Optimal. Leaf size=22 \[ -1+\frac {1}{2 (2-x) x^2}+\log \left (\frac {x}{5}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps
used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1608, 27, 12,
1634} \begin {gather*} \frac {1}{4 x^2}+\frac {1}{8 (2-x)}+\frac {1}{8 x}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 1608
Rule 1634
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4+3 x+8 x^2-8 x^3+2 x^4}{x^3 \left (8-8 x+2 x^2\right )} \, dx\\ &=\int \frac {-4+3 x+8 x^2-8 x^3+2 x^4}{2 (-2+x)^2 x^3} \, dx\\ &=\frac {1}{2} \int \frac {-4+3 x+8 x^2-8 x^3+2 x^4}{(-2+x)^2 x^3} \, dx\\ &=\frac {1}{2} \int \left (\frac {1}{4 (-2+x)^2}-\frac {1}{x^3}-\frac {1}{4 x^2}+\frac {2}{x}\right ) \, dx\\ &=\frac {1}{8 (2-x)}+\frac {1}{4 x^2}+\frac {1}{8 x}+\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.86 \begin {gather*} \frac {1}{2} \left (-\frac {1}{(-2+x) x^2}+2 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 21, normalized size = 0.95
method | result | size |
norman | \(-\frac {1}{2 x^{2} \left (x -2\right )}+\ln \left (x \right )\) | \(14\) |
risch | \(-\frac {1}{2 x^{2} \left (x -2\right )}+\ln \left (x \right )\) | \(14\) |
default | \(-\frac {1}{8 \left (x -2\right )}+\frac {1}{4 x^{2}}+\frac {1}{8 x}+\ln \left (x \right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 16, normalized size = 0.73 \begin {gather*} -\frac {1}{2 \, {\left (x^{3} - 2 \, x^{2}\right )}} + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 28, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (x\right ) - 1}{2 \, {\left (x^{3} - 2 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 14, normalized size = 0.64 \begin {gather*} \log {\left (x \right )} - \frac {1}{2 x^{3} - 4 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 14, normalized size = 0.64 \begin {gather*} -\frac {1}{2 \, {\left (x - 2\right )} x^{2}} + \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 = 0.05, size = 13, normalized size = 0.59 \begin {gather*} \ln \left (x\right )-\frac {1}{2\,x^2\,\left (x-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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