Integrand size = 61, antiderivative size = 21 \[ \int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx=\left (-1+\log \left (\left (10-\frac {2}{x}+\frac {x^2}{20}\right )^2\right )\right )^2 \] Output:
(ln((10+1/20*x^2-2/x)^2)-1)^2
Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(21)=42\).
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.33 \[ \int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx=8 \left (-\frac {1}{4} \log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )+\frac {1}{8} \log ^2\left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right ) \] Input:
Integrate[(-160 - 8*x^3 + (160 + 8*x^3)*Log[(1600 - 16000*x + 40000*x^2 - 80*x^3 + 400*x^4 + x^6)/(400*x^2)])/(-40*x + 200*x^2 + x^4),x]
Output:
8*(-1/4*Log[(-40 + 200*x + x^3)^2/(400*x^2)] + Log[(-40 + 200*x + x^3)^2/( 400*x^2)]^2/8)
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.
| \(\displaystyle \int \frac {-8 x^3+\left (8 x^3+160\right ) \log \left (\frac {x^6+400 x^4-80 x^3+40000 x^2-16000 x+1600}{400 x^2}\right )-160}{x^4+200 x^2-40 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-8 x^3+\left (8 x^3+160\right ) \log \left (\frac {x^6+400 x^4-80 x^3+40000 x^2-16000 x+1600}{400 x^2}\right )-160}{x \left (x^3+200 x-40\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {8 \left (x^3+20\right ) \left (1-\log \left (\frac {\left (x^3+200 x-40\right )^2}{400 x^2}\right )\right )}{x \left (-x^3-200 x+40\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 \int \frac {\left (x^3+20\right ) \left (1-\log \left (\frac {\left (-x^3-200 x+40\right )^2}{400 x^2}\right )\right )}{x \left (-x^3-200 x+40\right )}dx\) |
| \(\Big \downarrow \) 3008 |
| \(\displaystyle 8 \int \left (\frac {1-\log \left (\frac {\left (-x^3-200 x+40\right )^2}{400 x^2}\right )}{2 x}+\frac {\left (-3 x^2-200\right ) \left (1-\log \left (\frac {\left (-x^3-200 x+40\right )^2}{400 x^2}\right )\right )}{2 \left (x^3+200 x-40\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8 \left (-\frac {1}{2} \log ^2(x)+\frac {1}{2} \left (1-\log \left (\frac {\left (-x^3-200 x+40\right )^2}{400 x^2}\right )\right ) \log (x)-\frac {75 \sqrt [3]{5} 6^{5/6} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right ) \arctan \left (\frac {\sqrt [3]{30} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right )-6 x}{\sqrt {6 \left (600+3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}+\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )}}\right )}{\left (300-3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}-\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right ) \sqrt {600+3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}+\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}}}+\frac {150 \sqrt [3]{15} \sqrt [6]{\frac {2}{9+\sqrt {60081}}} \left (10\ 2^{2/3} \sqrt [3]{15}-\left (9+\sqrt {60081}\right )^{2/3}\right ) \arctan \left (\frac {\sqrt [6]{\frac {2}{9+\sqrt {60081}}} \left (-3 \sqrt [3]{2 \left (9+\sqrt {60081}\right )} x-\sqrt [3]{15} \left (9+\sqrt {60081}\right )^{2/3}+10\ 30^{2/3}\right )}{3 \sqrt {\sqrt [3]{2} \sqrt [6]{3} 5^{2/3} \left (3 \sqrt {3}+\sqrt {20027}\right )+1000\ 2^{2/3} \sqrt [3]{\frac {15}{9+\sqrt {60081}}}+200 \sqrt [3]{9+\sqrt {60081}}}}\right )}{\sqrt {\sqrt [3]{2} \sqrt [6]{3} 5^{2/3} \left (3 \sqrt {3}+\sqrt {20027}\right )+1000\ 2^{2/3} \sqrt [3]{\frac {15}{9+\sqrt {60081}}}+200 \sqrt [3]{9+\sqrt {60081}}} \left (300-3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}-\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )}-\frac {\left (600-\sqrt [3]{2} \left (\frac {3000 \sqrt [3]{30}}{\left (9+\sqrt {60081}\right )^{2/3}}+\left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )\right ) \log \left (3 x+\sqrt [3]{30} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right )\right )}{2 \left (300-3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}-\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )}+\frac {150 \log \left (3 x+\sqrt [3]{30} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right )\right )}{300-3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}-\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}}-\frac {75 \log \left (9 x^2+3\ 2^{2/3} \sqrt [3]{15 \left (9+\sqrt {60081}\right )} x-60\ 15^{2/3} \sqrt [3]{\frac {2}{9+\sqrt {60081}}} x+2 \left (300+3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}+\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )\right )}{300-3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}-\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}}-\frac {\left (150-\sqrt [3]{2} \left (\frac {3000 \sqrt [3]{30}}{\left (9+\sqrt {60081}\right )^{2/3}}+\left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )\right ) \log \left (3 \sqrt [3]{9+\sqrt {60081}} x^2+\sqrt [3]{15} \left (2 \left (9+\sqrt {60081}\right )\right )^{2/3} x-20 \sqrt [3]{2} 15^{2/3} x+2 \left (\sqrt [3]{2} \sqrt [6]{3} 5^{2/3} \left (3 \sqrt {3}+\sqrt {20027}\right )+1000\ 2^{2/3} \sqrt [3]{\frac {15}{9+\sqrt {60081}}}+100 \sqrt [3]{9+\sqrt {60081}}\right )\right )}{2 \left (300-3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}-\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )}+200 \int \frac {\log (x)}{x^3+200 x-40}dx+3 \int \frac {x^2 \log (x)}{x^3+200 x-40}dx+100 \int \frac {\log \left (\frac {\left (x^3+200 x-40\right )^2}{400 x^2}\right )}{x^3+200 x-40}dx+\frac {3}{2} \int \frac {x^2 \log \left (\frac {\left (x^3+200 x-40\right )^2}{400 x^2}\right )}{x^3+200 x-40}dx\right )\) |
Input:
Int[(-160 - 8*x^3 + (160 + 8*x^3)*Log[(1600 - 16000*x + 40000*x^2 - 80*x^3 + 400*x^4 + x^6)/(400*x^2)])/(-40*x + 200*x^2 + x^4),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(19)=38\).
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29
| method | result | size |
| risch | \(-4 \ln \left (x^{3}+200 x -40\right )+4 \ln \left (x \right )+\ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{400 x^{2}}\right )^{2}\) | \(48\) |
| parts | \(-4 \ln \left (x^{3}+200 x -40\right )+4 \ln \left (x \right )+\ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{400 x^{2}}\right )^{2}\) | \(48\) |
| norman | \(\ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{400 x^{2}}\right )^{2}-2 \ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{400 x^{2}}\right )\) | \(64\) |
| default | \(-4 \ln \left (x^{3}+200 x -40\right )+4 \ln \left (x \right )-16 \ln \left (20\right ) \left (\frac {\ln \left (x^{3}+200 x -40\right )}{2}-\frac {\ln \left (x \right )}{2}\right )+\ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{x^{2}}\right )^{2}\) | \(67\) |
Input:
int(((8*x^3+160)*ln(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+1600)/x^2) -8*x^3-160)/(x^4+200*x^2-40*x),x,method=_RETURNVERBOSE)
Output:
-4*ln(x^3+200*x-40)+4*ln(x)+ln(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x +1600)/x^2)^2
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (19) = 38\).
Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.00 \[ \int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx=\log \left (\frac {x^{6} + 400 \, x^{4} - 80 \, x^{3} + 40000 \, x^{2} - 16000 \, x + 1600}{400 \, x^{2}}\right )^{2} - 2 \, \log \left (\frac {x^{6} + 400 \, x^{4} - 80 \, x^{3} + 40000 \, x^{2} - 16000 \, x + 1600}{400 \, x^{2}}\right ) \] Input:
integrate(((8*x^3+160)*log(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+160 0)/x^2)-8*x^3-160)/(x^4+200*x^2-40*x),x, algorithm="fricas")
Output:
log(1/400*(x^6 + 400*x^4 - 80*x^3 + 40000*x^2 - 16000*x + 1600)/x^2)^2 - 2 *log(1/400*(x^6 + 400*x^4 - 80*x^3 + 40000*x^2 - 16000*x + 1600)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (15) = 30\).
