Optimal. Leaf size=14 \[ -\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^4}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 65, 213}
\begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {1+x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right )\\ &=-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^4}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 11, normalized size = 0.79
method | result | size |
default | \(-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(11\) |
elliptic | \(-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(11\) |
trager | \(-\frac {\ln \left (\frac {1+\sqrt {x^{4}+1}}{x^{2}}\right )}{2}\) | \(17\) |
meijerg | \(\frac {-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )+\left (-2 \ln \left (2\right )+4 \ln \left (x \right )\right ) \sqrt {\pi }}{4 \sqrt {\pi }}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (10) = 20\).
time = 0.29, size = 25, normalized size = 1.79 \begin {gather*} -\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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (10) = 20\).
time = 0.36, size = 25, normalized size = 1.79 \begin {gather*} -\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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Sympy [A]
time = 0.40, size = 8, normalized size = 0.57 \begin {gather*} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{2} \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. 25 vs.
\(2 (10) = 20\).
time = 1.31, size = 25, normalized size = 1.79 \begin {gather*} -\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 = 0.05, size = 10, normalized size = 0.71 \begin {gather*} -\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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