Optimal. Leaf size=35 \[ \frac {\sqrt {1+x^4}}{2 x^2}+\frac {1}{2} \log \left (x^2+\sqrt {1+x^4}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 0.71, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {462, 281, 221}
\begin {gather*} \frac {1}{2} \sinh ^{-1}\left (x^2\right )+\frac {\sqrt {x^4+1}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 281
Rule 462
Rubi steps
\begin {align*} \int \frac {-1+x^4}{x^3 \sqrt {1+x^4}} \, dx &=\frac {\sqrt {1+x^4}}{2 x^2}+\int \frac {x}{\sqrt {1+x^4}} \, dx\\ &=\frac {\sqrt {1+x^4}}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )\\ &=\frac {\sqrt {1+x^4}}{2 x^2}+\frac {1}{2} \sinh ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 32, normalized size = 0.91 \begin {gather*} \frac {1}{2} \left (\frac {\sqrt {1+x^4}}{x^2}+\tanh ^{-1}\left (\frac {x^2}{\sqrt {1+x^4}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 20, normalized size = 0.57
method | result | size |
default | \(\frac {\arcsinh \left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\) | \(20\) |
meijerg | \(\frac {\arcsinh \left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\) | \(20\) |
risch | \(\frac {\arcsinh \left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\) | \(20\) |
elliptic | \(\frac {\arcsinh \left (x^{2}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2 x^{2}}\) | \(20\) |
trager | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}}+\frac {\ln \left (-x^{2}-\sqrt {x^{4}+1}\right )}{2}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 45, normalized size = 1.29 \begin {gather*} \frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} + \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 38, normalized size = 1.09 \begin {gather*} -\frac {x^{2} \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) - x^{2} - \sqrt {x^{4} + 1}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.89, size = 19, normalized size = 0.54 \begin {gather*} \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} + \frac {\sqrt {x^{4} + 1}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 38, normalized size = 1.09 \begin {gather*} -\frac {1}{{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1} - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 19, normalized size = 0.54 \begin {gather*} \frac {\mathrm {asinh}\left (x^2\right )}{2}+\frac {\sqrt {x^4+1}}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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