Optimal. Leaf size=65 \[ \frac {3}{2 x^2}-\frac {1}{x}+3 \log (x)-\frac {1}{10} \left (15-\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right )-\frac {1}{10} \left (15+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1642, 646, 31}
\begin {gather*} \frac {3}{2 x^2}-\frac {1}{x}+3 \log (x)-\frac {1}{10} \left (15-\sqrt {5}\right ) \log \left (2 x-\sqrt {5}+1\right )-\frac {1}{10} \left (15+\sqrt {5}\right ) \log \left (2 x+\sqrt {5}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 1642
Rubi steps
\begin {align*} \int \frac {3-4 x-5 x^2+3 x^3}{x^3 \left (-1+x+x^2\right )} \, dx &=\int \left (-\frac {3}{x^3}+\frac {1}{x^2}+\frac {3}{x}+\frac {-1-3 x}{-1+x+x^2}\right ) \, dx\\ &=\frac {3}{2 x^2}-\frac {1}{x}+3 \log (x)+\int \frac {-1-3 x}{-1+x+x^2} \, dx\\ &=\frac {3}{2 x^2}-\frac {1}{x}+3 \log (x)+\frac {1}{10} \left (-15+\sqrt {5}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx-\frac {1}{10} \left (15+\sqrt {5}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx\\ &=\frac {3}{2 x^2}-\frac {1}{x}+3 \log (x)-\frac {1}{10} \left (15-\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 x\right )-\frac {1}{10} \left (15+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 58, normalized size = 0.89 \begin {gather*} \frac {1}{10} \left (\frac {15}{x^2}-\frac {10}{x}+\left (-15+\sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 x\right )+30 \log (x)-\left (15+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 41, normalized size = 0.63
method | result | size |
default | \(-\frac {3 \ln \left (x^{2}+x -1\right )}{2}-\frac {\arctanh \left (\frac {\left (2 x +1\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{5}-\frac {1}{x}+\frac {3}{2 x^{2}}+3 \ln \left (x \right )\) | \(41\) |
risch | \(\frac {-x +\frac {3}{2}}{x^{2}}+3 \ln \left (x \right )-\frac {3 \ln \left (x -\frac {\sqrt {5}}{2}+\frac {1}{2}\right )}{2}+\frac {\ln \left (x -\frac {\sqrt {5}}{2}+\frac {1}{2}\right ) \sqrt {5}}{10}-\frac {3 \ln \left (x +\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{2}-\frac {\ln \left (x +\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}{10}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 51, normalized size = 0.78 \begin {gather*} \frac {1}{10} \, \sqrt {5} \log \left (\frac {2 \, x - \sqrt {5} + 1}{2 \, x + \sqrt {5} + 1}\right ) - \frac {2 \, x - 3}{2 \, x^{2}} - \frac {3}{2} \, \log \left (x^{2} + x - 1\right ) + 3 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 66, normalized size = 1.02 \begin {gather*} \frac {\sqrt {5} x^{2} \log \left (\frac {2 \, x^{2} - \sqrt {5} {\left (2 \, x + 1\right )} + 2 \, x + 3}{x^{2} + x - 1}\right ) - 15 \, x^{2} \log \left (x^{2} + x - 1\right ) + 30 \, x^{2} \log \left (x\right ) - 10 \, x + 15}{10 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 99, normalized size = 1.52 \begin {gather*} 3 \log {\left (x \right )} + \left (- \frac {3}{2} + \frac {\sqrt {5}}{10}\right ) \log {\left (x - \frac {405}{202} - \frac {35 \sqrt {5}}{202} + \frac {110 \left (- \frac {3}{2} + \frac {\sqrt {5}}{10}\right )^{2}}{101} \right )} + \left (- \frac {3}{2} - \frac {\sqrt {5}}{10}\right ) \log {\left (x - \frac {405}{202} + \frac {35 \sqrt {5}}{202} + \frac {110 \left (- \frac {3}{2} - \frac {\sqrt {5}}{10}\right )^{2}}{101} \right )} + \frac {3 - 2 x}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.63, size = 55, normalized size = 0.85 \begin {gather*} \frac {1}{10} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x - \sqrt {5} + 1 \right |}}{{\left | 2 \, x + \sqrt {5} + 1 \right |}}\right ) - \frac {2 \, x - 3}{2 \, x^{2}} - \frac {3}{2} \, \log \left ({\left | x^{2} + x - 1 \right |}\right ) + 3 \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 48, normalized size = 0.74 \begin {gather*} 3\,\ln \left (x\right )-\frac {x-\frac {3}{2}}{x^2}+\ln \left (x-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{10}-\frac {3}{2}\right )-\ln \left (x+\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{10}+\frac {3}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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