Optimal. Leaf size=22 \[ \frac {1}{2 \left (x^2+1\right )}+\frac {1}{2} \log \left (x^2+1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {28, 266, 43} \begin {gather*} \frac {1}{2 \left (x^2+1\right )}+\frac {1}{2} \log \left (x^2+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{1+2 x^2+x^4} \, dx &=\int \frac {x^3}{\left (1+x^2\right )^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(1+x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2 \left (1+x^2\right )}+\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 0.82 \begin {gather*} \frac {1}{2} \left (\frac {1}{x^2+1}+\log \left (x^2+1\right )\right ) \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 {x^3}{1+2 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.86, size = 23, normalized size = 1.05 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 1}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 18, normalized size = 0.82 \begin {gather*} \frac {1}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \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 = 0.86 \begin {gather*} \frac {\ln \left (x^{2}+1\right )}{2}+\frac {1}{2 x^{2}+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 18, normalized size = 0.82 \begin {gather*} \frac {1}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 18, normalized size = 0.82 \begin {gather*} \frac {\ln \left (x^2+1\right )}{2}+\frac {1}{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 = 0.09, size = 15, normalized size = 0.68 \begin {gather*} \frac {\log {\left (x^{2} + 1 \right )}}{2} + \frac {1}{2 x^{2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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