Optimal. Leaf size=34 \[ -\frac {x \left (1-x^2\right )^2}{3 \left (1+x^2\right )^3}-\frac {2 x}{3 \left (1+x^2\right )} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {424, 21, 391}
\begin {gather*} -\frac {x \left (1-x^2\right )^2}{3 \left (x^2+1\right )^3}-\frac {2 x}{3 \left (x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 391
Rule 424
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right )^3}{\left (1+x^2\right )^4} \, dx &=-\frac {x \left (1-x^2\right )^2}{3 \left (1+x^2\right )^3}+\frac {1}{6} \int \frac {\left (-1+x^2\right ) \left (4+4 x^2\right )}{\left (1+x^2\right )^3} \, dx\\ &=-\frac {x \left (1-x^2\right )^2}{3 \left (1+x^2\right )^3}+\frac {2}{3} \int \frac {-1+x^2}{\left (1+x^2\right )^2} \, dx\\ &=-\frac {x \left (1-x^2\right )^2}{3 \left (1+x^2\right )^3}-\frac {2 x}{3 \left (1+x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.71 \begin {gather*} -\frac {x \left (3+2 x^2+3 x^4\right )}{3 \left (1+x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 23, normalized size = 0.68
method | result | size |
gosper | \(-\frac {x \left (3 x^{4}+2 x^{2}+3\right )}{3 \left (x^{2}+1\right )^{3}}\) | \(23\) |
default | \(\frac {-x^{5}-\frac {2}{3} x^{3}-x}{\left (x^{2}+1\right )^{3}}\) | \(23\) |
norman | \(\frac {-x^{5}-\frac {2}{3} x^{3}-x}{\left (x^{2}+1\right )^{3}}\) | \(23\) |
risch | \(\frac {-x^{5}-\frac {2}{3} x^{3}-x}{\left (x^{2}+1\right )^{3}}\) | \(23\) |
meijerg | \(-\frac {x \left (15 x^{4}+40 x^{2}+33\right )}{48 \left (x^{2}+1\right )^{3}}-\frac {x \left (231 x^{4}+280 x^{2}+105\right )}{336 \left (x^{2}+1\right )^{3}}+\frac {x \left (-15 x^{4}+40 x^{2}+15\right )}{80 \left (x^{2}+1\right )^{3}}-\frac {x \left (-3 x^{4}-8 x^{2}+3\right )}{16 \left (x^{2}+1\right )^{3}}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 33, normalized size = 0.97 \begin {gather*} -\frac {3 \, x^{5} + 2 \, x^{3} + 3 \, x}{3 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.72, size = 33, normalized size = 0.97 \begin {gather*} -\frac {3 \, x^{5} + 2 \, x^{3} + 3 \, x}{3 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 31, normalized size = 0.91 \begin {gather*} \frac {- 3 x^{5} - 2 x^{3} - 3 x}{3 x^{6} + 9 x^{4} + 9 x^{2} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.43, size = 20, normalized size = 0.59 \begin {gather*} -\frac {3 \, {\left (x + \frac {1}{x}\right )}^{2} - 4}{3 \, {\left (x + \frac {1}{x}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.00, size = 31, normalized size = 0.91 \begin {gather*} \frac {4\,x}{3\,{\left (x^2+1\right )}^2}-\frac {x}{x^2+1}-\frac {4\,x}{3\,{\left (x^2+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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