Optimal. Leaf size=66 \[ \frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {89 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{72 \sqrt {2}}+\frac {4 \log (x)}{9}-\frac {1}{9} \log \left (3+2 x^2+x^4\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {1677, 1660,
814, 648, 632, 210, 642} \begin {gather*} \frac {89 \text {ArcTan}\left (\frac {x^2+1}{\sqrt {2}}\right )}{72 \sqrt {2}}+\frac {25 \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac {1}{9} \log \left (x^4+2 x^2+3\right )+\frac {4 \log (x)}{9} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 814
Rule 1660
Rule 1677
Rubi steps
\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x \left (3+2 x^2+x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {4+x+3 x^2+5 x^3}{x \left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \text {Subst}\left (\int \frac {\frac {32}{3}+\frac {70 x}{3}}{x \left (3+2 x+x^2\right )} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {1}{16} \text {Subst}\left (\int \left (\frac {32}{9 x}-\frac {2 (-73+16 x)}{9 \left (3+2 x+x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {4 \log (x)}{9}-\frac {1}{72} \text {Subst}\left (\int \frac {-73+16 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {4 \log (x)}{9}-\frac {1}{9} \text {Subst}\left (\int \frac {2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )+\frac {89}{72} \text {Subst}\left (\int \frac {1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {4 \log (x)}{9}-\frac {1}{9} \log \left (3+2 x^2+x^4\right )-\frac {89}{36} \text {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac {25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac {89 \tan ^{-1}\left (\frac {1+x^2}{\sqrt {2}}\right )}{72 \sqrt {2}}+\frac {4 \log (x)}{9}-\frac {1}{9} \log \left (3+2 x^2+x^4\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.04, size = 93, normalized size = 1.41 \begin {gather*} \frac {1}{288} \left (-\frac {300 \left (-1+x^2\right )}{3+2 x^2+x^4}+128 \log (x)-\sqrt {2} \left (89 i+16 \sqrt {2}\right ) \log \left (1-i \sqrt {2}+x^2\right )+\sqrt {2} \left (89 i-16 \sqrt {2}\right ) \log \left (1+i \sqrt {2}+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 58, normalized size = 0.88
method | result | size |
default | \(-\frac {\frac {75 x^{2}}{4}-\frac {75}{4}}{18 \left (x^{4}+2 x^{2}+3\right )}-\frac {\ln \left (x^{4}+2 x^{2}+3\right )}{9}+\frac {89 \sqrt {2}\, \arctan \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4}\right )}{144}+\frac {4 \ln \left (x \right )}{9}\) | \(58\) |
risch | \(\frac {-\frac {25 x^{2}}{24}+\frac {25}{24}}{x^{4}+2 x^{2}+3}+\frac {4 \ln \left (x \right )}{9}-\frac {\ln \left (7921 x^{4}+15842 x^{2}+23763\right )}{9}+\frac {89 \sqrt {2}\, \arctan \left (\frac {\left (89 x^{2}+89\right ) \sqrt {2}}{178}\right )}{144}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 55, normalized size = 0.83 \begin {gather*} \frac {89}{144} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - \frac {25 \, {\left (x^{2} - 1\right )}}{24 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {1}{9} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac {2}{9} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 84, normalized size = 1.27 \begin {gather*} \frac {89 \, \sqrt {2} {\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) - 150 \, x^{2} - 16 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) + 64 \, {\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x\right ) + 150}{144 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 65, normalized size = 0.98 \begin {gather*} \frac {25 - 25 x^{2}}{24 x^{4} + 48 x^{2} + 72} + \frac {4 \log {\left (x \right )}}{9} - \frac {\log {\left (x^{4} + 2 x^{2} + 3 \right )}}{9} + \frac {89 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x^{2}}{2} + \frac {\sqrt {2}}{2} \right )}}{144} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.42, size = 62, normalized size = 0.94 \begin {gather*} \frac {89}{144} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{2} + 1\right )}\right ) + \frac {8 \, x^{4} - 59 \, x^{2} + 99}{72 \, {\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac {1}{9} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac {2}{9} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.91, size = 59, normalized size = 0.89 \begin {gather*} \frac {4\,\ln \left (x\right )}{9}-\frac {\ln \left (x^4+2\,x^2+3\right )}{9}-\frac {\frac {25\,x^2}{24}-\frac {25}{24}}{x^4+2\,x^2+3}+\frac {89\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x^2}{2}+\frac {\sqrt {2}}{2}\right )}{144} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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