Optimal. Leaf size=24 \[ \frac {1}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right )+\log (x) \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {266, 44} \[ \frac {1}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2 \left (1+x^2\right )}+\log (x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {1}{2 \left (x^2+1\right )}-\frac {1}{2} \log \left (x^2+1\right )+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 32, normalized size = 1.33 \[ -\frac {{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - 2 \, {\left (x^{2} + 1\right )} \log \relax (x) - 1}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 29, normalized size = 1.21 \[ \frac {x^{2} + 2}{2 \, {\left (x^{2} + 1\right )}} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.88 \[ \ln \relax (x )-\frac {\ln \left (x^{2}+1\right )}{2}+\frac {1}{2 x^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 24, normalized size = 1.00 \[ \frac {1}{2 \, {\left (x^{2} + 1\right )}} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 20, normalized size = 0.83 \[ \ln \relax (x)-\frac {\ln \left (x^2+1\right )}{2}+\frac {1}{2\,\left (x^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.79 \[ \log {\relax (x )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {1}{2 x^{2} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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