Optimal. Leaf size=24 \[ -\frac {48 x}{5}+\frac {3 (3-x)}{x \left (-1+\log \left (x^2\right )\right )} \]
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Rubi [A]
time = 0.49, antiderivative size = 33, normalized size of antiderivative = 1.38, number of steps
used = 13, number of rules used = 9, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.145, Rules used = {6820, 12,
6874, 2395, 2343, 2347, 2209, 2339, 30} \begin {gather*} \frac {3}{1-\log \left (x^2\right )}-\frac {9}{x \left (1-\log \left (x^2\right )\right )}-\frac {48 x}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2209
Rule 2339
Rule 2343
Rule 2347
Rule 2395
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-45+30 x-48 x^2+\left (-45+96 x^2\right ) \log \left (x^2\right )-48 x^2 \log ^2\left (x^2\right )}{5 x^2 \left (1-\log \left (x^2\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \frac {-45+30 x-48 x^2+\left (-45+96 x^2\right ) \log \left (x^2\right )-48 x^2 \log ^2\left (x^2\right )}{x^2 \left (1-\log \left (x^2\right )\right )^2} \, dx\\ &=\frac {1}{5} \int \left (-48+\frac {30 (-3+x)}{x^2 \left (-1+\log \left (x^2\right )\right )^2}-\frac {45}{x^2 \left (-1+\log \left (x^2\right )\right )}\right ) \, dx\\ &=-\frac {48 x}{5}+6 \int \frac {-3+x}{x^2 \left (-1+\log \left (x^2\right )\right )^2} \, dx-9 \int \frac {1}{x^2 \left (-1+\log \left (x^2\right )\right )} \, dx\\ &=-\frac {48 x}{5}+6 \int \left (-\frac {3}{x^2 \left (-1+\log \left (x^2\right )\right )^2}+\frac {1}{x \left (-1+\log \left (x^2\right )\right )^2}\right ) \, dx-\frac {\left (9 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {e^{-x/2}}{-1+x} \, dx,x,\log \left (x^2\right )\right )}{2 x}\\ &=-\frac {48 x}{5}-\frac {9 \sqrt {x^2} \text {Ei}\left (\frac {1}{2} \left (1-\log \left (x^2\right )\right )\right )}{2 \sqrt {e} x}+6 \int \frac {1}{x \left (-1+\log \left (x^2\right )\right )^2} \, dx-18 \int \frac {1}{x^2 \left (-1+\log \left (x^2\right )\right )^2} \, dx\\ &=-\frac {48 x}{5}-\frac {9 \sqrt {x^2} \text {Ei}\left (\frac {1}{2} \left (1-\log \left (x^2\right )\right )\right )}{2 \sqrt {e} x}-\frac {9}{x \left (1-\log \left (x^2\right )\right )}+3 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,-1+\log \left (x^2\right )\right )+9 \int \frac {1}{x^2 \left (-1+\log \left (x^2\right )\right )} \, dx\\ &=-\frac {48 x}{5}-\frac {9 \sqrt {x^2} \text {Ei}\left (\frac {1}{2} \left (1-\log \left (x^2\right )\right )\right )}{2 \sqrt {e} x}+\frac {3}{1-\log \left (x^2\right )}-\frac {9}{x \left (1-\log \left (x^2\right )\right )}+\frac {\left (9 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {e^{-x/2}}{-1+x} \, dx,x,\log \left (x^2\right )\right )}{2 x}\\ &=-\frac {48 x}{5}+\frac {3}{1-\log \left (x^2\right )}-\frac {9}{x \left (1-\log \left (x^2\right )\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{5} \left (-48 x-\frac {15 (-3+x)}{x \left (-1+\log \left (x^2\right )\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.47, size = 21, normalized size = 0.88
method | result | size |
risch | \(-\frac {48 x}{5}-\frac {3 \left (x -3\right )}{x \left (\ln \left (x^{2}\right )-1\right )}\) | \(21\) |
norman | \(\frac {9-3 x +\frac {48 x^{2}}{5}-\frac {48 x^{2} \ln \left (x^{2}\right )}{5}}{x \left (\ln \left (x^{2}\right )-1\right )}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 30, normalized size = 1.25 \begin {gather*} -\frac {3 \, {\left (32 \, x^{2} \log \left (x\right ) - 16 \, x^{2} + 5 \, x - 15\right )}}{5 \, {\left (2 \, x \log \left (x\right ) - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 33, normalized size = 1.38 \begin {gather*} -\frac {3 \, {\left (16 \, x^{2} \log \left (x^{2}\right ) - 16 \, x^{2} + 5 \, x - 15\right )}}{5 \, {\left (x \log \left (x^{2}\right ) - x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 17, normalized size = 0.71 \begin {gather*} - \frac {48 x}{5} + \frac {9 - 3 x}{x \log {\left (x^{2} \right )} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 21, normalized size = 0.88 \begin {gather*} -\frac {48}{5} \, x - \frac {3 \, {\left (x - 3\right )}}{x \log \left (x^{2}\right ) - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.07, size = 33, normalized size = 1.38 \begin {gather*} -\frac {\frac {48\,x^2}{5}+3\,x}{x}-\frac {3\,x-9}{x\,\left (\ln \left (x^2\right )-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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