Integrand size = 139, antiderivative size = 22 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^2}{\log \left ((4+x)^2 \left (-\frac {1}{x^2}+x\right )^2\right )} \]
[Out]
\[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x \left (8+x+4 x^3+2 x^4-\left (-4-x+4 x^3+x^4\right ) \log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )\right )}{\left (4+x-4 x^3-x^4\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx \\ & = 2 \int \frac {x \left (8+x+4 x^3+2 x^4-\left (-4-x+4 x^3+x^4\right ) \log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )\right )}{\left (4+x-4 x^3-x^4\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx \\ & = 2 \int \left (-\frac {x \left (8+x+4 x^3+2 x^4\right )}{(-1+x) (4+x) \left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx \\ & = -\left (2 \int \frac {x \left (8+x+4 x^3+2 x^4\right )}{(-1+x) (4+x) \left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\right )+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx \\ & = -\left (2 \int \left (-\frac {4}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {2 x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {16}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {1-x}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx\right )+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx \\ & = -\left (2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\right )-2 \int \frac {1-x}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx \\ & = -\left (2 \int \left (\frac {1}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}-\frac {x}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx\right )-2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx \\ & = -\left (2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\right )-2 \int \frac {1}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\left (1+x+x^2\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx \\ & = -\left (2 \int \left (\frac {2 i}{\sqrt {3} \left (-1+i \sqrt {3}-2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {2 i}{\sqrt {3} \left (1+i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx\right )+2 \int \left (\frac {1+\frac {i}{\sqrt {3}}}{\left (1-i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}+\frac {1-\frac {i}{\sqrt {3}}}{\left (1+i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )}\right ) \, dx-2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx \\ & = -\left (2 \int \frac {1}{(-1+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx\right )+2 \int \frac {x}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-4 \int \frac {x}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+8 \int \frac {1}{\log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-32 \int \frac {1}{(4+x) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx-\frac {(4 i) \int \frac {1}{\left (-1+i \sqrt {3}-2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx}{\sqrt {3}}-\frac {(4 i) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx}{\sqrt {3}}+\frac {1}{3} \left (2 \left (3-i \sqrt {3}\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx+\frac {1}{3} \left (2 \left (3+i \sqrt {3}\right )\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \log ^2\left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \, dx \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^2}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).
Time = 2.85 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18
method | result | size |
norman | \(\frac {x^{2}}{\ln \left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )}\) | \(48\) |
risch | \(\frac {x^{2}}{\ln \left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )}\) | \(48\) |
parallelrisch | \(\frac {x^{2}}{\ln \left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )}\) | \(48\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^{2}}{\log \left (\frac {x^{8} + 8 \, x^{7} + 16 \, x^{6} - 2 \, x^{5} - 16 \, x^{4} - 32 \, x^{3} + x^{2} + 8 \, x + 16}{x^{4}}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).
Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^{2}}{\log {\left (\frac {x^{8} + 8 x^{7} + 16 x^{6} - 2 x^{5} - 16 x^{4} - 32 x^{3} + x^{2} + 8 x + 16}{x^{4}} \right )}} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^{2}}{2 \, {\left (\log \left (x^{2} + x + 1\right ) + \log \left (x + 4\right ) + \log \left (x - 1\right ) - 2 \, \log \left (x\right )\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.54 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=\frac {x^{2}}{\log \left (\frac {x^{8} + 8 \, x^{7} + 16 \, x^{6} - 2 \, x^{5} - 16 \, x^{4} - 32 \, x^{3} + x^{2} + 8 \, x + 16}{x^{4}}\right )} \]
[In]
[Out]
Time = 18.00 (sec) , antiderivative size = 160, normalized size of antiderivative = 7.27 \[ \int \frac {-16 x-2 x^2-8 x^4-4 x^5+\left (-8 x-2 x^2+8 x^4+2 x^5\right ) \log \left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )}{\left (-4-x+4 x^3+x^4\right ) \log ^2\left (\frac {16+8 x+x^2-32 x^3-16 x^4-2 x^5+16 x^6+8 x^7+x^8}{x^4}\right )} \, dx=x+\frac {x^2+\frac {x^2\,\ln \left (\frac {x^8+8\,x^7+16\,x^6-2\,x^5-16\,x^4-32\,x^3+x^2+8\,x+16}{x^4}\right )\,\left (-x^4-4\,x^3+x+4\right )}{2\,x^4+4\,x^3+x+8}}{\ln \left (\frac {x^8+8\,x^7+16\,x^6-2\,x^5-16\,x^4-32\,x^3+x^2+8\,x+16}{x^4}\right )}-\frac {-\frac {13\,x^3}{4}+\frac {9\,x^2}{2}+3\,x-8}{x^4+2\,x^3+\frac {x}{2}+4}+\frac {x^2}{2} \]
[In]
[Out]