Optimal. Leaf size=23 \[ -\frac {1}{4 \left (1+x^2\right )^2}+\frac {2}{1+x^2}+\tan ^{-1}(x) \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1828, 12, 209}
\begin {gather*} \text {ArcTan}(x)+\frac {2}{x^2+1}-\frac {1}{4 \left (x^2+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 1828
Rubi steps
\begin {align*} \int \frac {1-3 x+2 x^2-4 x^3+x^4}{\left (1+x^2\right )^3} \, dx &=-\frac {1}{4 \left (1+x^2\right )^2}-\frac {1}{4} \int \frac {-4+16 x-4 x^2}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {1}{4 \left (1+x^2\right )^2}+\frac {2}{1+x^2}+\frac {1}{8} \int \frac {8}{1+x^2} \, dx\\ &=-\frac {1}{4 \left (1+x^2\right )^2}+\frac {2}{1+x^2}+\int \frac {1}{1+x^2} \, dx\\ &=-\frac {1}{4 \left (1+x^2\right )^2}+\frac {2}{1+x^2}+\tan ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} -\frac {1}{4 \left (1+x^2\right )^2}+\frac {2}{1+x^2}+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 19, normalized size = 0.83
method | result | size |
default | \(\frac {2 x^{2}+\frac {7}{4}}{\left (x^{2}+1\right )^{2}}+\arctan \left (x \right )\) | \(19\) |
risch | \(\frac {2 x^{2}+\frac {7}{4}}{\left (x^{2}+1\right )^{2}}+\arctan \left (x \right )\) | \(19\) |
meijerg | \(\frac {x \left (3 x^{2}+5\right )}{8 \left (x^{2}+1\right )^{2}}+\arctan \left (x \right )-\frac {x \left (25 x^{2}+15\right )}{40 \left (x^{2}+1\right )^{2}}-\frac {x^{4}}{\left (x^{2}+1\right )^{2}}-\frac {x \left (-3 x^{2}+3\right )}{12 \left (x^{2}+1\right )^{2}}-\frac {3 x^{2} \left (x^{2}+2\right )}{4 \left (x^{2}+1\right )^{2}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.93, size = 24, normalized size = 1.04 \begin {gather*} \frac {8 \, x^{2} + 7}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} + \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 35, normalized size = 1.52 \begin {gather*} \frac {8 \, x^{2} + 4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (x\right ) + 7}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 20, normalized size = 0.87 \begin {gather*} \frac {8 x^{2} + 7}{4 x^{4} + 8 x^{2} + 4} + \operatorname {atan}{\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.89, size = 19, normalized size = 0.83 \begin {gather*} \frac {8 \, x^{2} + 7}{4 \, {\left (x^{2} + 1\right )}^{2}} + \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 23, normalized size = 1.00 \begin {gather*} \mathrm {atan}\left (x\right )+\frac {2\,x^2+\frac {7}{4}}{x^4+2\,x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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