Optimal. Leaf size=49 \[ 5 x+\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {15}{2} \tan ^{-1}(x)-\frac {7 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1682, 1690,
1180, 209} \begin {gather*} -\frac {15 \text {ArcTan}(x)}{2}-\frac {7 \text {ArcTan}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {\left (25 x^2+24\right ) x}{2 \left (x^4+3 x^2+2\right )}+5 x \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1180
Rule 1682
Rule 1690
Rubi steps
\begin {align*} \int \frac {x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx &=\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \frac {48-2 x^2-20 x^4}{2+3 x^2+x^4} \, dx\\ &=\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{4} \int \left (-20+\frac {2 \left (44+29 x^2\right )}{2+3 x^2+x^4}\right ) \, dx\\ &=5 x+\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {1}{2} \int \frac {44+29 x^2}{2+3 x^2+x^4} \, dx\\ &=5 x+\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-7 \int \frac {1}{2+x^2} \, dx-\frac {15}{2} \int \frac {1}{1+x^2} \, dx\\ &=5 x+\frac {x \left (24+25 x^2\right )}{2 \left (2+3 x^2+x^4\right )}-\frac {15}{2} \tan ^{-1}(x)-\frac {7 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 50, normalized size = 1.02 \begin {gather*} 5 x+\frac {24 x+25 x^3}{2 \left (2+3 x^2+x^4\right )}-\frac {15}{2} \tan ^{-1}(x)-\frac {7 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 41, normalized size = 0.84
method | result | size |
default | \(5 x -\frac {x}{2 \left (x^{2}+1\right )}-\frac {15 \arctan \left (x \right )}{2}+\frac {13 x}{x^{2}+2}-\frac {7 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{2}\) | \(41\) |
risch | \(5 x +\frac {\frac {25}{2} x^{3}+12 x}{x^{4}+3 x^{2}+2}-\frac {15 \arctan \left (x \right )}{2}-\frac {7 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{2}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 43, normalized size = 0.88 \begin {gather*} -\frac {7}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 5 \, x + \frac {25 \, x^{3} + 24 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {15}{2} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 64, normalized size = 1.31 \begin {gather*} \frac {10 \, x^{5} + 55 \, x^{3} - 7 \, \sqrt {2} {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 15 \, {\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (x\right ) + 44 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 48, normalized size = 0.98 \begin {gather*} 5 x + \frac {25 x^{3} + 24 x}{2 x^{4} + 6 x^{2} + 4} - \frac {15 \operatorname {atan}{\left (x \right )}}{2} - \frac {7 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.62, size = 43, normalized size = 0.88 \begin {gather*} -\frac {7}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 5 \, x + \frac {25 \, x^{3} + 24 \, x}{2 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac {15}{2} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 42, normalized size = 0.86 \begin {gather*} 5\,x-\frac {15\,\mathrm {atan}\left (x\right )}{2}-\frac {7\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{2}+\frac {\frac {25\,x^3}{2}+12\,x}{x^4+3\,x^2+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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