Optimal. Leaf size=32 \[ \frac {1}{4 \left (2+2 x^2+x^4\right )}+\frac {1}{4} \log \left (2+2 x^2+x^4\right ) \]
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Rubi [A]
time = 0.02, 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 = {1261, 700, 642}
\begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 700
Rule 1261
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} \text {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} \text {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 = 0.01, size = 26, normalized size = 0.81 \begin {gather*} \frac {1}{4} \left (\frac {1}{1+\left (1+x^2\right )^2}+\log \left (1+\left (1+x^2\right )^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, 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 = 2.08, size = 28, normalized size = 0.88 \begin {gather*} \frac {1}{4 \, {\left (x^{4} + 2 \, x^{2} + 2\right )}} + \frac {1}{4} \, \log \left (x^{4} + 2 \, x^{2} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 38, normalized size = 1.19 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 26, normalized size = 0.81 \begin {gather*} \frac {\log {\left (x^{4} + 2 x^{2} + 2 \right )}}{4} + \frac {1}{4 x^{4} + 8 x^{2} + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.71, size = 32, normalized size = 1.00 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} \frac {\ln \left (x^4+2\,x^2+2\right )}{4}+\frac {1}{4\,\left (x^4+2\,x^2+2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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