Optimal. Leaf size=17 \[ -\frac {1}{2} \log \left (x^2+1\right )+4 \log (x)+\tan ^{-1}(x) \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1593, 1802, 635, 203, 260} \begin {gather*} -\frac {1}{2} \log \left (x^2+1\right )+4 \log (x)+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1593
Rule 1802
Rubi steps
\begin {align*} \int \frac {4+x+3 x^2}{x+x^3} \, dx &=\int \frac {4+x+3 x^2}{x \left (1+x^2\right )} \, dx\\ &=\int \left (\frac {4}{x}+\frac {1-x}{1+x^2}\right ) \, dx\\ &=4 \log (x)+\int \frac {1-x}{1+x^2} \, dx\\ &=4 \log (x)+\int \frac {1}{1+x^2} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=\tan ^{-1}(x)+4 \log (x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \log \left (x^2+1\right )+4 \log (x)+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4+x+3 x^2}{x+x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.26, size = 15, normalized size = 0.88 \begin {gather*} \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 4 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 16, normalized size = 0.94 \begin {gather*} \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 4 \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \arctan \relax (x )+4 \ln \relax (x )-\frac {\ln \left (x^{2}+1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 15, normalized size = 0.88 \begin {gather*} \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) + 4 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.28, size = 23, normalized size = 1.35 \begin {gather*} 4\,\ln \relax (x)+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 15, normalized size = 0.88 \begin {gather*} 4 \log {\relax (x )} - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \operatorname {atan}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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