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 = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2073, 261, 203} \begin {gather*} \frac {2}{x^2+1}-\frac {1}{4 \left (x^2+1\right )^2}+\tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 261
Rule 2073
Rubi steps
\begin {align*} \int \frac {1-3 x+2 x^2-4 x^3+x^4}{1+3 x^2+3 x^4+x^6} \, dx &=\int \left (\frac {x}{\left (1+x^2\right )^3}-\frac {4 x}{\left (1+x^2\right )^2}+\frac {1}{1+x^2}\right ) \, dx\\ &=-\left (4 \int \frac {x}{\left (1+x^2\right )^2} \, dx\right )+\int \frac {x}{\left (1+x^2\right )^3} \, dx+\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 {2}{x^2+1}-\frac {1}{4 \left (x^2+1\right )^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 {1-3 x+2 x^2-4 x^3+x^4}{1+3 x^2+3 x^4+x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.88, size = 35, normalized size = 1.52 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 19, normalized size = 0.83 \begin {gather*} \frac {8 \, x^{2} + 7}{4 \, {\left (x^{2} + 1\right )}^{2}} + \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 0.83 \begin {gather*} \arctan \relax (x )+\frac {2 x^{2}+\frac {7}{4}}{\left (x^{2}+1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 24, normalized size = 1.04 \begin {gather*} \frac {8 \, x^{2} + 7}{4 \, {\left (x^{4} + 2 \, x^{2} + 1\right )}} + \arctan \relax (x) \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}\relax (x)+\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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sympy [A] time = 0.13, size = 20, normalized size = 0.87 \begin {gather*} \frac {8 x^{2} + 7}{4 x^{4} + 8 x^{2} + 4} + \operatorname {atan}{\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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