Optimal. Leaf size=23 \[ \frac{1}{2} \log \left (x^2+4\right )-\frac{3}{2} \tan ^{-1}\left (\frac{x}{2}\right )+\tan ^{-1}(x) \]
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Rubi [A] time = 0.0309383, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1673, 1166, 203, 1247, 626, 31} \[ \frac{1}{2} \log \left (x^2+4\right )-\frac{3}{2} \tan ^{-1}\left (\frac{x}{2}\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1166
Rule 203
Rule 1247
Rule 626
Rule 31
Rubi steps
\begin{align*} \int \frac{1+x-2 x^2+x^3}{4+5 x^2+x^4} \, dx &=\int \frac{1-2 x^2}{4+5 x^2+x^4} \, dx+\int \frac{x \left (1+x^2\right )}{4+5 x^2+x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1+x}{4+5 x+x^2} \, dx,x,x^2\right )-3 \int \frac{1}{4+x^2} \, dx+\int \frac{1}{1+x^2} \, dx\\ &=-\frac{3}{2} \tan ^{-1}\left (\frac{x}{2}\right )+\tan ^{-1}(x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{4+x} \, dx,x,x^2\right )\\ &=-\frac{3}{2} \tan ^{-1}\left (\frac{x}{2}\right )+\tan ^{-1}(x)+\frac{1}{2} \log \left (4+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0096138, size = 23, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+4\right )-\frac{3}{2} \tan ^{-1}\left (\frac{x}{2}\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 18, normalized size = 0.8 \begin{align*} -{\frac{3}{2}\arctan \left ({\frac{x}{2}} \right ) }+\arctan \left ( x \right ) +{\frac{\ln \left ({x}^{2}+4 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41103, size = 23, normalized size = 1. \begin{align*} -\frac{3}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79896, size = 69, normalized size = 3. \begin{align*} -\frac{3}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.155445, size = 19, normalized size = 0.83 \begin{align*} \frac{\log{\left (x^{2} + 4 \right )}}{2} - \frac{3 \operatorname{atan}{\left (\frac{x}{2} \right )}}{2} + \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06224, size = 23, normalized size = 1. \begin{align*} -\frac{3}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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