Optimal. Leaf size=27 \[ \sinh ^{-1}(x)-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {399, 221, 385,
213} \begin {gather*} \sinh ^{-1}(x)-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 221
Rule 385
Rule 399
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2}}{-1+x^2} \, dx &=2 \int \frac {1}{\left (-1+x^2\right ) \sqrt {1+x^2}} \, dx+\int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=\sinh ^{-1}(x)+2 \text {Subst}\left (\int \frac {1}{-1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^2}}\right )\\ &=\sinh ^{-1}(x)-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 45, normalized size = 1.67 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {1-x^2+x \sqrt {1+x^2}}{\sqrt {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. \(83\) vs.
\(2(21)=42\).
time = 0.19, size = 84, normalized size = 3.11
method | result | size |
trager | \(-\ln \left (-\sqrt {x^{2}+1}+x \right )-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{2}+1}\, x +\RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (x -1\right ) \left (x +1\right )}\right )}{2}\) | \(65\) |
default | \(-\frac {\sqrt {\left (x +1\right )^{2}-2 x}}{2}+\arcsinh \left (x \right )+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2-2 x \right ) \sqrt {2}}{4 \sqrt {\left (x +1\right )^{2}-2 x}}\right )}{2}+\frac {\sqrt {\left (x -1\right )^{2}+2 x}}{2}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2+2 x \right ) \sqrt {2}}{4 \sqrt {\left (x -1\right )^{2}+2 x}}\right )}{2}\) | \(84\) |
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. 59 vs.
\(2 (21) = 42\).
time = 0.50, size = 59, normalized size = 2.19 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \operatorname {arsinh}\left (\frac {2 \, x}{{\left | 2 \, x + 2 \right |}} - \frac {2}{{\left | 2 \, x + 2 \right |}}\right ) - \frac {1}{2} \, \sqrt {2} \operatorname {arsinh}\left (\frac {2 \, x}{{\left | 2 \, x - 2 \right |}} + \frac {2}{{\left | 2 \, x - 2 \right |}}\right ) + \operatorname {arsinh}\left (x\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. 67 vs.
\(2 (21) = 42\).
time = 0.43, size = 67, normalized size = 2.48 \begin {gather*} \frac {1}{2} \, \sqrt {2} \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 ) - \log \left (-x + \sqrt {x^{2} + 1}\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 {\sqrt {x^{2} + 1}}{\left (x - 1\right ) \left (x + 1\right )}\, 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. 70 vs.
\(2 (21) = 42\).
time = 0.87, size = 70, normalized size = 2.59 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \log \left (\frac {{\left | 2 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 4 \, \sqrt {2} - 6 \right |}}{{\left | 2 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 4 \, \sqrt {2} - 6 \right |}}\right ) - \log \left (-x + \sqrt {x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 59, normalized size = 2.19 \begin {gather*} \mathrm {asinh}\left (x\right )+\frac {\sqrt {2}\,\left (\ln \left (x-1\right )-\ln \left (x+\sqrt {2}\,\sqrt {x^2+1}+1\right )\right )}{2}-\frac {\sqrt {2}\,\left (\ln \left (x+1\right )-\ln \left (\sqrt {2}\,\sqrt {x^2+1}-x+1\right )\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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