Optimal. Leaf size=21 \[ -\left (-\frac {2}{x^2}+\log (3)\right )^2+\log \left (\frac {2+x}{3}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 17, normalized size of antiderivative = 0.81, number of steps
used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1607, 1634}
\begin {gather*} -\frac {4}{x^4}+\frac {4 \log (3)}{x^2}+\log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 1607
Rule 1634
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {32+16 x+x^5+\left (-16 x^2-8 x^3\right ) \log (3)}{x^5 (2+x)} \, dx\\ &=\int \left (\frac {16}{x^5}+\frac {1}{2+x}-\frac {8 \log (3)}{x^3}\right ) \, dx\\ &=-\frac {4}{x^4}+\frac {4 \log (3)}{x^2}+\log (2+x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 0.81 \begin {gather*} -\frac {4}{x^4}+\frac {4 \log (3)}{x^2}+\log (2+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 18, normalized size = 0.86
method | result | size |
default | \(\ln \left (2+x \right )-\frac {4}{x^{4}}+\frac {4 \ln \left (3\right )}{x^{2}}\) | \(18\) |
norman | \(\frac {-4+4 x^{2} \ln \left (3\right )}{x^{4}}+\ln \left (2+x \right )\) | \(19\) |
risch | \(\frac {-4+4 x^{2} \ln \left (3\right )}{x^{4}}+\ln \left (2+x \right )\) | \(19\) |
meijerg | \(-\frac {4}{x^{4}}-2 \ln \left (3\right ) \left (\ln \left (1+\frac {x}{2}\right )-\ln \left (x \right )+\ln \left (2\right )-\frac {2}{x}\right )-2 \ln \left (3\right ) \left (-\ln \left (1+\frac {x}{2}\right )+\ln \left (x \right )-\ln \left (2\right )-\frac {2}{x^{2}}+\frac {2}{x}\right )+\ln \left (1+\frac {x}{2}\right )\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 18, normalized size = 0.86 \begin {gather*} \frac {4 \, {\left (x^{2} \log \left (3\right ) - 1\right )}}{x^{4}} + \log \left (x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 21, normalized size = 1.00 \begin {gather*} \frac {x^{4} \log \left (x + 2\right ) + 4 \, x^{2} \log \left (3\right ) - 4}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 17, normalized size = 0.81 \begin {gather*} \log {\left (x + 2 \right )} + \frac {4 x^{2} \log {\left (3 \right )} - 4}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 19, normalized size = 0.90 \begin {gather*} \frac {4 \, {\left (x^{2} \log \left (3\right ) - 1\right )}}{x^{4}} + \log \left ({\left | x + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 18, normalized size = 0.86 \begin {gather*} \ln \left (x+2\right )+\frac {4\,x^2\,\ln \left (3\right )-4}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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