Optimal. Leaf size=30 \[ -\sin ^{-1}(x)+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {399, 222, 385,
209} \begin {gather*} \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )-\text {ArcSin}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 222
Rule 385
Rule 399
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}}{1+x^2} \, dx &=2 \int \frac {1}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx-\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\sin ^{-1}(x)+2 \text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )\\ &=-\sin ^{-1}(x)+\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 46, normalized size = 1.53 \begin {gather*} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )+2 \tan ^{-1}\left (\frac {\sqrt {1-x^2}}{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 33, normalized size = 1.10
method | result | size |
default | \(-\arcsin \left (x \right )-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-x^{2}+1}\, x}{x^{2}-1}\right )\) | \(33\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )-\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {-x^{2}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+2\right )}{x^{2}+1}\right )}{2}\) | \(78\) |
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 [A]
time = 0.47, size = 42, normalized size = 1.40 \begin {gather*} -\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + 1}}{2 \, x}\right ) + 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\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 {- \left (x - 1\right ) \left (x + 1\right )}}{x^{2} + 1}\, 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. 95 vs.
\(2 (24) = 48\).
time = 0.90, size = 95, normalized size = 3.17 \begin {gather*} -\frac {1}{2} \, \pi \mathrm {sgn}\left (x\right ) + \frac {1}{2} \, \sqrt {2} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (-\frac {\sqrt {2} x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{4 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} - \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 83, normalized size = 2.77 \begin {gather*} -\mathrm {asin}\left (x\right )+\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (-1+x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\mathrm {i}}\right )\,1{}\mathrm {i}}{2}-\frac {\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (1+x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+1{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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