Optimal. Leaf size=48 \[ \frac {3 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {x^2+1}}\right )}{4 \sqrt {2}}-\frac {x \sqrt {x^2+1}}{4 \left (x^2+2\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {382, 377, 206} \[ \frac {3 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {x^2+1}}\right )}{4 \sqrt {2}}-\frac {x \sqrt {x^2+1}}{4 \left (x^2+2\right )} \]
Antiderivative was successfully verified.
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Rule 206
Rule 377
Rule 382
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x^2} \left (2+x^2\right )^2} \, dx &=-\frac {x \sqrt {1+x^2}}{4 \left (2+x^2\right )}+\frac {3}{4} \int \frac {1}{\sqrt {1+x^2} \left (2+x^2\right )} \, dx\\ &=-\frac {x \sqrt {1+x^2}}{4 \left (2+x^2\right )}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {x}{\sqrt {1+x^2}}\right )\\ &=-\frac {x \sqrt {1+x^2}}{4 \left (2+x^2\right )}+\frac {3 \tanh ^{-1}\left (\frac {x}{\sqrt {2} \sqrt {1+x^2}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 74, normalized size = 1.54 \[ \frac {\sqrt {x^2+1} \left (3 \sqrt {2} \sqrt {\frac {x^2}{x^2+1}} \left (x^2+2\right ) \tanh ^{-1}\left (\sqrt {\frac {x^2}{2 x^2+2}}\right )-2 x^2\right )}{8 x \left (x^2+2\right )} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 64, normalized size = 1.33 \[ \frac {3 \tanh ^{-1}\left (\frac {x^2}{\sqrt {2}}-\frac {\sqrt {x^2+1} x}{\sqrt {2}}+\sqrt {2}\right )}{4 \sqrt {2}}-\frac {x \sqrt {x^2+1}}{4 \left (x^2+2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 83, normalized size = 1.73 \[ \frac {3 \, \sqrt {2} {\left (x^{2} + 2\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {2} {\left (3 \, x^{2} + 2\right )} + 2 \, \sqrt {x^{2} + 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} + 6}{x^{2} + 2}\right ) - 4 \, x^{2} - 4 \, \sqrt {x^{2} + 1} x - 8}{16 \, {\left (x^{2} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.64, size = 101, normalized size = 2.10 \[ -\frac {3}{16} \, \sqrt {2} \log \left (\frac {{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 2 \, \sqrt {2} + 3}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 2 \, \sqrt {2} + 3}\right ) - \frac {3 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 1}{2 \, {\left ({\left (x - \sqrt {x^{2} + 1}\right )}^{4} + 6 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 38, normalized size = 0.79
method | result | size |
risch | \(\frac {3 \arctanh \left (\frac {x \sqrt {2}}{2 \sqrt {x^{2}+1}}\right ) \sqrt {2}}{8}-\frac {x \sqrt {x^{2}+1}}{4 \left (x^{2}+2\right )}\) | \(38\) |
default | \(\frac {x}{4 \sqrt {x^{2}+1}\, \left (\frac {x^{2}}{x^{2}+1}-2\right )}+\frac {3 \arctanh \left (\frac {x \sqrt {2}}{2 \sqrt {x^{2}+1}}\right ) \sqrt {2}}{8}\) | \(46\) |
trager | \(-\frac {x \sqrt {x^{2}+1}}{4 \left (x^{2}+2\right )}-\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-4 x \sqrt {x^{2}+1}+2 \RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{2}+2}\right )}{16}\) | \(67\) |
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 \[ \int \frac {1}{{\left (x^{2} + 2\right )}^{2} \sqrt {x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 117, normalized size = 2.44 \[ -\frac {3\,\sqrt {2}\,\left (\ln \left (x-\sqrt {2}\,1{}\mathrm {i}\right )-\ln \left (1+\sqrt {2}\,x\,1{}\mathrm {i}+\sqrt {x^2+1}\,1{}\mathrm {i}\right )\right )}{16}+\frac {3\,\sqrt {2}\,\left (\ln \left (x+\sqrt {2}\,1{}\mathrm {i}\right )-\ln \left (1-\sqrt {2}\,x\,1{}\mathrm {i}+\sqrt {x^2+1}\,1{}\mathrm {i}\right )\right )}{16}-\frac {\sqrt {x^2+1}}{8\,\left (x-\sqrt {2}\,1{}\mathrm {i}\right )}-\frac {\sqrt {x^2+1}}{8\,\left (x+\sqrt {2}\,1{}\mathrm {i}\right )} \]
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 \[ \int \frac {1}{\sqrt {x^{2} + 1} \left (x^{2} + 2\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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