Optimal. Leaf size=34 \[ -\frac {x \left (1-x^2\right )^2}{3 \left (x^2+1\right )^3}-\frac {2 x}{3 \left (x^2+1\right )} \]
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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 = {413, 21, 383} \[ -\frac {x \left (1-x^2\right )^2}{3 \left (x^2+1\right )^3}-\frac {2 x}{3 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 21
Rule 383
Rule 413
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 \[ -\frac {x \left (3 x^4+2 x^2+3\right )}{3 \left (x^2+1\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 33, normalized size = 0.97 \[ -\frac {3 \, x^{5} + 2 \, x^{3} + 3 \, x}{3 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 20, normalized size = 0.59 \[ -\frac {3 \, {\left (x + \frac {1}{x}\right )}^{2} - 4}{3 \, {\left (x + \frac {1}{x}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 23, normalized size = 0.68 \[ \frac {-x^{5}-\frac {2}{3} x^{3}-x}{\left (x^{2}+1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 33, normalized size = 0.97 \[ -\frac {3 \, x^{5} + 2 \, x^{3} + 3 \, x}{3 \, {\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )}} \]
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 \[ \frac {4\,x}{3\,{\left (x^2+1\right )}^2}-\frac {x}{x^2+1}-\frac {4\,x}{3\,{\left (x^2+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 31, normalized size = 0.91 \[ \frac {- 3 x^{5} - 2 x^{3} - 3 x}{3 x^{6} + 9 x^{4} + 9 x^{2} + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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