Integrand size = 158, antiderivative size = 25 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=(4+x) \left (2+2 x+\log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )\right ) \]
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=10 x+2 x^2+4 \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )+x \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right ) \]
Integrate[(16 - 76*x - 52*x^2 + 22*x^4 + 8*x^5 + (-40*x - 26*x^2 - 4*x^3 + 10*x^4 + 4*x^5)*Log[(4 + x - x^3)/x] + (-8*x - 2*x^2 + 2*x^4 + (-4*x - x^ 2 + x^4)*Log[(4 + x - x^3)/x])*Log[2 + Log[(4 + x - x^3)/x]])/(-8*x - 2*x^ 2 + 2*x^4 + (-4*x - x^2 + x^4)*Log[(4 + x - x^3)/x]),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.
\(\displaystyle \int \frac {8 x^5+22 x^4-52 x^2+\left (2 x^4-2 x^2+\left (x^4-x^2-4 x\right ) \log \left (\frac {-x^3+x+4}{x}\right )-8 x\right ) \log \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )+\left (4 x^5+10 x^4-4 x^3-26 x^2-40 x\right ) \log \left (\frac {-x^3+x+4}{x}\right )-76 x+16}{2 x^4-2 x^2+\left (x^4-x^2-4 x\right ) \log \left (\frac {-x^3+x+4}{x}\right )-8 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-8 x^5-22 x^4+52 x^2-\left (2 x^4-2 x^2+\left (x^4-x^2-4 x\right ) \log \left (\frac {-x^3+x+4}{x}\right )-8 x\right ) \log \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )-\left (4 x^5+10 x^4-4 x^3-26 x^2-40 x\right ) \log \left (\frac {-x^3+x+4}{x}\right )+76 x-16}{x \left (-x^3+x+4\right ) \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {22 x^3}{\left (x^3-x-4\right ) \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}-\frac {52 x}{\left (x^3-x-4\right ) \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}+\log \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )-\frac {76}{\left (x^3-x-4\right ) \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}+\frac {16}{\left (x^3-x-4\right ) x \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}+\frac {8 x^4}{\left (x^3-x-4\right ) \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}+\frac {2 (2 x+5) \log \left (-x^2+\frac {4}{x}+1\right )}{\log \left (\frac {-x^3+x+4}{x}\right )+2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 22 \int \frac {1}{\log \left (\frac {-x^3+x+4}{x}\right )+2}dx-4 \int \frac {1}{x \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}dx+8 \int \frac {x}{\log \left (\frac {-x^3+x+4}{x}\right )+2}dx+8 \int \frac {1}{\left (x^3-x-4\right ) \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}dx+2 \int \frac {x}{\left (x^3-x-4\right ) \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}dx+\int \log \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )dx+12 \int \frac {x^2}{\left (x^3-x-4\right ) \left (\log \left (\frac {-x^3+x+4}{x}\right )+2\right )}dx+10 \int \frac {\log \left (-x^2+1+\frac {4}{x}\right )}{\log \left (\frac {-x^3+x+4}{x}\right )+2}dx+4 \int \frac {x \log \left (-x^2+1+\frac {4}{x}\right )}{\log \left (\frac {-x^3+x+4}{x}\right )+2}dx\) |
