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 = 0.0111564, antiderivative size = 28, normalized size of antiderivative = 1., 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 266
Rule 51
Rule 63
Rule 207
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*}
Mathematica [C] time = 0.0046358, 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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Maple [A] time = 0.013, size = 21, normalized size = 0.8 \begin{align*}{\frac{1}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{4}+1}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04491, size = 46, normalized size = 1.64 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47157, size = 140, normalized size = 5. \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.87887, size = 87, normalized size = 3.11 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17101, size = 46, normalized size = 1.64 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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