Optimal. Leaf size=43 \[ \frac {3 x \sqrt {-1+x^2}}{8 \left (4-3 x^2\right )}+\frac {5}{16} \tanh ^{-1}\left (\frac {x}{2 \sqrt {-1+x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {390, 385, 213}
\begin {gather*} \frac {3 \sqrt {x^2-1} x}{8 \left (4-3 x^2\right )}+\frac {5}{16} \tanh ^{-1}\left (\frac {x}{2 \sqrt {x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 385
Rule 390
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x^2} \left (-4+3 x^2\right )^2} \, dx &=\frac {3 x \sqrt {-1+x^2}}{8 \left (4-3 x^2\right )}-\frac {5}{8} \int \frac {1}{\sqrt {-1+x^2} \left (-4+3 x^2\right )} \, dx\\ &=\frac {3 x \sqrt {-1+x^2}}{8 \left (4-3 x^2\right )}-\frac {5}{8} \text {Subst}\left (\int \frac {1}{-4+x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )\\ &=\frac {3 x \sqrt {-1+x^2}}{8 \left (4-3 x^2\right )}+\frac {5}{16} \tanh ^{-1}\left (\frac {x}{2 \sqrt {-1+x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 71, normalized size = 1.65 \begin {gather*} -\frac {3 x \sqrt {-1+x^2}}{8 \left (-4+3 x^2\right )}+\frac {5}{32} \log \left (2-x^2+x \sqrt {-1+x^2}\right )-\frac {5}{32} \log \left (2-3 x^2+3 x \sqrt {-1+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(171\) vs.
\(2(33)=66\).
time = 0.18, size = 172, normalized size = 4.00
method | result | size |
trager | \(-\frac {3 x \sqrt {x^{2}-1}}{8 \left (3 x^{2}-4\right )}-\frac {5 \ln \left (-\frac {4 \sqrt {x^{2}-1}\, x -5 x^{2}+4}{3 x^{2}-4}\right )}{32}\) | \(52\) |
risch | \(-\frac {3 x \sqrt {x^{2}-1}}{8 \left (3 x^{2}-4\right )}+\frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}+\frac {4 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x -\frac {2 \sqrt {3}}{3}\right )^{2}+12 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}-\frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}-\frac {4 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x +\frac {2 \sqrt {3}}{3}\right )^{2}-12 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}\) | \(119\) |
default | \(-\frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}-\frac {4 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x +\frac {2 \sqrt {3}}{3}\right )^{2}-12 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}-\frac {\sqrt {\left (x -\frac {2 \sqrt {3}}{3}\right )^{2}+\frac {4 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )}{3}+\frac {1}{3}}}{16 \left (x -\frac {2 \sqrt {3}}{3}\right )}+\frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}+\frac {4 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x -\frac {2 \sqrt {3}}{3}\right )^{2}+12 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}-\frac {\sqrt {\left (x +\frac {2 \sqrt {3}}{3}\right )^{2}-\frac {4 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )}{3}+\frac {1}{3}}}{16 \left (x +\frac {2 \sqrt {3}}{3}\right )}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 80 vs.
\(2 (33) = 66\).
time = 0.54, size = 80, normalized size = 1.86 \begin {gather*} -\frac {12 \, x^{2} + 5 \, {\left (3 \, x^{2} - 4\right )} \log \left (3 \, x^{2} - 3 \, \sqrt {x^{2} - 1} x - 2\right ) - 5 \, {\left (3 \, x^{2} - 4\right )} \log \left (x^{2} - \sqrt {x^{2} - 1} x - 2\right ) + 12 \, \sqrt {x^{2} - 1} x - 16}{32 \, {\left (3 \, x^{2} - 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \left (3 x^{2} - 4\right )^{2}}\, dx \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. 94 vs.
\(2 (33) = 66\).
time = 0.74, size = 94, normalized size = 2.19 \begin {gather*} \frac {5 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 3}{4 \, {\left (3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{4} - 10 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 3\right )}} - \frac {5}{32} \, \log \left ({\left | 3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 1 \right |}\right ) + \frac {5}{32} \, \log \left ({\left | {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 3 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x^2-1}\,{\left (3\,x^2-4\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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