Optimal. Leaf size=23 \[ \frac {2}{x^2+1}-\frac {1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]
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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 = {1814, 12, 203} \[ \frac {2}{x^2+1}-\frac {1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 1814
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 \[ \frac {2}{x^2+1}-\frac {1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 35, normalized size = 1.52 \[ \frac {8 \, x^{2} + 4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \relax (x) + 7}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 19, normalized size = 0.83 \[ \frac {8 \, x^{2} + 7}{4 \, {\left (x^{2} + 1\right )}^{2}} + \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 19, normalized size = 0.83 \[ \arctan \relax (x )+\frac {2 x^{2}+\frac {7}{4}}{\left (x^{2}+1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 24, normalized size = 1.04 \[ \frac {8 \, x^{2} + 7}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} + \arctan \relax (x) \]
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 \[ \mathrm {atan}\relax (x)+\frac {2\,x^2+\frac {7}{4}}{x^4+2\,x^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 20, normalized size = 0.87 \[ \frac {8 x^{2} + 7}{4 x^{4} + 8 x^{2} + 4} + \operatorname {atan}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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