Optimal. Leaf size=28 \[ \frac {1}{2 \sqrt {x^4+1}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \]
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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, 51, 63, 207} \[ \frac {1}{2 \sqrt {x^4+1}}-\frac {1}{2} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{4} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{2 \sqrt {1+x^4}}+\frac {1}{2} \operatorname {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 [C] time = 0.01, size = 26, normalized size = 0.93 \[ \frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};x^4+1\right )}{2 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 52, normalized size = 1.86 \[ -\frac {{\left (x^{4} + 1\right )} \log \left (\sqrt {x^{4} + 1} + 1\right ) - {\left (x^{4} + 1\right )} \log \left (\sqrt {x^{4} + 1} - 1\right ) - 2 \, \sqrt {x^{4} + 1}}{4 \, {\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 34, normalized size = 1.21 \[ \frac {1}{2 \, \sqrt {x^{4} + 1}} - \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.75 \[ -\frac {\arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}+\frac {1}{2 \sqrt {x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 34, normalized size = 1.21 \[ \frac {1}{2 \, \sqrt {x^{4} + 1}} - \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
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 \[ \frac {1}{2\,\sqrt {x^4+1}}-\frac {\mathrm {atanh}\left (\sqrt {x^4+1}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.89, size = 87, normalized size = 3.11 \[ \frac {x^{4} \log {\left (x^{4} \right )}}{4 x^{4} + 4} - \frac {2 x^{4} \log {\left (\sqrt {x^{4} + 1} + 1 \right )}}{4 x^{4} + 4} + \frac {2 \sqrt {x^{4} + 1}}{4 x^{4} + 4} + \frac {\log {\left (x^{4} \right )}}{4 x^{4} + 4} - \frac {2 \log {\left (\sqrt {x^{4} + 1} + 1 \right )}}{4 x^{4} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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