Optimal. Leaf size=27 \[ -\frac {1}{2} \tan ^{-1}(x)+\log (x)-\frac {1}{2} \log (1+x)-\frac {1}{4} \log \left (1+x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {908, 649, 209,
266} \begin {gather*} -\frac {\text {ArcTan}(x)}{2}-\frac {1}{4} \log \left (x^2+1\right )+\log (x)-\frac {1}{2} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 908
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*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tan ^{-1}(x)+\log (x)-\frac {1}{2} \log (1+x)-\frac {1}{4} \log \left (1+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 22, normalized size = 0.81
method | result | size |
default | \(-\frac {\arctan \left (x \right )}{2}+\ln \left (x \right )-\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
risch | \(-\frac {\arctan \left (x \right )}{2}+\ln \left (x \right )-\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.74, size = 21, normalized size = 0.78 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 21, normalized size = 0.78 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 22, normalized size = 0.81 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 23, normalized size = 0.85 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 27, normalized size = 1.00 \begin {gather*} \ln \left (x\right )-\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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