Int[(16 - 76*x - 52*x^2 + 22*x^4 + 8*x^5 + (-40*x - 26*x^2 - 4*x^3 + 10*x^ 4 + 4*x^5)*Log[(4 + x - x^3)/x] + (-8*x - 2*x^2 + 2*x^4 + (-4*x - x^2 + x^ 4)*Log[(4 + x - x^3)/x])*Log[2 + Log[(4 + x - x^3)/x]])/(-8*x - 2*x^2 + 2* x^4 + (-4*x - x^2 + x^4)*Log[(4 + x - x^3)/x]),x]
3.15.71.3.1 Defintions of rubi rules used
Time = 1.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96
method | result | size |
parallelrisch | \(-39+2 x^{2}+\ln \left (\ln \left (-\frac {x^{3}-x -4}{x}\right )+2\right ) x +4 \ln \left (\ln \left (-\frac {x^{3}-x -4}{x}\right )+2\right )+10 x\) | \(49\) |
int((((x^4-x^2-4*x)*ln((-x^3+x+4)/x)+2*x^4-2*x^2-8*x)*ln(ln((-x^3+x+4)/x)+ 2)+(4*x^5+10*x^4-4*x^3-26*x^2-40*x)*ln((-x^3+x+4)/x)+8*x^5+22*x^4-52*x^2-7 6*x+16)/((x^4-x^2-4*x)*ln((-x^3+x+4)/x)+2*x^4-2*x^2-8*x),x,method=_RETURNV ERBOSE)
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=2 \, x^{2} + {\left (x + 4\right )} \log \left (\log \left (-\frac {x^{3} - x - 4}{x}\right ) + 2\right ) + 10 \, x \]
integrate((((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x)*log(log((-x^3 +x+4)/x)+2)+(4*x^5+10*x^4-4*x^3-26*x^2-40*x)*log((-x^3+x+4)/x)+8*x^5+22*x^ 4-52*x^2-76*x+16)/((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x),x, alg orithm=\
Time = 0.40 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=2 x^{2} + x \log {\left (\log {\left (\frac {- x^{3} + x + 4}{x} \right )} + 2 \right )} + 10 x + 4 \log {\left (\log {\left (\frac {- x^{3} + x + 4}{x} \right )} + 2 \right )} \]
integrate((((x**4-x**2-4*x)*ln((-x**3+x+4)/x)+2*x**4-2*x**2-8*x)*ln(ln((-x **3+x+4)/x)+2)+(4*x**5+10*x**4-4*x**3-26*x**2-40*x)*ln((-x**3+x+4)/x)+8*x* *5+22*x**4-52*x**2-76*x+16)/((x**4-x**2-4*x)*ln((-x**3+x+4)/x)+2*x**4-2*x* *2-8*x),x)
Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=2 \, x^{2} + {\left (x + 4\right )} \log \left (\log \left (-x^{3} + x + 4\right ) - \log \left (x\right ) + 2\right ) + 10 \, x \]
integrate((((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x)*log(log((-x^3 +x+4)/x)+2)+(4*x^5+10*x^4-4*x^3-26*x^2-40*x)*log((-x^3+x+4)/x)+8*x^5+22*x^ 4-52*x^2-76*x+16)/((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x),x, alg orithm=\
Time = 0.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=2 \, x^{2} + x \log \left (\log \left (-\frac {x^{3} - x - 4}{x}\right ) + 2\right ) + 10 \, x + 4 \, \log \left (\log \left (-x^{3} + x + 4\right ) - \log \left (x\right ) + 2\right ) \]
integrate((((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x)*log(log((-x^3 +x+4)/x)+2)+(4*x^5+10*x^4-4*x^3-26*x^2-40*x)*log((-x^3+x+4)/x)+8*x^5+22*x^ 4-52*x^2-76*x+16)/((x^4-x^2-4*x)*log((-x^3+x+4)/x)+2*x^4-2*x^2-8*x),x, alg orithm=\
Time = 8.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {16-76 x-52 x^2+22 x^4+8 x^5+\left (-40 x-26 x^2-4 x^3+10 x^4+4 x^5\right ) \log \left (\frac {4+x-x^3}{x}\right )+\left (-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )\right ) \log \left (2+\log \left (\frac {4+x-x^3}{x}\right )\right )}{-8 x-2 x^2+2 x^4+\left (-4 x-x^2+x^4\right ) \log \left (\frac {4+x-x^3}{x}\right )} \, dx=10\,x+4\,\ln \left (\ln \left (\frac {-x^3+x+4}{x}\right )+2\right )+x\,\ln \left (\ln \left (\frac {-x^3+x+4}{x}\right )+2\right )+2\,x^2 \]