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 = 0.113411, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {6725, 203, 199} \[ -\frac{13 x}{24 \left (x^2+4\right )}+\frac{25}{144} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 6725
Rule 203
Rule 199
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*}
Mathematica [A] time = 0.0161005, size = 29, normalized size = 1. \[ -\frac{13 x}{24 \left (x^2+4\right )}+\frac{25}{144} \tan ^{-1}\left (\frac{x}{2}\right )+\frac{1}{9} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 22, normalized size = 0.8 \begin{align*} -{\frac{13\,x}{24\,{x}^{2}+96}}+{\frac{25}{144}\arctan \left ({\frac{x}{2}} \right ) }+{\frac{\arctan \left ( x \right ) }{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40065, size = 28, normalized size = 0.97 \begin{align*} -\frac{13 \, x}{24 \,{\left (x^{2} + 4\right )}} + \frac{25}{144} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{9} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90528, size = 105, normalized size = 3.62 \begin{align*} \frac{25 \,{\left (x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, x\right ) + 16 \,{\left (x^{2} + 4\right )} \arctan \left (x\right ) - 78 \, x}{144 \,{\left (x^{2} + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.156617, size = 22, normalized size = 0.76 \begin{align*} - \frac{13 x}{24 x^{2} + 96} + \frac{25 \operatorname{atan}{\left (\frac{x}{2} \right )}}{144} + \frac{\operatorname{atan}{\left (x \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04939, size = 28, normalized size = 0.97 \begin{align*} -\frac{13 \, x}{24 \,{\left (x^{2} + 4\right )}} + \frac{25}{144} \, \arctan \left (\frac{1}{2} \, x\right ) + \frac{1}{9} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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