Optimal. Leaf size=71 \[ -\frac {\sqrt {4+2 x^2+x^4}}{16 x^4}+\frac {3 \sqrt {4+2 x^2+x^4}}{64 x^2}+\frac {1}{128} \tanh ^{-1}\left (\frac {4+x^2}{2 \sqrt {4+2 x^2+x^4}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1128, 758, 820,
738, 212} \begin {gather*} \frac {3 \sqrt {x^4+2 x^2+4}}{64 x^2}-\frac {\sqrt {x^4+2 x^2+4}}{16 x^4}+\frac {1}{128} \tanh ^{-1}\left (\frac {x^2+4}{2 \sqrt {x^4+2 x^2+4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 758
Rule 820
Rule 1128
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {4+2 x^2+x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {4+2 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {4+2 x^2+x^4}}{16 x^4}-\frac {1}{16} \text {Subst}\left (\int \frac {3+x}{x^2 \sqrt {4+2 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {4+2 x^2+x^4}}{16 x^4}+\frac {3 \sqrt {4+2 x^2+x^4}}{64 x^2}-\frac {1}{64} \text {Subst}\left (\int \frac {1}{x \sqrt {4+2 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {4+2 x^2+x^4}}{16 x^4}+\frac {3 \sqrt {4+2 x^2+x^4}}{64 x^2}+\frac {1}{32} \text {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\frac {2 \left (4+x^2\right )}{\sqrt {4+2 x^2+x^4}}\right )\\ &=-\frac {\sqrt {4+2 x^2+x^4}}{16 x^4}+\frac {3 \sqrt {4+2 x^2+x^4}}{64 x^2}+\frac {1}{128} \tanh ^{-1}\left (\frac {4+x^2}{2 \sqrt {4+2 x^2+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 57, normalized size = 0.80 \begin {gather*} \frac {1}{64} \left (\frac {\left (-4+3 x^2\right ) \sqrt {4+2 x^2+x^4}}{x^4}-\tanh ^{-1}\left (\frac {1}{2} \left (x^2-\sqrt {4+2 x^2+x^4}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 60, normalized size = 0.85
method | result | size |
trager | \(\frac {\left (3 x^{2}-4\right ) \sqrt {x^{4}+2 x^{2}+4}}{64 x^{4}}-\frac {\ln \left (\frac {-x^{2}+2 \sqrt {x^{4}+2 x^{2}+4}-4}{x^{2}}\right )}{128}\) | \(54\) |
default | \(-\frac {\sqrt {x^{4}+2 x^{2}+4}}{16 x^{4}}+\frac {3 \sqrt {x^{4}+2 x^{2}+4}}{64 x^{2}}+\frac {\arctanh \left (\frac {2 x^{2}+8}{4 \sqrt {x^{4}+2 x^{2}+4}}\right )}{128}\) | \(60\) |
risch | \(\frac {3 x^{6}+2 x^{4}+4 x^{2}-16}{64 x^{4} \sqrt {x^{4}+2 x^{2}+4}}+\frac {\arctanh \left (\frac {2 x^{2}+8}{4 \sqrt {x^{4}+2 x^{2}+4}}\right )}{128}\) | \(60\) |
elliptic | \(-\frac {\sqrt {x^{4}+2 x^{2}+4}}{16 x^{4}}+\frac {3 \sqrt {x^{4}+2 x^{2}+4}}{64 x^{2}}+\frac {\arctanh \left (\frac {2 x^{2}+8}{4 \sqrt {x^{4}+2 x^{2}+4}}\right )}{128}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.48, size = 52, normalized size = 0.73 \begin {gather*} \frac {3 \, \sqrt {x^{4} + 2 \, x^{2} + 4}}{64 \, x^{2}} - \frac {\sqrt {x^{4} + 2 \, x^{2} + 4}}{16 \, x^{4}} + \frac {1}{128} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} + \frac {4 \, \sqrt {3}}{3 \, x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 81, normalized size = 1.14 \begin {gather*} \frac {x^{4} \log \left (-x^{2} + \sqrt {x^{4} + 2 \, x^{2} + 4} + 2\right ) - x^{4} \log \left (-x^{2} + \sqrt {x^{4} + 2 \, x^{2} + 4} - 2\right ) + 6 \, x^{4} + 2 \, \sqrt {x^{4} + 2 \, x^{2} + 4} {\left (3 \, x^{2} - 4\right )}}{128 \, x^{4}} \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}{x^{5} \sqrt {x^{4} + 2 x^{2} + 4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 112, normalized size = 1.58 \begin {gather*} \frac {{\left (x^{2} - \sqrt {x^{4} + 2 \, x^{2} + 4}\right )}^{3} + 36 \, x^{2} - 36 \, \sqrt {x^{4} + 2 \, x^{2} + 4} + 64}{32 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 2 \, x^{2} + 4}\right )}^{2} - 4\right )}^{2}} - \frac {1}{128} \, \log \left (x^{2} - \sqrt {x^{4} + 2 \, x^{2} + 4} + 2\right ) + \frac {1}{128} \, \log \left (-x^{2} + \sqrt {x^{4} + 2 \, x^{2} + 4} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^5\,\sqrt {x^4+2\,x^2+4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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