Optimal. Leaf size=28 \[ \frac {1}{2 \sqrt {1+x^4}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^4}\right ) \]
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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 = {272, 53, 65,
213} \begin {gather*} \frac {1}{2 \sqrt {x^4+1}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 213
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x (1+x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{2 \sqrt {1+x^4}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{2 \sqrt {1+x^4}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right )\\ &=\frac {1}{2 \sqrt {1+x^4}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^4}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} \frac {1}{2 \sqrt {1+x^4}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {1+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 21, normalized size = 0.75
method | result | size |
default | \(\frac {1}{2 \sqrt {x^{4}+1}}-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(21\) |
risch | \(\frac {1}{2 \sqrt {x^{4}+1}}-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(21\) |
elliptic | \(\frac {1}{2 \sqrt {x^{4}+1}}-\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}\) | \(21\) |
trager | \(\frac {1}{2 \sqrt {x^{4}+1}}-\frac {\ln \left (\frac {1+\sqrt {x^{4}+1}}{x^{2}}\right )}{2}\) | \(27\) |
meijerg | \(\frac {-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {x^{4}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+4 \ln \left (x \right )\right ) \sqrt {\pi }}{2}}{2 \sqrt {\pi }}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs.
\(2 (20) = 40\).
time = 0.36, size = 52, normalized size = 1.86 \begin {gather*} -\frac {{\left (x^{4} + 1\right )} \log \left (\sqrt {x^{4} + 1} + 1\right ) - {\left (x^{4} + 1\right )} \log \left (\sqrt {x^{4} + 1} - 1\right ) - 2 \, \sqrt {x^{4} + 1}}{4 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (22) = 44\).
time = 0.64, size = 87, normalized size = 3.11 \begin {gather*} \frac {x^{4} \log {\left (x^{4} \right )}}{4 x^{4} + 4} - \frac {2 x^{4} \log {\left (\sqrt {x^{4} + 1} + 1 \right )}}{4 x^{4} + 4} + \frac {2 \sqrt {x^{4} + 1}}{4 x^{4} + 4} + \frac {\log {\left (x^{4} \right )}}{4 x^{4} + 4} - \frac {2 \log {\left (\sqrt {x^{4} + 1} + 1 \right )}}{4 x^{4} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.61, 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 = 1.19, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{2\,\sqrt {x^4+1}}-\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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