Optimal. Leaf size=26 \[ 4-\left (3-\frac {1}{x}\right ) \left (-5+x+\log ^2\left (x+\frac {x}{(-1+x)^4}\right )\right ) \]
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Rubi [F]
time = 5.77, 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 {-10+30 x-44 x^2+32 x^3+5 x^4-25 x^5+15 x^6-3 x^7+\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )+\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{-2 x^2+6 x^3-10 x^4+10 x^5-5 x^6+x^7} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10-30 x+44 x^2-32 x^3-5 x^4+25 x^5-15 x^6+3 x^7-\left (-4+16 x-32 x^2+80 x^3-70 x^4+32 x^5-6 x^6\right ) \log \left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )-\left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log ^2\left (\frac {2 x-4 x^2+6 x^3-4 x^4+x^5}{1-4 x+6 x^2-4 x^3+x^4}\right )}{x^2 \left (2-6 x+10 x^2-10 x^3+5 x^4-x^5\right )} \, dx\\ &=\int \left (-\frac {44}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {10}{(-1+x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )}+\frac {30}{(-1+x) x \left (2-4 x+6 x^2-4 x^3+x^4\right )}+\frac {32 x}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}+\frac {5 x^2}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {25 x^3}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}+\frac {15 x^4}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {3 x^5}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {2 (-1+3 x) \left (-2+2 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{(-1+x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )}-\frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {(-1+3 x) \left (-2+2 x-10 x^2+10 x^3-5 x^4+x^5\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{(-1+x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx\right )-3 \int \frac {x^5}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+5 \int \frac {x^2}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-10 \int \frac {1}{(-1+x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+15 \int \frac {x^4}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-25 \int \frac {x^3}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+30 \int \frac {1}{(-1+x) x \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx+32 \int \frac {x}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-44 \int \frac {1}{(-1+x) \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-\int \frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2} \, dx\\ &=\frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )}{x}-2 \int \frac {\left (2-2 x+10 x^2-10 x^3+5 x^4-x^5\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{(1-x) x^2 \left (2-4 x+6 x^2-4 x^3+x^4\right )} \, dx-2 \int \left (-\frac {8 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{-1+x}-\frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2}+\frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x}+\frac {2 \left (-8+18 x-16 x^2+5 x^3\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-3 \int \left (1+\frac {1}{-1+x}+\frac {2 x \left (1-x+2 x^2\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx+5 \int \left (\frac {1}{-1+x}+\frac {2-2 x+3 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-10 \int \left (\frac {1}{-1+x}-\frac {1}{2 x^2}-\frac {3}{2 x}+\frac {(-2+x)^2 (-1+x)}{2 \left (2-4 x+6 x^2-4 x^3+x^4\right )}\right ) \, dx+15 \int \left (\frac {1}{-1+x}+\frac {2 \left (1-x+2 x^2\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-25 \int \left (\frac {1}{-1+x}+\frac {2-2 x+4 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx+30 \int \left (\frac {1}{-1+x}-\frac {1}{2 x}+\frac {-2+2 x^2-x^3}{2 \left (2-4 x+6 x^2-4 x^3+x^4\right )}\right ) \, dx+32 \int \left (\frac {1}{-1+x}+\frac {2-3 x+3 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-44 \int \left (\frac {1}{-1+x}-\frac {(-1+x)^3}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx\\ &=-\frac {5}{x}-3 x+\frac {\log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(1-x)^4}\right )}{x}+2 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2} \, dx-2 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x} \, dx-2 \int \left (-\frac {4 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{-1+x}+\frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x^2}+\frac {2 \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x}+\frac {2 \left (2-2 x^2+x^3\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4}\right ) \, dx-4 \int \frac {\left (-8+18 x-16 x^2+5 x^3\right ) \log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx-5 \int \frac {(-2+x)^2 (-1+x)}{2-4 x+6 x^2-4 x^3+x^4} \, dx+5 \int \frac {2-2 x+3 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx-6 \int \frac {x \left (1-x+2 x^2\right )}{2-4 x+6 x^2-4 x^3+x^4} \, dx+15 \int \frac {-2+2 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx+16 \int \frac {\log \left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{-1+x} \, dx-25 \int \frac {2-2 x+4 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx+30 \int \frac {1-x+2 x^2}{2-4 x+6 x^2-4 x^3+x^4} \, dx+32 \int \frac {2-3 x+3 x^2-x^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx+44 \int \frac {(-1+x)^3}{2-4 x+6 x^2-4 x^3+x^4} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 47, normalized size = 1.81 \begin {gather*} -\frac {5}{x}-3 x-\frac {(-1+3 x) \log ^2\left (\frac {x \left (2-4 x+6 x^2-4 x^3+x^4\right )}{(-1+x)^4}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(106\) vs.
\(2(26)=52\).
time = 0.19, size = 107, normalized size = 4.12
method | result | size |
norman | \(\frac {-5+\ln \left (\frac {x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+2 x}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}\right )^{2}-3 x^{2}-3 x \ln \left (\frac {x^{5}-4 x^{4}+6 x^{3}-4 x^{2}+2 x}{x^{4}-4 x^{3}+6 x^{2}-4 x +1}\right )^{2}}{x}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs.
\(2 (23) = 46\).
time = 0.49, size = 117, normalized size = 4.50 \begin {gather*} -\frac {{\left (3 \, x - 1\right )} \log \left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 2\right )^{2} + 16 \, {\left (3 \, x - 1\right )} \log \left (x - 1\right )^{2} - 8 \, {\left (3 \, x - 1\right )} \log \left (x - 1\right ) \log \left (x\right ) + {\left (3 \, x - 1\right )} \log \left (x\right )^{2} + 3 \, x^{2} - 2 \, {\left (4 \, {\left (3 \, x - 1\right )} \log \left (x - 1\right ) - {\left (3 \, x - 1\right )} \log \left (x\right )\right )} \log \left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 2\right ) + 5}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs.
\(2 (23) = 46\).
time = 0.39, size = 64, normalized size = 2.46 \begin {gather*} -\frac {{\left (3 \, x - 1\right )} \log \left (\frac {x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + 2 \, x}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right )^{2} + 3 \, x^{2} + 5}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (20) = 40\).
time = 0.16, size = 56, normalized size = 2.15 \begin {gather*} - 3 x + \frac {\left (1 - 3 x\right ) \log {\left (\frac {x^{5} - 4 x^{4} + 6 x^{3} - 4 x^{2} + 2 x}{x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1} \right )}^{2}}{x} - \frac {5}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.59, size = 61, normalized size = 2.35 \begin {gather*} {\ln \left (\frac {x^5-4\,x^4+6\,x^3-4\,x^2+2\,x}{x^4-4\,x^3+6\,x^2-4\,x+1}\right )}^2\,\left (\frac {1}{x}-3\right )-3\,x-\frac {5}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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