Optimal. Leaf size=22 \[ \frac {x^2}{\log \left ((4+x)^2 \left (-\frac {1}{x^2}+x\right )^2\right )} \]
________________________________________________________________________________________
Rubi [F] time = 2.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 26, normalized size = 1.18 \begin {gather*} \frac {x^2}{\log \left (\frac {\left (-4-x+4 x^3+x^4\right )^2}{x^4}\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.62, size = 47, normalized size = 2.14 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.43, size = 47, normalized size = 2.14 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.08, size = 48, normalized size = 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 | \(\frac {x^{2}}{\ln \left (\frac {x^{8}+8 x^{7}+16 x^{6}-2 x^{5}-16 x^{4}-32 x^{3}+x^{2}+8 x +16}{x^{4}}\right )}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 27, normalized size = 1.23 \begin {gather*} \frac {x^{2}}{2 \, {\left (\log \left (x^{2} + x + 1\right ) + \log \left (x + 4\right ) + \log \left (x - 1\right ) - 2 \, \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.47, size = 160, normalized size = 7.27 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.20, size = 44, normalized size = 2.00 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________