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx=4 \log {\left (x \right )} + \log {\left (\frac {\frac {x^{6}}{400} + x^{4} - \frac {x^{3}}{5} + 100 x^{2} - 40 x + 4}{x^{2}} \right )}^{2} - 4 \log {\left (x^{3} + 200 x - 40 \right )} \] Input:
integrate(((8*x**3+160)*ln(1/400*(x**6+400*x**4-80*x**3+40000*x**2-16000*x +1600)/x**2)-8*x**3-160)/(x**4+200*x**2-40*x),x)
Output:
4*log(x) + log((x**6/400 + x**4 - x**3/5 + 100*x**2 - 40*x + 4)/x**2)**2 - 4*log(x**3 + 200*x - 40)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (19) = 38\).
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.81 \[ \int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx=-4 \, {\left (2 \, \log \left (5\right ) + 4 \, \log \left (2\right ) + 2 \, \log \left (x\right ) + 1\right )} \log \left (x^{3} + 200 \, x - 40\right ) + 4 \, \log \left (x^{3} + 200 \, x - 40\right )^{2} + 4 \, {\left (2 \, \log \left (5\right ) + 4 \, \log \left (2\right ) + 1\right )} \log \left (x\right ) + 4 \, \log \left (x\right )^{2} \] Input:
integrate(((8*x^3+160)*log(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+160 0)/x^2)-8*x^3-160)/(x^4+200*x^2-40*x),x, algorithm="maxima")
Output:
-4*(2*log(5) + 4*log(2) + 2*log(x) + 1)*log(x^3 + 200*x - 40) + 4*log(x^3 + 200*x - 40)^2 + 4*(2*log(5) + 4*log(2) + 1)*log(x) + 4*log(x)^2
\[ \int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx=\int { -\frac {8 \, {\left (x^{3} - {\left (x^{3} + 20\right )} \log \left (\frac {x^{6} + 400 \, x^{4} - 80 \, x^{3} + 40000 \, x^{2} - 16000 \, x + 1600}{400 \, x^{2}}\right ) + 20\right )}}{x^{4} + 200 \, x^{2} - 40 \, x} \,d x } \] Input:
integrate(((8*x^3+160)*log(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+160 0)/x^2)-8*x^3-160)/(x^4+200*x^2-40*x),x, algorithm="giac")
Output:
integrate(-8*(x^3 - (x^3 + 20)*log(1/400*(x^6 + 400*x^4 - 80*x^3 + 40000*x ^2 - 16000*x + 1600)/x^2) + 20)/(x^4 + 200*x^2 - 40*x), x)
Time = 1.70 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx={\ln \left (\frac {\frac {x^6}{400}+x^4-\frac {x^3}{5}+100\,x^2-40\,x+4}{x^2}\right )}^2-4\,\ln \left (x^3+200\,x-40\right )+4\,\ln \left (x\right ) \] Input:
int(-(8*x^3 - log((100*x^2 - 40*x - x^3/5 + x^4 + x^6/400 + 4)/x^2)*(8*x^3 + 160) + 160)/(200*x^2 - 40*x + x^4),x)
Output:
4*log(x) - 4*log(200*x + x^3 - 40) + log((100*x^2 - 40*x - x^3/5 + x^4 + x ^6/400 + 4)/x^2)^2
Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24 \[ \int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx=-4 \,\mathrm {log}\left (x^{3}+200 x -40\right )+\mathrm {log}\left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{400 x^{2}}\right )^{2}+4 \,\mathrm {log}\left (x \right ) \] Input:
int(((8*x^3+160)*log(1/400*(x^6+400*x^4-80*x^3+40000*x^2-16000*x+1600)/x^2 )-8*x^3-160)/(x^4+200*x^2-40*x),x)
Output:
- 4*log(x**3 + 200*x - 40) + log((x**6 + 400*x**4 - 80*x**3 + 40000*x**2 - 16000*x + 1600)/(400*x**2))**2 + 4*log(x)