Optimal. Leaf size=14 \[ \frac {x}{x^2+1}+2 \tan ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {385, 203} \begin {gather*} \frac {x}{x^2+1}+2 \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 385
Rubi steps
\begin {align*} \int \frac {3+x^2}{\left (1+x^2\right )^2} \, dx &=\frac {x}{1+x^2}+2 \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{1+x^2}+2 \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{x^2+1}+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 {3+x^2}{\left (1+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 19, normalized size = 1.36 \begin {gather*} \frac {2 \, {\left (x^{2} + 1\right )} \arctan \relax (x) + x}{x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{x^{2} + 1} + 2 \, \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 = 15, normalized size = 1.07 \begin {gather*} \frac {x}{x^{2}+1}+2 \arctan \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.29, size = 14, normalized size = 1.00 \begin {gather*} \frac {x}{x^{2} + 1} + 2 \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 14, normalized size = 1.00 \begin {gather*} 2\,\mathrm {atan}\relax (x)+\frac {x}{x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 10, normalized size = 0.71 \begin {gather*} \frac {x}{x^{2} + 1} + 2 \operatorname {atan}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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