}\right )\right )}dx-\frac {104 \int \frac {1}{\left (2 x+\sqrt {21}+5\right ) \log ^2\left (2 \left (x+5+\frac {1}{x}\right )\right )}dx}{\sqrt {21}}+6 \int \frac {1}{\log \left (2 \left (x+5+\frac {1}{x}\right )\right )}dx-2 \int \frac {x}{\log \left (2 \left (x+5+\frac {1}{x}\right )\right )}dx\) |
Int[(-7 + 6*x + 6*x^2 - 6*x^3 + x^4 + (6*x + 28*x^2 - 4*x^3 - 2*x^4)*Log[( 2 + 10*x + 2*x^2)/x])/((x + 5*x^2 + x^3)*Log[(2 + 10*x + 2*x^2)/x]^2),x]
3.11.88.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[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 = 1.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87
method | result | size |
parallelrisch | \(-\frac {x^{2}-6 x +7}{\ln \left (\frac {2 x^{2}+10 x +2}{x}\right )}\) | \(27\) |
risch | \(-\frac {x^{2}-6 x +7}{\ln \left (\frac {2 x^{2}+10 x +2}{x}\right )}\) | \(28\) |
norman | \(\frac {-x^{2}+6 x -7}{\ln \left (\frac {2 x^{2}+10 x +2}{x}\right )}\) | \(29\) |
int(((-2*x^4-4*x^3+28*x^2+6*x)*ln((2*x^2+10*x+2)/x)+x^4-6*x^3+6*x^2+6*x-7) /(x^3+5*x^2+x)/ln((2*x^2+10*x+2)/x)^2,x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-7+6 x+6 x^2-6 x^3+x^4+\left (6 x+28 x^2-4 x^3-2 x^4\right ) \log \left (\frac {2+10 x+2 x^2}{x}\right )}{\left (x+5 x^2+x^3\right ) \log ^2\left (\frac {2+10 x+2 x^2}{x}\right )} \, dx=-\frac {x^{2} - 6 \, x + 7}{\log \left (\frac {2 \, {\left (x^{2} + 5 \, x + 1\right )}}{x}\right )} \]
integrate(((-2*x^4-4*x^3+28*x^2+6*x)*log((2*x^2+10*x+2)/x)+x^4-6*x^3+6*x^2 +6*x-7)/(x^3+5*x^2+x)/log((2*x^2+10*x+2)/x)^2,x, algorithm=\
Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {-7+6 x+6 x^2-6 x^3+x^4+\left (6 x+28 x^2-4 x^3-2 x^4\right ) \log \left (\frac {2+10 x+2 x^2}{x}\right )}{\left (x+5 x^2+x^3\right ) \log ^2\left (\frac {2+10 x+2 x^2}{x}\right )} \, dx=\frac {- x^{2} + 6 x - 7}{\log {\left (\frac {2 x^{2} + 10 x + 2}{x} \right )}} \]
integrate(((-2*x**4-4*x**3+28*x**2+6*x)*ln((2*x**2+10*x+2)/x)+x**4-6*x**3+ 6*x**2+6*x-7)/(x**3+5*x**2+x)/ln((2*x**2+10*x+2)/x)**2,x)
Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {-7+6 x+6 x^2-6 x^3+x^4+\left (6 x+28 x^2-4 x^3-2 x^4\right ) \log \left (\frac {2+10 x+2 x^2}{x}\right )}{\left (x+5 x^2+x^3\right ) \log ^2\left (\frac {2+10 x+2 x^2}{x}\right )} \, dx=-\frac {x^{2} - 6 \, x + 7}{\log \left (2\right ) + \log \left (x^{2} + 5 \, x + 1\right ) - \log \left (x\right )} \]
integrate(((-2*x^4-4*x^3+28*x^2+6*x)*log((2*x^2+10*x+2)/x)+x^4-6*x^3+6*x^2 +6*x-7)/(x^3+5*x^2+x)/log((2*x^2+10*x+2)/x)^2,x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-7+6 x+6 x^2-6 x^3+x^4+\left (6 x+28 x^2-4 x^3-2 x^4\right ) \log \left (\frac {2+10 x+2 x^2}{x}\right )}{\left (x+5 x^2+x^3\right ) \log ^2\left (\frac {2+10 x+2 x^2}{x}\right )} \, dx=-\frac {x^{2} - 6 \, x + 7}{\log \left (\frac {2 \, {\left (x^{2} + 5 \, x + 1\right )}}{x}\right )} \]
integrate(((-2*x^4-4*x^3+28*x^2+6*x)*log((2*x^2+10*x+2)/x)+x^4-6*x^3+6*x^2 +6*x-7)/(x^3+5*x^2+x)/log((2*x^2+10*x+2)/x)^2,x, algorithm=\
Time = 13.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-7+6 x+6 x^2-6 x^3+x^4+\left (6 x+28 x^2-4 x^3-2 x^4\right ) \log \left (\frac {2+10 x+2 x^2}{x}\right )}{\left (x+5 x^2+x^3\right ) \log ^2\left (\frac {2+10 x+2 x^2}{x}\right )} \, dx=-\frac {x^2-6\,x+7}{\ln \left (\frac {2\,x^2+10\,x+2}{x}\right )} \]