Optimal. Leaf size=13 \[ \log (x)-\frac{1}{4} \log \left (x^4+1\right ) \]
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Rubi [A] time = 0.0049037, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {266, 36, 29, 31} \[ \log (x)-\frac{1}{4} \log \left (x^4+1\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \left (1+x^4\right )} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x (1+x)} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^4\right )-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,x^4\right )\\ &=\log (x)-\frac{1}{4} \log \left (1+x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0030564, size = 13, normalized size = 1. \[ \log (x)-\frac{1}{4} \log \left (x^4+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 12, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -{\frac{\ln \left ({x}^{4}+1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.929044, size = 20, normalized size = 1.54 \begin{align*} -\frac{1}{4} \, \log \left (x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02057, size = 38, normalized size = 2.92 \begin{align*} -\frac{1}{4} \, \log \left (x^{4} + 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09011, size = 10, normalized size = 0.77 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{4} + 1 \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04758, size = 20, normalized size = 1.54 \begin{align*} -\frac{1}{4} \, \log \left (x^{4} + 1\right ) + \frac{1}{4} \, \log \left (x^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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