Optimal. Leaf size=28 \[ \frac {\sqrt {x^4+1}}{2}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 50, 63, 207} \begin {gather*} \frac {\sqrt {x^4+1}}{2}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^4}}{x} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^4\right )\\ &=\frac {\sqrt {1+x^4}}{2}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {\sqrt {1+x^4}}{2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right )\\ &=\frac {\sqrt {1+x^4}}{2}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^4}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^4+1}}{2}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^4}}{2}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 34, normalized size = 1.21 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 1} - \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 34, normalized size = 1.21 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 1} - \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 21, normalized size = 0.75
method | result | size |
default | \(\frac {\sqrt {x^{4}+1}}{2}-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(21\) |
elliptic | \(\frac {\sqrt {x^{4}+1}}{2}-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(21\) |
trager | \(\frac {\sqrt {x^{4}+1}}{2}+\frac {\ln \left (\frac {-1+\sqrt {x^{4}+1}}{x^{2}}\right )}{2}\) | \(27\) |
meijerg | \(-\frac {-2 \left (2-2 \ln \relax (2)+4 \ln \relax (x )\right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {x^{4}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )}{8 \sqrt {\pi }}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 34, normalized size = 1.21 \begin {gather*} \frac {1}{2} \, \sqrt {x^{4} + 1} - \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 20, normalized size = 0.71 \begin {gather*} \frac {\sqrt {x^4+1}}{2}-\frac {\mathrm {atanh}\left (\sqrt {x^4+1}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.96, size = 39, normalized size = 1.39 \begin {gather*} \frac {x^{2}}{2 \sqrt {1 + \frac {1}{x^{4}}}} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{2} + \frac {1}{2 x^{2} \sqrt {1 + \frac {1}{x^{4}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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