Optimal. Leaf size=43 \[ \frac {3 \sqrt {x^2-1} x}{8 \left (4-3 x^2\right )}+\frac {5}{16} \tanh ^{-1}\left (\frac {x}{2 \sqrt {x^2-1}}\right ) \]
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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 = {382, 377, 207} \[ \frac {3 \sqrt {x^2-1} x}{8 \left (4-3 x^2\right )}+\frac {5}{16} \tanh ^{-1}\left (\frac {x}{2 \sqrt {x^2-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 377
Rule 382
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} \operatorname {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 [C] time = 3.76, size = 167, normalized size = 3.88 \[ -\frac {x \sqrt {x^2-1} \left (\frac {8 x^2 \left (x^2-1\right ) \, _2F_1\left (2,3;\frac {7}{2};\frac {x^2}{4-3 x^2}\right )}{45 x^2-60}-\frac {x^2 \left (2 x^2-3\right ) \sqrt {\frac {x^2-1}{3 x^2-4}} \left (2 \sqrt {\frac {x^2-x^4}{\left (4-3 x^2\right )^2}}-\sin ^{-1}\left (\sqrt {\frac {x^2}{4-3 x^2}}\right )\right )}{4 \left (\frac {x^2}{4-3 x^2}\right )^{5/2} \left (x^2-1\right )}\right )}{16 \left (1-\frac {3 x^2}{4}\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.53, size = 80, normalized size = 1.86 \[ -\frac {12 \, x^{2} + 5 \, {\left (3 \, x^{2} - 4\right )} \log \left (3 \, x^{2} - 3 \, \sqrt {x^{2} - 1} x - 2\right ) - 5 \, {\left (3 \, x^{2} - 4\right )} \log \left (x^{2} - \sqrt {x^{2} - 1} x - 2\right ) + 12 \, \sqrt {x^{2} - 1} x - 16}{32 \, {\left (3 \, x^{2} - 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.94, size = 94, normalized size = 2.19 \[ \frac {5 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 3}{4 \, {\left (3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{4} - 10 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 3\right )}} - \frac {5}{32} \, \log \left ({\left | 3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 1 \right |}\right ) + \frac {5}{32} \, \log \left ({\left | {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 172, normalized size = 4.00 \[ \frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}+\frac {4 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x -\frac {2 \sqrt {3}}{3}\right )^{2}+12 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}-\frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}-\frac {4 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x +\frac {2 \sqrt {3}}{3}\right )^{2}-12 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}-\frac {\sqrt {\left (x +\frac {2 \sqrt {3}}{3}\right )^{2}-\frac {4 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )}{3}+\frac {1}{3}}}{16 \left (x +\frac {2 \sqrt {3}}{3}\right )}-\frac {\sqrt {\left (x -\frac {2 \sqrt {3}}{3}\right )^{2}+\frac {4 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )}{3}+\frac {1}{3}}}{16 \left (x -\frac {2 \sqrt {3}}{3}\right )} \]
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 \[ \int \frac {1}{{\left (3 \, x^{2} - 4\right )}^{2} \sqrt {x^{2} - 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {x^2-1}\,{\left (3\,x^2-4\right )}^2} \,d x \]
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 \[ \int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \left (3 x^{2} - 4\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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