Optimal. Leaf size=47 \[ \frac {x \sqrt {1-x^2}}{1+x^2}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 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 = {541, 12, 385,
209} \begin {gather*} 2 \sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x}{\sqrt {1-x^2}}\right )+\frac {\sqrt {1-x^2} x}{x^2+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 385
Rule 541
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 \text {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 = 47, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 70, normalized size = 1.49
method | result | size |
risch | \(-\frac {x \left (x^{2}-1\right )}{\left (x^{2}+1\right ) \sqrt {-x^{2}+1}}-2 \sqrt {2}\, \arctan \left (\frac {x \sqrt {-x^{2}+1}\, \sqrt {2}}{x^{2}-1}\right )\) | \(53\) |
trager | \(\frac {x \sqrt {-x^{2}+1}}{x^{2}+1}+\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 x \sqrt {-x^{2}+1}+\RootOf \left (\textit {\_Z}^{2}+2\right )}{x^{2}+1}\right )\) | \(66\) |
default | \(-\frac {x \sqrt {-x^{2}+1}}{2 \left (x^{2}-1\right ) \left (\frac {x^{2} \left (-x^{2}+1\right )}{\left (x^{2}-1\right )^{2}}+\frac {1}{2}\right )}-2 \sqrt {2}\, \arctan \left (\frac {x \sqrt {-x^{2}+1}\, \sqrt {2}}{x^{2}-1}\right )\) | \(70\) |
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 \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 = 0.49, size = 50, normalized size = 1.06 \begin {gather*} -\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} \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 {x^{2} + 5}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right )^{2}}\, 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. 123 vs.
\(2 (39) = 78\).
time = 0.61, size = 123, normalized size = 2.62 \begin {gather*} \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 )} - \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} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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