Optimal. Leaf size=27 \[ -\frac{1}{4} \log \left (x^2+1\right )+\log (x)-\frac{1}{2} \log (x+1)-\frac{1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.0248596, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {894, 635, 203, 260} \[ -\frac{1}{4} \log \left (x^2+1\right )+\log (x)-\frac{1}{2} \log (x+1)-\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 894
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{x (1+x) \left (1+x^2\right )} \, dx &=\int \left (\frac{1}{x}-\frac{1}{2 (1+x)}+\frac{-1-x}{2 \left (1+x^2\right )}\right ) \, dx\\ &=\log (x)-\frac{1}{2} \log (1+x)+\frac{1}{2} \int \frac{-1-x}{1+x^2} \, dx\\ &=\log (x)-\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)+\log (x)-\frac{1}{2} \log (1+x)-\frac{1}{4} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0067695, size = 27, normalized size = 1. \[ -\frac{1}{4} \log \left (x^2+1\right )+\log (x)-\frac{1}{2} \log (x+1)-\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 22, normalized size = 0.8 \begin{align*} -{\frac{\arctan \left ( x \right ) }{2}}+\ln \left ( x \right ) -{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38402, size = 28, normalized size = 1.04 \begin{align*} -\frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) - \frac{1}{2} \, \log \left (x + 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02233, size = 82, normalized size = 3.04 \begin{align*} -\frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) - \frac{1}{2} \, \log \left (x + 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 = 0.157123, size = 22, normalized size = 0.81 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x + 1 \right )}}{2} - \frac{\log{\left (x^{2} + 1 \right )}}{4} - \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06179, size = 31, normalized size = 1.15 \begin{align*} -\frac{1}{2} \, \arctan \left (x\right ) - \frac{1}{4} \, \log \left (x^{2} + 1\right ) - \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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