Optimal. Leaf size=31 \[ \frac {1}{3} \tanh ^{-1}(x)+\frac {1}{3} \tanh ^{-1}(2 x)-\frac {\tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2082, 213}
\begin {gather*} \frac {1}{3} \tanh ^{-1}(x)+\frac {1}{3} \tanh ^{-1}(2 x)-\frac {\tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 2082
Rubi steps
\begin {align*} \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx &=\int \left (-\frac {1}{3 \left (-1+x^2\right )}+\frac {2}{-3+4 x^2}-\frac {2}{3 \left (-1+4 x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {1}{-1+x^2} \, dx\right )-\frac {2}{3} \int \frac {1}{-1+4 x^2} \, dx+2 \int \frac {1}{-3+4 x^2} \, dx\\ &=\frac {1}{3} \tanh ^{-1}(x)+\frac {1}{3} \tanh ^{-1}(2 x)-\frac {\tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 62, normalized size = 2.00 \begin {gather*} \frac {1}{6} \left (\sqrt {3} \log \left (\sqrt {3}-2 x\right )-\sqrt {3} \log \left (\sqrt {3}+2 x\right )-\log \left (1-3 x+2 x^2\right )+\log \left (1+3 x+2 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 42, normalized size = 1.35
method | result | size |
default | \(-\frac {\arctanh \left (\frac {2 x \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\ln \left (2 x +1\right )}{6}-\frac {\ln \left (-1+x \right )}{6}-\frac {\ln \left (2 x -1\right )}{6}+\frac {\ln \left (1+x \right )}{6}\) | \(42\) |
risch | \(\frac {\sqrt {3}\, \ln \left (2 x -\sqrt {3}\right )}{6}-\frac {\sqrt {3}\, \ln \left (2 x +\sqrt {3}\right )}{6}+\frac {\ln \left (2 x^{2}+3 x +1\right )}{6}-\frac {\ln \left (2 x^{2}-3 x +1\right )}{6}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (23) = 46\).
time = 0.48, size = 54, normalized size = 1.74 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3}}{2 \, x + \sqrt {3}}\right ) + \frac {1}{6} \, \log \left (2 \, x + 1\right ) - \frac {1}{6} \, \log \left (2 \, x - 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{6} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (23) = 46\).
time = 0.39, size = 56, normalized size = 1.81 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {4 \, x^{2} - 4 \, \sqrt {3} x + 3}{4 \, x^{2} - 3}\right ) + \frac {1}{6} \, \log \left (2 \, x^{2} + 3 \, x + 1\right ) - \frac {1}{6} \, \log \left (2 \, x^{2} - 3 \, x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (29) = 58\).
time = 0.05, size = 63, normalized size = 2.03 \begin {gather*} \frac {\sqrt {3} \log {\left (x - \frac {\sqrt {3}}{2} \right )}}{6} - \frac {\sqrt {3} \log {\left (x + \frac {\sqrt {3}}{2} \right )}}{6} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {1}{2} \right )}}{6} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {1}{2} \right )}}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs.
\(2 (23) = 46\).
time = 4.91, size = 62, normalized size = 2.00 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 8 \, x - 4 \, \sqrt {3} \right |}}{{\left | 8 \, x + 4 \, \sqrt {3} \right |}}\right ) + \frac {1}{6} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 27, normalized size = 0.87 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {x}{4608\,\left (\frac {x^2}{6912}+\frac {1}{13824}\right )}\right )}{3}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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