Optimal. Leaf size=36 \[ -\frac {4}{x}-\frac {7 x}{4 \left (1+x^2\right )^2}-\frac {25 x}{8 \left (1+x^2\right )}-\frac {57}{8} \tan ^{-1}(x) \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1274, 467, 464,
209} \begin {gather*} -\frac {57 \text {ArcTan}(x)}{8}-\frac {25 x}{8 \left (x^2+1\right )}-\frac {7 x}{4 \left (x^2+1\right )^2}-\frac {4}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 464
Rule 467
Rule 1274
Rubi steps
\begin {align*} \int \frac {4+3 x^4}{x^2 \left (1+x^2\right )^3} \, dx &=-\frac {7 x}{4 \left (1+x^2\right )^2}-\frac {1}{4} \int \frac {-16+9 x^2}{x^2 \left (1+x^2\right )^2} \, dx\\ &=-\frac {7 x}{4 \left (1+x^2\right )^2}-\frac {25 x}{8 \left (1+x^2\right )}+\frac {1}{8} \int \frac {32-25 x^2}{x^2 \left (1+x^2\right )} \, dx\\ &=-\frac {4}{x}-\frac {7 x}{4 \left (1+x^2\right )^2}-\frac {25 x}{8 \left (1+x^2\right )}-\frac {57}{8} \int \frac {1}{1+x^2} \, dx\\ &=-\frac {4}{x}-\frac {7 x}{4 \left (1+x^2\right )^2}-\frac {25 x}{8 \left (1+x^2\right )}-\frac {57}{8} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.92 \begin {gather*} -\frac {32+103 x^2+57 x^4}{8 x \left (1+x^2\right )^2}-\frac {57}{8} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 29, normalized size = 0.81
method | result | size |
default | \(-\frac {4}{x}-\frac {\frac {25}{8} x^{3}+\frac {39}{8} x}{\left (x^{2}+1\right )^{2}}-\frac {57 \arctan \left (x \right )}{8}\) | \(29\) |
risch | \(\frac {-\frac {57}{8} x^{4}-\frac {103}{8} x^{2}-4}{\left (x^{2}+1\right )^{2} x}-\frac {57 \arctan \left (x \right )}{8}\) | \(29\) |
meijerg | \(-\frac {15 x^{4}+25 x^{2}+8}{2 x \left (x^{2}+1\right )^{2}}-\frac {57 \arctan \left (x \right )}{8}-\frac {x \left (-3 x^{2}+3\right )}{8 \left (x^{2}+1\right )^{2}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.94, size = 31, normalized size = 0.86 \begin {gather*} -\frac {57 \, x^{4} + 103 \, x^{2} + 32}{8 \, {\left (x^{5} + 2 \, x^{3} + x\right )}} - \frac {57}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 40, normalized size = 1.11 \begin {gather*} -\frac {57 \, x^{4} + 103 \, x^{2} + 57 \, {\left (x^{5} + 2 \, x^{3} + x\right )} \arctan \left (x\right ) + 32}{8 \, {\left (x^{5} + 2 \, x^{3} + x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 32, normalized size = 0.89 \begin {gather*} \frac {- 57 x^{4} - 103 x^{2} - 32}{8 x^{5} + 16 x^{3} + 8 x} - \frac {57 \operatorname {atan}{\left (x \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 28, normalized size = 0.78 \begin {gather*} -\frac {25 \, x^{3} + 39 \, x}{8 \, {\left (x^{2} + 1\right )}^{2}} - \frac {4}{x} - \frac {57}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 29, normalized size = 0.81 \begin {gather*} -\frac {57\,\mathrm {atan}\left (x\right )}{8}-\frac {\frac {57\,x^4}{8}+\frac {103\,x^2}{8}+4}{x\,{\left (x^2+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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