Optimal. Leaf size=70 \[ \frac {\sqrt {x^2+x-1}}{2 \left (1-x^2\right )}-\frac {1}{8} \tan ^{-1}\left (\frac {x+3}{2 \sqrt {x^2+x-1}}\right )-\frac {5}{8} \tanh ^{-1}\left (\frac {1-3 x}{2 \sqrt {x^2+x-1}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {976, 1033, 724, 206, 204} \[ \frac {\sqrt {x^2+x-1}}{2 \left (1-x^2\right )}-\frac {1}{8} \tan ^{-1}\left (\frac {x+3}{2 \sqrt {x^2+x-1}}\right )-\frac {5}{8} \tanh ^{-1}\left (\frac {1-3 x}{2 \sqrt {x^2+x-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 724
Rule 976
Rule 1033
Rubi steps
\begin {align*} \int \frac {1}{\left (-1+x^2\right )^2 \sqrt {-1+x+x^2}} \, dx &=\frac {\sqrt {-1+x+x^2}}{2 \left (1-x^2\right )}-\frac {1}{4} \int \frac {3+2 x}{\left (-1+x^2\right ) \sqrt {-1+x+x^2}} \, dx\\ &=\frac {\sqrt {-1+x+x^2}}{2 \left (1-x^2\right )}+\frac {1}{8} \int \frac {1}{(1+x) \sqrt {-1+x+x^2}} \, dx-\frac {5}{8} \int \frac {1}{(-1+x) \sqrt {-1+x+x^2}} \, dx\\ &=\frac {\sqrt {-1+x+x^2}}{2 \left (1-x^2\right )}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,\frac {-3-x}{\sqrt {-1+x+x^2}}\right )+\frac {5}{4} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+3 x}{\sqrt {-1+x+x^2}}\right )\\ &=\frac {\sqrt {-1+x+x^2}}{2 \left (1-x^2\right )}-\frac {1}{8} \tan ^{-1}\left (\frac {3+x}{2 \sqrt {-1+x+x^2}}\right )-\frac {5}{8} \tanh ^{-1}\left (\frac {1-3 x}{2 \sqrt {-1+x+x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 66, normalized size = 0.94 \[ \frac {1}{8} \left (-\frac {4 \sqrt {x^2+x-1}}{x^2-1}-\tan ^{-1}\left (\frac {x+3}{2 \sqrt {x^2+x-1}}\right )-5 \tanh ^{-1}\left (\frac {1-3 x}{2 \sqrt {x^2+x-1}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 82, normalized size = 1.17 \[ \frac {2 \, {\left (x^{2} - 1\right )} \arctan \left (-x + \sqrt {x^{2} + x - 1} - 1\right ) + 5 \, {\left (x^{2} - 1\right )} \log \left (-x + \sqrt {x^{2} + x - 1} + 2\right ) - 5 \, {\left (x^{2} - 1\right )} \log \left (-x + \sqrt {x^{2} + x - 1}\right ) - 4 \, \sqrt {x^{2} + x - 1}}{8 \, {\left (x^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 143, normalized size = 2.04 \[ \frac {2 \, {\left (x - \sqrt {x^{2} + x - 1}\right )}^{3} + 3 \, {\left (x - \sqrt {x^{2} + x - 1}\right )}^{2} - x + \sqrt {x^{2} + x - 1} - 1}{2 \, {\left ({\left (x - \sqrt {x^{2} + x - 1}\right )}^{4} - 2 \, {\left (x - \sqrt {x^{2} + x - 1}\right )}^{2} - 4 \, x + 4 \, \sqrt {x^{2} + x - 1}\right )}} + \frac {1}{4} \, \arctan \left (-x + \sqrt {x^{2} + x - 1} - 1\right ) + \frac {5}{8} \, \log \left ({\left | -x + \sqrt {x^{2} + x - 1} + 2 \right |}\right ) - \frac {5}{8} \, \log \left ({\left | -x + \sqrt {x^{2} + x - 1} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 1.20 \[ \frac {5 \arctanh \left (\frac {3 x -1}{2 \sqrt {3 x +\left (x -1\right )^{2}-2}}\right )}{8}+\frac {\arctan \left (\frac {-x -3}{2 \sqrt {-x +\left (x +1\right )^{2}-2}}\right )}{8}+\frac {\sqrt {-x +\left (x +1\right )^{2}-2}}{4 x +4}-\frac {\sqrt {3 x +\left (x -1\right )^{2}-2}}{4 \left (x -1\right )} \]
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 \[ \int \frac {1}{\sqrt {x^{2} + x - 1} {\left (x^{2} - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (x^2-1\right )}^2\,\sqrt {x^2+x-1}} \,d x \]
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 \[ \int \frac {1}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \sqrt {x^{2} + x - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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