Optimal. Leaf size=36 \[ -\frac {25 x}{8 \left (x^2+1\right )}-\frac {7 x}{4 \left (x^2+1\right )^2}-\frac {4}{x}-\frac {57}{8} \tan ^{-1}(x) \]
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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 = {1260, 456, 453, 203} \[ -\frac {25 x}{8 \left (x^2+1\right )}-\frac {7 x}{4 \left (x^2+1\right )^2}-\frac {4}{x}-\frac {57}{8} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 453
Rule 456
Rule 1260
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 = 0.02, size = 33, normalized size = 0.92 \[ -\frac {57 x^4+103 x^2+32}{8 x \left (x^2+1\right )^2}-\frac {57}{8} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 33, normalized size = 0.92 \[ \frac {-57 x^4-103 x^2-32}{8 x \left (x^2+1\right )^2}-\frac {57}{8} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 40, normalized size = 1.11 \[ -\frac {57 \, x^{4} + 103 \, x^{2} + 57 \, {\left (x^{5} + 2 \, x^{3} + x\right )} \arctan \relax (x) + 32}{8 \, {\left (x^{5} + 2 \, x^{3} + x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 28, normalized size = 0.78 \[ -\frac {25 \, x^{3} + 39 \, x}{8 \, {\left (x^{2} + 1\right )}^{2}} - \frac {4}{x} - \frac {57}{8} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 29, normalized size = 0.81
method | result | size |
default | \(-\frac {4}{x}-\frac {\frac {25}{8} x^{3}+\frac {39}{8} x}{\left (x^{2}+1\right )^{2}}-\frac {57 \arctan \relax (x )}{8}\) | \(29\) |
risch | \(\frac {-\frac {57}{8} x^{4}-\frac {103}{8} x^{2}-4}{\left (x^{2}+1\right )^{2} x}-\frac {57 \arctan \relax (x )}{8}\) | \(29\) |
meijerg | \(-\frac {15 x^{4}+25 x^{2}+8}{2 x \left (x^{2}+1\right )^{2}}-\frac {57 \arctan \relax (x )}{8}-\frac {x \left (-3 x^{2}+3\right )}{8 \left (x^{2}+1\right )^{2}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 31, normalized size = 0.86 \[ -\frac {57 \, x^{4} + 103 \, x^{2} + 32}{8 \, {\left (x^{5} + 2 \, x^{3} + x\right )}} - \frac {57}{8} \, \arctan \relax (x) \]
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 \[ -\frac {57\,\mathrm {atan}\relax (x)}{8}-\frac {\frac {57\,x^4}{8}+\frac {103\,x^2}{8}+4}{x\,{\left (x^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 32, normalized size = 0.89 \[ \frac {- 57 x^{4} - 103 x^{2} - 32}{8 x^{5} + 16 x^{3} + 8 x} - \frac {57 \operatorname {atan}{\relax (x )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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