Optimal. Leaf size=48 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )}{4 \sqrt {2}}-\frac {x \sqrt {x^2-1}}{4 \left (x^2+1\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {382, 377, 206} \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )}{4 \sqrt {2}}-\frac {x \sqrt {x^2-1}}{4 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 377
Rule 382
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x^2} \left (1+x^2\right )^2} \, dx &=-\frac {x \sqrt {-1+x^2}}{4 \left (1+x^2\right )}+\frac {3}{4} \int \frac {1}{\sqrt {-1+x^2} \left (1+x^2\right )} \, dx\\ &=-\frac {x \sqrt {-1+x^2}}{4 \left (1+x^2\right )}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )\\ &=-\frac {x \sqrt {-1+x^2}}{4 \left (1+x^2\right )}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 75, normalized size = 1.56 \[ \frac {\sqrt {x^2-1} \left (3 \sqrt {2} \sqrt {\frac {x^2}{x^2-1}} \left (x^2+1\right ) \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {x^2}{x^2-1}}\right )-2 x^2\right )}{8 \left (x^3+x\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 83, normalized size = 1.73 \[ \frac {3 \, \sqrt {2} {\left (x^{2} + 1\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {2} {\left (3 \, x^{2} - 1\right )} + 2 \, \sqrt {x^{2} - 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} - 3}{x^{2} + 1}\right ) - 4 \, x^{2} - 4 \, \sqrt {x^{2} - 1} x - 4}{16 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.96, size = 101, normalized size = 2.10 \[ -\frac {3}{16} \, \sqrt {2} \log \left (\frac {{\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 2 \, \sqrt {2} + 3}{{\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 2 \, \sqrt {2} + 3}\right ) - \frac {3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1}{2 \, {\left ({\left (x - \sqrt {x^{2} - 1}\right )}^{4} + 6 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 45, normalized size = 0.94 \[ -\frac {x}{8 \sqrt {x^{2}-1}\, \left (\frac {x^{2}}{x^{2}-1}-\frac {1}{2}\right )}+\frac {3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, x}{\sqrt {x^{2}-1}}\right )}{8} \]
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 \[ \int \frac {1}{{\left (x^{2} + 1\right )}^{2} \sqrt {x^{2} - 1}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {x^2-1}\,{\left (x^2+1\right )}^2} \,d x \]
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 \[ \int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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