Optimal. Leaf size=32 \[ \frac {x}{5 \left (3 x-\log \left (2 x+\frac {2 x^2}{-\frac {5}{2}+x^2}\right )\right )} \]
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Rubi [A]
time = 0.30, antiderivative size = 36, normalized size of antiderivative = 1.12, number of steps
used = 4, number of rules used = 4, integrand size = 188, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6820, 12,
6843, 32} \begin {gather*} -\frac {1}{15 \left (1-\frac {3 x}{\log \left (\frac {2 x \left (-2 x^2-2 x+5\right )}{5-2 x^2}\right )}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 6820
Rule 6843
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )}{5 \left (25-10 x-20 x^2+4 x^3+4 x^4\right ) \left (3 x-\log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )}{\left (25-10 x-20 x^2+4 x^3+4 x^4\right ) \left (3 x-\log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )\right )^2} \, dx\\ &=-\left (\frac {1}{5} \text {Subst}\left (\int \frac {1}{(-1+3 x)^2} \, dx,x,\frac {x}{\log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )}\right )\right )\\ &=-\frac {1}{15 \left (1-\frac {3 x}{\log \left (\frac {2 x \left (5-2 x-2 x^2\right )}{5-2 x^2}\right )}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 34, normalized size = 1.06 \begin {gather*} -\frac {x}{5 \left (-3 x+\log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 37, normalized size = 1.16
method | result | size |
norman | \(\frac {x}{15 x -5 \ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}\) | \(37\) |
risch | \(\frac {x}{15 x -5 \ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 38, normalized size = 1.19 \begin {gather*} \frac {x}{5 \, {\left (3 \, x - \log \left (2\right ) - \log \left (2 \, x^{2} + 2 \, x - 5\right ) + \log \left (2 \, x^{2} - 5\right ) - \log \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 37, normalized size = 1.16 \begin {gather*} \frac {x}{5 \, {\left (3 \, x - \log \left (\frac {2 \, {\left (2 \, x^{3} + 2 \, x^{2} - 5 \, x\right )}}{2 \, x^{2} - 5}\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 29, normalized size = 0.91 \begin {gather*} - \frac {x}{- 15 x + 5 \log {\left (\frac {4 x^{3} + 4 x^{2} - 10 x}{2 x^{2} - 5} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 37, normalized size = 1.16 \begin {gather*} \frac {x}{5 \, {\left (3 \, x - \log \left (\frac {2 \, {\left (2 \, x^{3} + 2 \, x^{2} - 5 \, x\right )}}{2 \, x^{2} - 5}\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.33, size = 35, normalized size = 1.09 \begin {gather*} \frac {x}{5\,\left (3\,x-\ln \left (\frac {4\,x^3+4\,x^2-10\,x}{2\,x^2-5}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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