Optimal. Leaf size=10 \[ \frac {1}{x^2+1}+\log (x) \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1252, 894} \[ \frac {1}{x^2+1}+\log (x) \]
Antiderivative was successfully verified.
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Rule 894
Rule 1252
Rubi steps
\begin {align*} \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{x (1+x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x}-\frac {2}{(1+x)^2}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{1+x^2}+\log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ \frac {1}{x^2+1}+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 18, normalized size = 1.80 \[ \frac {{\left (x^{2} + 1\right )} \log \relax (x) + 1}{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 14, normalized size = 1.40 \[ \frac {1}{x^{2} + 1} + \frac {1}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 11, normalized size = 1.10 \[ \ln \relax (x )+\frac {1}{x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 14, normalized size = 1.40 \[ \frac {1}{x^{2} + 1} + \frac {1}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 10, normalized size = 1.00 \[ \ln \relax (x)+\frac {1}{x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 8, normalized size = 0.80 \[ \log {\relax (x )} + \frac {1}{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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