Optimal. Leaf size=25 \[ \frac {1}{4} \log \left (x^2+1\right )-\frac {1}{2} \log (x+1)+\frac {1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {801, 635, 203, 260} \[ \frac {1}{4} \log \left (x^2+1\right )-\frac {1}{2} \log (x+1)+\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 801
Rubi steps
\begin {align*} \int \frac {x}{(1+x) \left (1+x^2\right )} \, dx &=\int \left (-\frac {1}{2 (1+x)}+\frac {1+x}{2 \left (1+x^2\right )}\right ) \, dx\\ &=-\frac {1}{2} \log (1+x)+\frac {1}{2} \int \frac {1+x}{1+x^2} \, dx\\ &=-\frac {1}{2} \log (1+x)+\frac {1}{2} \int \frac {1}{1+x^2} \, dx+\frac {1}{2} \int \frac {x}{1+x^2} \, dx\\ &=\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \log (1+x)+\frac {1}{4} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {1}{4} \log \left (x^2+1\right )-\frac {1}{2} \log (x+1)+\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 19, normalized size = 0.76 \[ \frac {1}{2} \, \arctan \relax (x) + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 20, normalized size = 0.80 \[ \frac {1}{2} \, \arctan \relax (x) + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.80 \[ \frac {\arctan \relax (x )}{2}-\frac {\ln \left (x +1\right )}{2}+\frac {\ln \left (x^{2}+1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 19, normalized size = 0.76 \[ \frac {1}{2} \, \arctan \relax (x) + \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.21, size = 25, normalized size = 1.00 \[ -\frac {\ln \left (x+1\right )}{2}+\ln \left (x-\mathrm {i}\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 19, normalized size = 0.76 \[ - \frac {\log {\left (x + 1 \right )}}{2} + \frac {\log {\left (x^{2} + 1 \right )}}{4} + \frac {\operatorname {atan}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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