Optimal. Leaf size=32 \[ \frac {1}{x^2+4}+\log \left (x^2+4\right )-2 \log (x)+\frac {1}{2} \tan ^{-1}\left (\frac {x}{2}\right )+2 \tan ^{-1}(x) \]
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Rubi [A] time = 0.25, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {6725, 203, 261, 635, 260} \begin {gather*} \frac {1}{x^2+4}+\log \left (x^2+4\right )-2 \log (x)+\frac {1}{2} \tan ^{-1}\left (\frac {x}{2}\right )+2 \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 261
Rule 635
Rule 6725
Rubi steps
\begin {align*} \int \frac {-32+36 x-42 x^2+21 x^3-10 x^4+3 x^5}{x \left (1+x^2\right ) \left (4+x^2\right )^2} \, dx &=\int \left (-\frac {2}{x}+\frac {2}{1+x^2}-\frac {2 x}{\left (4+x^2\right )^2}+\frac {1+2 x}{4+x^2}\right ) \, dx\\ &=-2 \log (x)+2 \int \frac {1}{1+x^2} \, dx-2 \int \frac {x}{\left (4+x^2\right )^2} \, dx+\int \frac {1+2 x}{4+x^2} \, dx\\ &=\frac {1}{4+x^2}+2 \tan ^{-1}(x)-2 \log (x)+2 \int \frac {x}{4+x^2} \, dx+\int \frac {1}{4+x^2} \, dx\\ &=\frac {1}{4+x^2}+\frac {1}{2} \tan ^{-1}\left (\frac {x}{2}\right )+2 \tan ^{-1}(x)-2 \log (x)+\log \left (4+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 1.00 \begin {gather*} \frac {1}{x^2+4}+\log \left (x^2+4\right )-2 \log (x)+\frac {1}{2} \tan ^{-1}\left (\frac {x}{2}\right )+2 \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 {-32+36 x-42 x^2+21 x^3-10 x^4+3 x^5}{x \left (1+x^2\right ) \left (4+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.22, size = 52, normalized size = 1.62 \begin {gather*} \frac {{\left (x^{2} + 4\right )} \arctan \left (\frac {1}{2} \, x\right ) + 4 \, {\left (x^{2} + 4\right )} \arctan \relax (x) + 2 \, {\left (x^{2} + 4\right )} \log \left (x^{2} + 4\right ) - 4 \, {\left (x^{2} + 4\right )} \log \relax (x) + 2}{2 \, {\left (x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 29, normalized size = 0.91 \begin {gather*} \frac {1}{x^{2} + 4} + \frac {1}{2} \, \arctan \left (\frac {1}{2} \, x\right ) + 2 \, \arctan \relax (x) + \log \left (x^{2} + 4\right ) - 2 \, \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 = 0.01, size = 29, normalized size = 0.91 \begin {gather*} 2 \arctan \relax (x )+\frac {\arctan \left (\frac {x}{2}\right )}{2}-2 \ln \relax (x )+\ln \left (x^{2}+4\right )+\frac {1}{x^{2}+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.02, size = 28, normalized size = 0.88 \begin {gather*} \frac {1}{x^{2} + 4} + \frac {1}{2} \, \arctan \left (\frac {1}{2} \, x\right ) + 2 \, \arctan \relax (x) + \log \left (x^{2} + 4\right ) - 2 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 44, normalized size = 1.38 \begin {gather*} \frac {1}{x^2+4}-2\,\ln \relax (x)-2\,\mathrm {atan}\left (\frac {328000}{7\,\left (36288\,x-19584\right )}+\frac {34}{63}\right )+\ln \left (x-2{}\mathrm {i}\right )\,\left (1-\frac {1}{4}{}\mathrm {i}\right )+\ln \left (x+2{}\mathrm {i}\right )\,\left (1+\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 29, normalized size = 0.91 \begin {gather*} - 2 \log {\relax (x )} + \log {\left (x^{2} + 4 \right )} + \frac {\operatorname {atan}{\left (\frac {x}{2} \right )}}{2} + 2 \operatorname {atan}{\relax (x )} + \frac {1}{x^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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