Optimal. Leaf size=10 \[ \frac {1}{1+x^2}+\log (x) \]
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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 = {1266, 908}
\begin {gather*} \frac {1}{x^2+1}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 908
Rule 1266
Rubi steps
\begin {align*} \int \frac {1+x^4}{x \left (1+x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{x (1+x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {1}{1+x^2}+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 11, normalized size = 1.10
method | result | size |
default | \(\frac {1}{x^{2}+1}+\ln \left (x \right )\) | \(11\) |
norman | \(\frac {1}{x^{2}+1}+\ln \left (x \right )\) | \(11\) |
risch | \(\frac {1}{x^{2}+1}+\ln \left (x \right )\) | \(11\) |
meijerg | \(-\frac {x^{2}}{2 \left (x^{2}+1\right )}+\frac {1}{2}+\ln \left (x \right )-\frac {x^{2}}{2 x^{2}+2}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.60, size = 14, normalized size = 1.40 \begin {gather*} \frac {1}{x^{2} + 1} + \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.33, size = 18, normalized size = 1.80 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \log \left (x\right ) + 1}{x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.80 \begin {gather*} \log {\left (x \right )} + \frac {1}{x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 14, normalized size = 1.40 \begin {gather*} \frac {1}{x^{2} + 1} + \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}
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 \begin {gather*} \ln \left (x\right )+\frac {1}{x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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