Integrand size = 152, antiderivative size = 35 \[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx=\frac {\left (5+\frac {x-\log \left (\frac {5}{2 x \log ^2(x)}\right )}{\log (3)}\right )^2}{-\frac {4}{x}+x} \] Output:
(5+(x-ln(5/2/x/ln(x)^2))/ln(3))^2/(x-4/x)
\[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx=\int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx \] Input:
Integrate[(-16*x + 4*x^3 + (-80 + 20*x^2)*Log[3] + (-8*x - 12*x^2 + 2*x^3 + x^4 + (-40 - 80*x + 10*x^2)*Log[3] + (-100 - 25*x^2)*Log[3]^2)*Log[x] + (16 - 4*x^2 + (8 + 16*x - 2*x^2 + (40 + 10*x^2)*Log[3])*Log[x])*Log[5/(2*x *Log[x]^2)] + (-4 - x^2)*Log[x]*Log[5/(2*x*Log[x]^2)]^2)/((16 - 8*x^2 + x^ 4)*Log[3]^2*Log[x]),x]
Output:
Integrate[(-16*x + 4*x^3 + (-80 + 20*x^2)*Log[3] + (-8*x - 12*x^2 + 2*x^3 + x^4 + (-40 - 80*x + 10*x^2)*Log[3] + (-100 - 25*x^2)*Log[3]^2)*Log[x] + (16 - 4*x^2 + (8 + 16*x - 2*x^2 + (40 + 10*x^2)*Log[3])*Log[x])*Log[5/(2*x *Log[x]^2)] + (-4 - x^2)*Log[x]*Log[5/(2*x*Log[x]^2)]^2)/((16 - 8*x^2 + x^ 4)*Log[x]), x]/Log[3]^2
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.
| \(\displaystyle \int \frac {4 x^3+\left (-x^2-4\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4 x^2+\left (-2 x^2+\left (10 x^2+40\right ) \log (3)+16 x+8\right ) \log (x)+16\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (20 x^2-80\right ) \log (3)+\left (x^4+2 x^3-12 x^2+\left (-25 x^2-100\right ) \log ^2(3)+\left (10 x^2-80 x-40\right ) \log (3)-8 x\right ) \log (x)-16 x}{\left (x^4-8 x^2+16\right ) \log ^2(3) \log (x)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {-4 x^3+16 x+\left (x^2+4\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )+\left (-x^4-2 x^3+12 x^2+8 x+25 \left (x^2+4\right ) \log ^2(3)+10 \left (-x^2+8 x+4\right ) \log (3)\right ) \log (x)-2 \left (-2 x^2+\left (-x^2+8 x+5 \left (x^2+4\right ) \log (3)+4\right ) \log (x)+8\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+20 \left (4-x^2\right ) \log (3)}{\left (x^4-8 x^2+16\right ) \log (x)}dx}{\log ^2(3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {-4 x^3+16 x+\left (x^2+4\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )+\left (-x^4-2 x^3+12 x^2+8 x+25 \left (x^2+4\right ) \log ^2(3)+10 \left (-x^2+8 x+4\right ) \log (3)\right ) \log (x)-2 \left (-2 x^2+\left (-x^2+8 x+5 \left (x^2+4\right ) \log (3)+4\right ) \log (x)+8\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+20 \left (4-x^2\right ) \log (3)}{\left (x^4-8 x^2+16\right ) \log (x)}dx}{\log ^2(3)}\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle -\frac {\int \frac {-4 x^3+16 x+\left (x^2+4\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )+\left (-x^4-2 x^3+12 x^2+8 x+25 \left (x^2+4\right ) \log ^2(3)+10 \left (-x^2+8 x+4\right ) \log (3)\right ) \log (x)-2 \left (-2 x^2+\left (-x^2+8 x+5 \left (x^2+4\right ) \log (3)+4\right ) \log (x)+8\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+20 \left (4-x^2\right ) \log (3)}{\left (4-x^2\right )^2 \log (x)}dx}{\log ^2(3)}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {\int \frac {\left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) \left (-4 \left (x^2-4\right )-\log (x) \left (x^3+(2-5 \log (3)) x^2-12 x+\left (x^2+4\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )-4 (2+\log (243))\right )\right )}{\left (4-x^2\right )^2 \log (x)}dx}{\log ^2(3)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (-\frac {\left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) x^3}{\left (x^2-4\right )^2}+\frac {(-2+\log (243)) \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) x^2}{\left (x^2-4\right )^2}+\frac {12 \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) x}{\left (x^2-4\right )^2}-\frac {4 \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{\left (x^2-4\right ) \log (x)}+\frac {4 (2+\log (243)) \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right )}{\left (x^2-4\right )^2}-\frac {\left (x^2+4\right ) \left (x-\log \left (\frac {5}{486 x \log ^2(x)}\right )\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )}{\left (x^2-4\right )^2}\right )dx}{\log ^2(3)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {x^3}{2 \left (4-x^2\right )}+\frac {6 x}{4-x^2}-\frac {3 x}{2}+\frac {1}{8} (2+\log (243)) \log (2-x)-\frac {1}{8} (2-\log (243)) \log (2-x)+\frac {1}{4} \log (2-x)-2 \log (x)-\frac {1}{8} (2+\log (243)) \log (x+2)+\frac {1}{8} (2-\log (243)) \log (x+2)+\frac {1}{4} \log (x+2)-\frac {1}{2} (2-\log (243)) \log \left (4-x^2\right )+\frac {3}{4} \log \left (4-x^2\right )+\frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{2 (2-x)}+\frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{2 (x+2)}-\frac {6 \log \left (\frac {5}{486 x \log ^2(x)}\right )}{4-x^2}-\frac {(2+\log (243)) \log \left (\frac {5}{486 x \log ^2(x)}\right )}{4 (2-x)}+\frac {(2+\log (243)) \log \left (\frac {5}{486 x \log ^2(x)}\right )}{4 (x+2)}+\frac {(2-\log (243)) \log \left (\frac {5}{486 x \log ^2(x)}\right )}{4 (2-x)}-\frac {(2-\log (243)) \log \left (\frac {5}{486 x \log ^2(x)}\right )}{4 (x+2)}-\frac {\log \left (\frac {5}{2 x \log ^2(x)}\right )}{2-x}-\frac {\log \left (\frac {5}{2 x \log ^2(x)}\right )}{x+2}+\frac {1}{2} (2+\log (243)) \int \frac {1}{(x-2) x \log (x)}dx-\frac {1}{2} (2-\log (243)) \int \frac {1}{(x-2) x \log (x)}dx+\int \frac {1}{(x-2) x \log (x)}dx+\frac {1}{2} (2+\log (243)) \int \frac {1}{x (x+2) \log (x)}dx-\frac {1}{2} (2-\log (243)) \int \frac {1}{x (x+2) \log (x)}dx-\int \frac {1}{x (x+2) \log (x)}dx+12 \int \frac {1}{x \left (x^2-4\right ) \log (x)}dx-4 \int \frac {x}{\left (x^2-4\right ) \log (x)}dx+\frac {1}{8} (2+\log (243)) \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{x-2}dx+\frac {1}{8} (2-\log (243)) \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{x-2}dx+\frac {1}{2} \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{x-2}dx-\frac {1}{8} (2+\log (243)) \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{x+2}dx-\frac {1}{8} (2-\log (243)) \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{x+2}dx+\frac {1}{2} \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{x+2}dx+\int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{(x-2) \log (x)}dx-\int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right )}{(x+2) \log (x)}dx-\frac {1}{2} \int \frac {\log \left (\frac {5}{2 x \log ^2(x)}\right )}{x-2}dx-\frac {1}{2} \int \frac {\log \left (\frac {5}{2 x \log ^2(x)}\right )}{x+2}dx+\frac {1}{2} \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )}{(x-2)^2}dx+\frac {1}{2} \int \frac {\log \left (\frac {5}{486 x \log ^2(x)}\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )}{(x+2)^2}dx+\frac {2 (2+\log (243))}{4-x^2}-\frac {2 (2-\log (243))}{4-x^2}}{\log ^2(3)}\) |
Input:
Int[(-16*x + 4*x^3 + (-80 + 20*x^2)*Log[3] + (-8*x - 12*x^2 + 2*x^3 + x^4 + (-40 - 80*x + 10*x^2)*Log[3] + (-100 - 25*x^2)*Log[3]^2)*Log[x] + (16 - 4*x^2 + (8 + 16*x - 2*x^2 + (40 + 10*x^2)*Log[3])*Log[x])*Log[5/(2*x*Log[x ]^2)] + (-4 - x^2)*Log[x]*Log[5/(2*x*Log[x]^2)]^2)/((16 - 8*x^2 + x^4)*Log [3]^2*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(33)=66\).
