Optimal. Leaf size=47 \[ \frac {\sqrt {1-x^2} x}{x^2+1}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {527, 12, 377, 203} \[ \frac {\sqrt {1-x^2} x}{x^2+1}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 377
Rule 527
Rubi steps
\begin {align*} \int \frac {5+x^2}{\sqrt {1-x^2} \left (1+x^2\right )^2} \, dx &=\frac {x \sqrt {1-x^2}}{1+x^2}-\frac {1}{4} \int -\frac {16}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx\\ &=\frac {x \sqrt {1-x^2}}{1+x^2}+4 \int \frac {1}{\sqrt {1-x^2} \left (1+x^2\right )} \, dx\\ &=\frac {x \sqrt {1-x^2}}{1+x^2}+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}}{1+x^2}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.16, size = 85, normalized size = 1.81 \[ \frac {\sqrt {1-x^2} x}{x^2+1}+\frac {\tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )}{\sqrt {2}}+\frac {3 x \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {x^2}{x^2-1}}\right )}{\sqrt {2} \sqrt {-x^2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 54, normalized size = 1.15 \[ \frac {x \sqrt {1-x^2}}{x^2+1}-2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {1-x^2}}{x^2-1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 50, normalized size = 1.06 \[ -\frac {2 \, \sqrt {2} {\left (x^{2} + 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + 1}}{2 \, x}\right ) - \sqrt {-x^{2} + 1} x}{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.65, size = 123, normalized size = 2.62 \[ \sqrt {2} {\left (\pi \mathrm {sgn}\relax (x) + 2 \, \arctan \left (-\frac {\sqrt {2} x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{4 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} - \frac {2 \, {\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}}{{\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}^{2} + 8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 53, normalized size = 1.13
method | result | size |
risch | \(-\frac {x \left (x^{2}-1\right )}{\left (x^{2}+1\right ) \sqrt {-x^{2}+1}}-2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-x^{2}+1}\, x}{x^{2}-1}\right )\) | \(53\) |
trager | \(\frac {x \sqrt {-x^{2}+1}}{x^{2}+1}+\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {-x^{2}+1}\, x +\RootOf \left (\textit {\_Z}^{2}+2\right )}{x^{2}+1}\right )\) | \(66\) |
default | \(-2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {-x^{2}+1}\, x}{x^{2}-1}\right )-\frac {\sqrt {-x^{2}+1}\, x}{2 \left (x^{2}-1\right ) \left (\frac {\left (-x^{2}+1\right ) x^{2}}{\left (x^{2}-1\right )^{2}}+\frac {1}{2}\right )}\) | \(70\) |
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 \[ \int \frac {x^{2} + 5}{{\left (x^{2} + 1\right )}^{2} \sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 115, normalized size = 2.45 \[ \sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (-1+x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\mathrm {i}}\right )\,1{}\mathrm {i}-\sqrt {2}\,\ln \left (\frac {\frac {\sqrt {2}\,\left (1+x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+1{}\mathrm {i}}\right )\,1{}\mathrm {i}+\frac {\sqrt {1-x^2}}{2\,\left (x-\mathrm {i}\right )}+\frac {\sqrt {1-x^2}}{2\,\left (x+1{}\mathrm {i}\right )} \]
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 \[ \int \frac {x^{2} + 5}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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