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 = 0.02, 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 = {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 203
Rule 260
Rule 635
Rule 894
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 \[ -\frac {1}{4} \log \left (x^2+1\right )+\log (x)-\frac {1}{2} \log (x+1)-\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 27, normalized size = 1.00 \[ -\frac {1}{4} \log \left (x^2+1\right )+\log (x)-\frac {1}{2} \log (x+1)-\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 21, normalized size = 0.78 \[ -\frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left (x + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.78, size = 23, normalized size = 0.85 \[ -\frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 22, normalized size = 0.81
method | result | size |
default | \(-\frac {\arctan \relax (x )}{2}+\ln \relax (x )-\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
risch | \(-\frac {\arctan \relax (x )}{2}+\ln \relax (x )-\frac {\ln \left (1+x \right )}{2}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 21, normalized size = 0.78 \[ -\frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) - \frac {1}{2} \, \log \left (x + 1\right ) + \log \relax (x) \]
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 \[ \ln \relax (x)-\frac {\ln \left (x+1\right )}{2}+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 22, normalized size = 0.81 \[ \log {\relax (x )} - \frac {\log {\left (x + 1 \right )}}{2} - \frac {\log {\left (x^{2} + 1 \right )}}{4} - \frac {\operatorname {atan}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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