Time = 3.85 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.06
| method | result | size |
| parallelrisch | \(\frac {25 x \ln \left (3\right )^{2}-10 \ln \left (3\right ) \ln \left (\frac {5}{2 x \ln \left (x \right )^{2}}\right ) x +x^{3}-2 \ln \left (\frac {5}{2 x \ln \left (x \right )^{2}}\right ) x^{2}+\ln \left (\frac {5}{2 x \ln \left (x \right )^{2}}\right )^{2} x +40 \ln \left (3\right )}{\ln \left (3\right )^{2} \left (x^{2}-4\right )}\) | \(72\) |
| risch | \(\text {Expression too large to display}\) | \(2034\) |
Input:
int(((-x^2-4)*ln(x)*ln(5/2/x/ln(x)^2)^2+(((10*x^2+40)*ln(3)-2*x^2+16*x+8)* ln(x)-4*x^2+16)*ln(5/2/x/ln(x)^2)+((-25*x^2-100)*ln(3)^2+(10*x^2-80*x-40)* ln(3)+x^4+2*x^3-12*x^2-8*x)*ln(x)+(20*x^2-80)*ln(3)+4*x^3-16*x)/(x^4-8*x^2 +16)/ln(3)^2/ln(x),x,method=_RETURNVERBOSE)
Output:
1/ln(3)^2*(25*x*ln(3)^2-10*ln(3)*ln(5/2/x/ln(x)^2)*x+x^3-2*ln(5/2/x/ln(x)^ 2)*x^2+ln(5/2/x/ln(x)^2)^2*x+40*ln(3))/(x^2-4)
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.77 \[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx=\frac {x^{3} + 25 \, x \log \left (3\right )^{2} + x \log \left (\frac {5}{2 \, x \log \left (x\right )^{2}}\right )^{2} - 2 \, {\left (x^{2} + 5 \, x \log \left (3\right )\right )} \log \left (\frac {5}{2 \, x \log \left (x\right )^{2}}\right ) + 40 \, \log \left (3\right )}{{\left (x^{2} - 4\right )} \log \left (3\right )^{2}} \] Input:
integrate(((-x^2-4)*log(x)*log(5/2/x/log(x)^2)^2+(((10*x^2+40)*log(3)-2*x^ 2+16*x+8)*log(x)-4*x^2+16)*log(5/2/x/log(x)^2)+((-25*x^2-100)*log(3)^2+(10 *x^2-80*x-40)*log(3)+x^4+2*x^3-12*x^2-8*x)*log(x)+(20*x^2-80)*log(3)+4*x^3 -16*x)/(x^4-8*x^2+16)/log(3)^2/log(x),x, algorithm="fricas")
Output:
(x^3 + 25*x*log(3)^2 + x*log(5/2/(x*log(x)^2))^2 - 2*(x^2 + 5*x*log(3))*lo g(5/2/(x*log(x)^2)) + 40*log(3))/((x^2 - 4)*log(3)^2)
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (24) = 48\).
Time = 1.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.60 \[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx=\frac {x}{\log {\left (3 \right )}^{2}} + \frac {x \log {\left (\frac {5}{2 x \log {\left (x \right )}^{2}} \right )}^{2}}{x^{2} \log {\left (3 \right )}^{2} - 4 \log {\left (3 \right )}^{2}} + \frac {x \left (4 + 25 \log {\left (3 \right )}^{2}\right ) + 40 \log {\left (3 \right )}}{x^{2} \log {\left (3 \right )}^{2} - 4 \log {\left (3 \right )}^{2}} + \frac {\left (- 10 x \log {\left (3 \right )} - 8\right ) \log {\left (\frac {5}{2 x \log {\left (x \right )}^{2}} \right )}}{x^{2} \log {\left (3 \right )}^{2} - 4 \log {\left (3 \right )}^{2}} + \frac {2 \log {\left (x \right )}}{\log {\left (3 \right )}^{2}} + \frac {4 \log {\left (\log {\left (x \right )} \right )}}{\log {\left (3 \right )}^{2}} \] Input:
integrate(((-x**2-4)*ln(x)*ln(5/2/x/ln(x)**2)**2+(((10*x**2+40)*ln(3)-2*x* *2+16*x+8)*ln(x)-4*x**2+16)*ln(5/2/x/ln(x)**2)+((-25*x**2-100)*ln(3)**2+(1 0*x**2-80*x-40)*ln(3)+x**4+2*x**3-12*x**2-8*x)*ln(x)+(20*x**2-80)*ln(3)+4* x**3-16*x)/(x**4-8*x**2+16)/ln(3)**2/ln(x),x)
Output:
x/log(3)**2 + x*log(5/(2*x*log(x)**2))**2/(x**2*log(3)**2 - 4*log(3)**2) + (x*(4 + 25*log(3)**2) + 40*log(3))/(x**2*log(3)**2 - 4*log(3)**2) + (-10* x*log(3) - 8)*log(5/(2*x*log(x)**2))/(x**2*log(3)**2 - 4*log(3)**2) + 2*lo g(x)/log(3)**2 + 4*log(log(x))/log(3)**2
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (33) = 66\).
Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 3.57 \[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx=\frac {x^{3} + x \log \left (x\right )^{2} + 4 \, x \log \left (\log \left (x\right )\right )^{2} + {\left (\log \left (5\right )^{2} - 10 \, \log \left (5\right ) \log \left (3\right ) + 25 \, \log \left (3\right )^{2} - 2 \, {\left (\log \left (5\right ) - 5 \, \log \left (3\right )\right )} \log \left (2\right ) + \log \left (2\right )^{2}\right )} x + 2 \, {\left (x^{2} - x {\left (\log \left (5\right ) - 5 \, \log \left (3\right ) - \log \left (2\right )\right )}\right )} \log \left (x\right ) + 4 \, {\left (x^{2} - x {\left (\log \left (5\right ) - 5 \, \log \left (3\right ) - \log \left (2\right )\right )} + x \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) - 8 \, \log \left (5\right ) + 40 \, \log \left (3\right ) + 8 \, \log \left (2\right )}{{\left (x^{2} - 4\right )} \log \left (3\right )^{2}} \] Input:
integrate(((-x^2-4)*log(x)*log(5/2/x/log(x)^2)^2+(((10*x^2+40)*log(3)-2*x^ 2+16*x+8)*log(x)-4*x^2+16)*log(5/2/x/log(x)^2)+((-25*x^2-100)*log(3)^2+(10 *x^2-80*x-40)*log(3)+x^4+2*x^3-12*x^2-8*x)*log(x)+(20*x^2-80)*log(3)+4*x^3 -16*x)/(x^4-8*x^2+16)/log(3)^2/log(x),x, algorithm="maxima")
Output:
(x^3 + x*log(x)^2 + 4*x*log(log(x))^2 + (log(5)^2 - 10*log(5)*log(3) + 25* log(3)^2 - 2*(log(5) - 5*log(3))*log(2) + log(2)^2)*x + 2*(x^2 - x*(log(5) - 5*log(3) - log(2)))*log(x) + 4*(x^2 - x*(log(5) - 5*log(3) - log(2)) + x*log(x))*log(log(x)) - 8*log(5) + 40*log(3) + 8*log(2))/((x^2 - 4)*log(3) ^2)
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (33) = 66\).
Time = 0.31 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.29 \[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx=\frac {2 \, {\left (\frac {x \log \left (x\right )}{x^{2} - 4} - \frac {x \log \left (5\right ) - 5 \, x \log \left (3\right ) - 4}{x^{2} - 4}\right )} \log \left (2 \, \log \left (x\right )^{2}\right ) + \frac {x \log \left (2 \, \log \left (x\right )^{2}\right )^{2}}{x^{2} - 4} + \frac {x \log \left (x\right )^{2}}{x^{2} - 4} + x - \frac {2 \, {\left (x \log \left (5\right ) - 5 \, x \log \left (3\right ) - 4\right )} \log \left (x\right )}{x^{2} - 4} + \frac {x \log \left (5\right )^{2} - 10 \, x \log \left (5\right ) \log \left (3\right ) + 25 \, x \log \left (3\right )^{2} + 4 \, x - 8 \, \log \left (5\right ) + 40 \, \log \left (3\right )}{x^{2} - 4} + 2 \, \log \left (x\right ) + 4 \, \log \left (\log \left (x\right )\right )}{\log \left (3\right )^{2}} \] Input:
integrate(((-x^2-4)*log(x)*log(5/2/x/log(x)^2)^2+(((10*x^2+40)*log(3)-2*x^ 2+16*x+8)*log(x)-4*x^2+16)*log(5/2/x/log(x)^2)+((-25*x^2-100)*log(3)^2+(10 *x^2-80*x-40)*log(3)+x^4+2*x^3-12*x^2-8*x)*log(x)+(20*x^2-80)*log(3)+4*x^3 -16*x)/(x^4-8*x^2+16)/log(3)^2/log(x),x, algorithm="giac")
Output:
