Optimal. Leaf size=19 \[ \frac {x}{2 \left (1+x^2\right )}+\frac {1}{2} \tan ^{-1}(x) \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {205, 209}
\begin {gather*} \frac {x}{2 \left (x^2+1\right )}+\frac {1}{2} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 209
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x^2\right )^2} \, dx &=\frac {x}{2 \left (1+x^2\right )}+\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{2 \left (1+x^2\right )}+\frac {1}{2} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{2} \left (\frac {x}{1+x^2}+\tan ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.78, size = 19, normalized size = 1.00 \begin {gather*} \frac {x+\text {ArcTan}\left [x\right ] \left (1+x^2\right )}{2+2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 16, normalized size = 0.84
method | result | size |
default | \(\frac {x}{2 x^{2}+2}+\frac {\arctan \left (x \right )}{2}\) | \(16\) |
risch | \(\frac {x}{2 x^{2}+2}+\frac {\arctan \left (x \right )}{2}\) | \(16\) |
meijerg | \(\frac {x}{2 x^{2}+2}+\frac {\arctan \left (x \right )}{2}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 12, normalized size = 0.63 \begin {gather*} \frac {x}{2 x^{2} + 2} + \frac {\operatorname {atan}{\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} \frac {x}{2 \left (x^{2}+1\right )}+\frac {\arctan x}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 16, normalized size = 0.84 \begin {gather*} \frac {\mathrm {atan}\left (x\right )}{2}+\frac {x}{2\,\left (x^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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