Optimal. Leaf size=29 \[ -\frac {13 x}{24 \left (x^2+4\right )}+\frac {25}{144} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{9} \tan ^{-1}(x) \]
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Rubi [A] time = 0.11, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6725, 203, 199} \begin {gather*} -\frac {13 x}{24 \left (x^2+4\right )}+\frac {25}{144} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{9} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 203
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^2+x^4}{\left (1+x^2\right ) \left (4+x^2\right )^2} \, dx &=\int \left (\frac {1}{9 \left (1+x^2\right )}-\frac {13}{3 \left (4+x^2\right )^2}+\frac {8}{9 \left (4+x^2\right )}\right ) \, dx\\ &=\frac {1}{9} \int \frac {1}{1+x^2} \, dx+\frac {8}{9} \int \frac {1}{4+x^2} \, dx-\frac {13}{3} \int \frac {1}{\left (4+x^2\right )^2} \, dx\\ &=-\frac {13 x}{24 \left (4+x^2\right )}+\frac {4}{9} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{9} \tan ^{-1}(x)-\frac {13}{24} \int \frac {1}{4+x^2} \, dx\\ &=-\frac {13 x}{24 \left (4+x^2\right )}+\frac {25}{144} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{9} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.00 \begin {gather*} -\frac {13 x}{24 \left (x^2+4\right )}+\frac {25}{144} \tan ^{-1}\left (\frac {x}{2}\right )+\frac {1}{9} \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 {1+x^2+x^4}{\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 = 1.06, size = 33, normalized size = 1.14 \begin {gather*} \frac {25 \, {\left (x^{2} + 4\right )} \arctan \left (\frac {1}{2} \, x\right ) + 16 \, {\left (x^{2} + 4\right )} \arctan \relax (x) - 78 \, x}{144 \, {\left (x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 21, normalized size = 0.72 \begin {gather*} -\frac {13 \, x}{24 \, {\left (x^{2} + 4\right )}} + \frac {25}{144} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{9} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.76 \begin {gather*} -\frac {13 x}{24 \left (x^{2}+4\right )}+\frac {\arctan \relax (x )}{9}+\frac {25 \arctan \left (\frac {x}{2}\right )}{144} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.09, size = 21, normalized size = 0.72 \begin {gather*} -\frac {13 \, x}{24 \, {\left (x^{2} + 4\right )}} + \frac {25}{144} \, \arctan \left (\frac {1}{2} \, x\right ) + \frac {1}{9} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 23, normalized size = 0.79 \begin {gather*} \frac {25\,\mathrm {atan}\left (\frac {x}{2}\right )}{144}+\frac {\mathrm {atan}\relax (x)}{9}-\frac {13\,x}{24\,\left (x^2+4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 22, normalized size = 0.76 \begin {gather*} - \frac {13 x}{24 x^{2} + 96} + \frac {25 \operatorname {atan}{\left (\frac {x}{2} \right )}}{144} + \frac {\operatorname {atan}{\relax (x )}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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