(2*(x*log(x)/(x^2 - 4) - (x*log(5) - 5*x*log(3) - 4)/(x^2 - 4))*log(2*log( x)^2) + x*log(2*log(x)^2)^2/(x^2 - 4) + x*log(x)^2/(x^2 - 4) + x - 2*(x*lo g(5) - 5*x*log(3) - 4)*log(x)/(x^2 - 4) + (x*log(5)^2 - 10*x*log(5)*log(3) + 25*x*log(3)^2 + 4*x - 8*log(5) + 40*log(3))/(x^2 - 4) + 2*log(x) + 4*lo g(log(x)))/log(3)^2
Time = 4.47 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.54 \[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx=\frac {2\,\ln \left (x\right )}{{\ln \left (3\right )}^2}+\frac {4\,\ln \left (\ln \left (x\right )\right )}{{\ln \left (3\right )}^2}+\frac {x}{{\ln \left (3\right )}^2}+\frac {40\,\ln \left (3\right )+x\,\left (25\,{\ln \left (3\right )}^2+4\right )}{x^2\,{\ln \left (3\right )}^2-4\,{\ln \left (3\right )}^2}+\frac {x\,{\ln \left (\frac {5}{2\,x\,{\ln \left (x\right )}^2}\right )}^2}{x^2\,{\ln \left (3\right )}^2-4\,{\ln \left (3\right )}^2}-\frac {\ln \left (\frac {5}{2\,x\,{\ln \left (x\right )}^2}\right )\,\left (10\,x\,\ln \left (3\right )+8\right )}{x^2\,{\ln \left (3\right )}^2-4\,{\ln \left (3\right )}^2} \] Input:
int(-(16*x + log(x)*(8*x + log(3)*(80*x - 10*x^2 + 40) + log(3)^2*(25*x^2 + 100) + 12*x^2 - 2*x^3 - x^4) - log(3)*(20*x^2 - 80) - 4*x^3 - log(5/(2*x *log(x)^2))*(log(x)*(16*x + log(3)*(10*x^2 + 40) - 2*x^2 + 8) - 4*x^2 + 16 ) + log(x)*log(5/(2*x*log(x)^2))^2*(x^2 + 4))/(log(3)^2*log(x)*(x^4 - 8*x^ 2 + 16)),x)
Output:
(2*log(x))/log(3)^2 + (4*log(log(x)))/log(3)^2 + x/log(3)^2 + (40*log(3) + x*(25*log(3)^2 + 4))/(x^2*log(3)^2 - 4*log(3)^2) + (x*log(5/(2*x*log(x)^2 ))^2)/(x^2*log(3)^2 - 4*log(3)^2) - (log(5/(2*x*log(x)^2))*(10*x*log(3) + 8))/(x^2*log(3)^2 - 4*log(3)^2)
Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.91 \[ \int \frac {-16 x+4 x^3+\left (-80+20 x^2\right ) \log (3)+\left (-8 x-12 x^2+2 x^3+x^4+\left (-40-80 x+10 x^2\right ) \log (3)+\left (-100-25 x^2\right ) \log ^2(3)\right ) \log (x)+\left (16-4 x^2+\left (8+16 x-2 x^2+\left (40+10 x^2\right ) \log (3)\right ) \log (x)\right ) \log \left (\frac {5}{2 x \log ^2(x)}\right )+\left (-4-x^2\right ) \log (x) \log ^2\left (\frac {5}{2 x \log ^2(x)}\right )}{\left (16-8 x^2+x^4\right ) \log ^2(3) \log (x)} \, dx=\frac {x \left (\mathrm {log}\left (\frac {5}{2 \mathrm {log}\left (x \right )^{2} x}\right )^{2}-10 \,\mathrm {log}\left (\frac {5}{2 \mathrm {log}\left (x \right )^{2} x}\right ) \mathrm {log}\left (3\right )-2 \,\mathrm {log}\left (\frac {5}{2 \mathrm {log}\left (x \right )^{2} x}\right ) x +25 \mathrm {log}\left (3\right )^{2}+10 \,\mathrm {log}\left (3\right ) x +x^{2}\right )}{\mathrm {log}\left (3\right )^{2} \left (x^{2}-4\right )} \] Input:
int(((-x^2-4)*log(x)*log(5/2/x/log(x)^2)^2+(((10*x^2+40)*log(3)-2*x^2+16*x +8)*log(x)-4*x^2+16)*log(5/2/x/log(x)^2)+((-25*x^2-100)*log(3)^2+(10*x^2-8 0*x-40)*log(3)+x^4+2*x^3-12*x^2-8*x)*log(x)+(20*x^2-80)*log(3)+4*x^3-16*x) /(x^4-8*x^2+16)/log(3)^2/log(x),x)
Output:
(x*(log(5/(2*log(x)**2*x))**2 - 10*log(5/(2*log(x)**2*x))*log(3) - 2*log(5 /(2*log(x)**2*x))*x + 25*log(3)**2 + 10*log(3)*x + x**2))/(log(3)**2*(x**2 - 4))