Optimal. Leaf size=79 \[ -\frac {1}{2 x}+\frac {x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (547+347 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac {189}{8} \tan ^{-1}(x)-\frac {1119 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{32 \sqrt {2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1683, 1678,
209} \begin {gather*} \frac {189 \text {ArcTan}(x)}{8}-\frac {1119 \text {ArcTan}\left (\frac {x}{\sqrt {2}}\right )}{32 \sqrt {2}}+\frac {x \left (11 x^2+9\right )}{8 \left (x^4+3 x^2+2\right )^2}-\frac {x \left (347 x^2+547\right )}{32 \left (x^4+3 x^2+2\right )}-\frac {1}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 1678
Rule 1683
Rubi steps
\begin {align*} \int \frac {4+x^2+3 x^4+5 x^6}{x^2 \left (2+3 x^2+x^4\right )^3} \, dx &=\frac {x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac {1}{8} \int \frac {-16+29 x^2-55 x^4}{x^2 \left (2+3 x^2+x^4\right )^2} \, dx\\ &=\frac {x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (547+347 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \frac {32+441 x^2-347 x^4}{x^2 \left (2+3 x^2+x^4\right )} \, dx\\ &=\frac {x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (547+347 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac {1}{32} \int \left (\frac {16}{x^2}+\frac {756}{1+x^2}-\frac {1119}{2+x^2}\right ) \, dx\\ &=-\frac {1}{2 x}+\frac {x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (547+347 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac {189}{8} \int \frac {1}{1+x^2} \, dx-\frac {1119}{32} \int \frac {1}{2+x^2} \, dx\\ &=-\frac {1}{2 x}+\frac {x \left (9+11 x^2\right )}{8 \left (2+3 x^2+x^4\right )^2}-\frac {x \left (547+347 x^2\right )}{32 \left (2+3 x^2+x^4\right )}+\frac {189}{8} \tan ^{-1}(x)-\frac {1119 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{32 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 63, normalized size = 0.80 \begin {gather*} \frac {1}{64} \left (-\frac {2 \left (64+1250 x^2+2499 x^4+1684 x^6+363 x^8\right )}{x \left (2+3 x^2+x^4\right )^2}+1512 \tan ^{-1}(x)-1119 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 58, normalized size = 0.73
method | result | size |
risch | \(\frac {-\frac {363}{32} x^{8}-\frac {421}{8} x^{6}-\frac {2499}{32} x^{4}-\frac {625}{16} x^{2}-2}{x \left (x^{4}+3 x^{2}+2\right )^{2}}+\frac {189 \arctan \left (x \right )}{8}-\frac {1119 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{64}\) | \(56\) |
default | \(\frac {-\frac {35}{8} x^{3}-\frac {37}{8} x}{\left (x^{2}+1\right )^{2}}+\frac {189 \arctan \left (x \right )}{8}-\frac {1}{2 x}-\frac {\frac {207}{16} x^{3}+\frac {233}{8} x}{2 \left (x^{2}+2\right )^{2}}-\frac {1119 \arctan \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{64}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 65, normalized size = 0.82 \begin {gather*} -\frac {1119}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {363 \, x^{8} + 1684 \, x^{6} + 2499 \, x^{4} + 1250 \, x^{2} + 64}{32 \, {\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )}} + \frac {189}{8} \, \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 = 108, normalized size = 1.37 \begin {gather*} -\frac {726 \, x^{8} + 3368 \, x^{6} + 4998 \, x^{4} + 1119 \, \sqrt {2} {\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 2500 \, x^{2} - 1512 \, {\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )} \arctan \left (x\right ) + 128}{64 \, {\left (x^{9} + 6 \, x^{7} + 13 \, x^{5} + 12 \, x^{3} + 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 71, normalized size = 0.90 \begin {gather*} \frac {- 363 x^{8} - 1684 x^{6} - 2499 x^{4} - 1250 x^{2} - 64}{32 x^{9} + 192 x^{7} + 416 x^{5} + 384 x^{3} + 128 x} + \frac {189 \operatorname {atan}{\left (x \right )}}{8} - \frac {1119 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{64} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.46, size = 55, normalized size = 0.70 \begin {gather*} -\frac {1119}{64} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {347 \, x^{7} + 1588 \, x^{5} + 2291 \, x^{3} + 1058 \, x}{32 \, {\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac {1}{2 \, x} + \frac {189}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.92, size = 65, normalized size = 0.82 \begin {gather*} \frac {189\,\mathrm {atan}\left (x\right )}{8}-\frac {1119\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{64}-\frac {\frac {363\,x^8}{32}+\frac {421\,x^6}{8}+\frac {2499\,x^4}{32}+\frac {625\,x^2}{16}+2}{x^9+6\,x^7+13\,x^5+12\,x^3+4\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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