Optimal. Leaf size=32 \[ \frac {1}{4 \left (x^4+2 x^2+2\right )}+\frac {1}{4} \log \left (x^4+2 x^2+2\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1247, 686, 628} \[ \frac {1}{4} \log \left (x^4+2 x^2+2\right )-\frac {\left (x^2+1\right )^2}{4 \left (x^4+2 x^2+2\right )} \]
Antiderivative was successfully verified.
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Rule 628
Rule 686
Rule 1247
Rubi steps
\begin {align*} \int \frac {x \left (1+x^2\right )^3}{\left (2+2 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1+x)^3}{\left (2+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {\left (1+x^2\right )^2}{4 \left (2+2 x^2+x^4\right )}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x}{2+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (1+x^2\right )^2}{4 \left (2+2 x^2+x^4\right )}+\frac {1}{4} \log \left (2+2 x^2+x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 0.81 \[ \frac {1}{4} \left (\frac {1}{\left (x^2+1\right )^2+1}+\log \left (\left (x^2+1\right )^2+1\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 32, normalized size = 1.00 \[ \frac {1}{4 \left (x^4+2 x^2+2\right )}+\frac {1}{4} \log \left (x^4+2 x^2+2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 38, normalized size = 1.19 \[ \frac {{\left (x^{4} + 2 \, x^{2} + 2\right )} \log \left (x^{4} + 2 \, x^{2} + 2\right ) + 1}{4 \, {\left (x^{4} + 2 \, x^{2} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.61, size = 32, normalized size = 1.00 \[ \frac {1}{4 \, {\left (x^{4} + 2 \, x^{2} + 2\right )}} - \frac {1}{4} \, \log \left (\frac {1}{2 \, {\left (x^{4} + 2 \, x^{2} + 2\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 29, normalized size = 0.91
method | result | size |
default | \(\frac {1}{4 x^{4}+8 x^{2}+8}+\frac {\ln \left (x^{4}+2 x^{2}+2\right )}{4}\) | \(29\) |
norman | \(\frac {1}{4 x^{4}+8 x^{2}+8}+\frac {\ln \left (x^{4}+2 x^{2}+2\right )}{4}\) | \(29\) |
risch | \(\frac {1}{4 x^{4}+8 x^{2}+8}+\frac {\ln \left (x^{4}+2 x^{2}+2\right )}{4}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 28, normalized size = 0.88 \[ \frac {1}{4 \, {\left (x^{4} + 2 \, x^{2} + 2\right )}} + \frac {1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 28, normalized size = 0.88 \[ \frac {\ln \left (x^4+2\,x^2+2\right )}{4}+\frac {1}{4\,\left (x^4+2\,x^2+2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 26, normalized size = 0.81 \[ \frac {\log {\left (x^{4} + 2 x^{2} + 2 \right )}}{4} + \frac {1}{4 x^{4} + 8 x^{2} + 8} \]
Verification of antiderivative is not currently implemented for this CAS.
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