Optimal. Leaf size=25 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {385, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 385
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx &=\text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 25, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 28, normalized size = 1.12
method | result | size |
default | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-x^{2}+1}\, x}{x^{2}-1}\right )}{2}\) | \(28\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {-x^{2}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+2\right )}{x^{2}+1}\right )}{4}\) | \(48\) |
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 = 1.01, size = 23, normalized size = 0.92 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + 1}}{2 \, 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 {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 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. 51 vs.
\(2 (20) = 40\).
time = 0.77, size = 51, normalized size = 2.04 \begin {gather*} \frac {1}{4} \, \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 79, normalized size = 3.16 \begin {gather*} \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}}{4}-\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}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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