Optimal. Leaf size=12 \[ \tan ^{-1}(x)-\frac {1}{x^2+1} \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.83, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {27, 723, 203} \[ \tan ^{-1}(x)-\frac {(1-x) (x+1)}{2 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 27
Rule 203
Rule 723
Rubi steps
\begin {align*} \int \frac {1+2 x+x^2}{\left (1+x^2\right )^2} \, dx &=\int \frac {(1+x)^2}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {(1-x) (1+x)}{2 \left (1+x^2\right )}+\int \frac {1}{1+x^2} \, dx\\ &=-\frac {(1-x) (1+x)}{2 \left (1+x^2\right )}+\tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ \tan ^{-1}(x)-\frac {1}{x^2+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 18, normalized size = 1.50 \[ \frac {{\left (x^{2} + 1\right )} \arctan \relax (x) - 1}{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 12, normalized size = 1.00 \[ -\frac {1}{x^{2} + 1} + \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 1.08 \[ \arctan \relax (x )-\frac {1}{x^{2}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 12, normalized size = 1.00 \[ -\frac {1}{x^{2} + 1} + \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 12, normalized size = 1.00 \[ \mathrm {atan}\relax (x)-\frac {1}{x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 8, normalized size = 0.67 \[ \operatorname {atan}{\relax (x )} - \frac {1}{x^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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