Optimal. Leaf size=47 \[ \frac {x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac {3 x \left (1-x^2\right )}{8 \left (1+x^2\right )^2}+\frac {3}{8} \tan ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {424, 21, 209}
\begin {gather*} \frac {3 \text {ArcTan}(x)}{8}+\frac {x \left (1-x^2\right )^3}{4 \left (x^2+1\right )^4}+\frac {3 x \left (1-x^2\right )}{8 \left (x^2+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 209
Rule 424
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right )^4}{\left (1+x^2\right )^5} \, dx &=\frac {x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac {1}{8} \int \frac {\left (-1+x^2\right )^2 \left (6+6 x^2\right )}{\left (1+x^2\right )^4} \, dx\\ &=\frac {x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac {3}{4} \int \frac {\left (-1+x^2\right )^2}{\left (1+x^2\right )^3} \, dx\\ &=\frac {x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac {3 x \left (1-x^2\right )}{8 \left (1+x^2\right )^2}+\frac {3}{16} \int \frac {2+2 x^2}{\left (1+x^2\right )^2} \, dx\\ &=\frac {x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac {3 x \left (1-x^2\right )}{8 \left (1+x^2\right )^2}+\frac {3}{8} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x \left (1-x^2\right )^3}{4 \left (1+x^2\right )^4}+\frac {3 x \left (1-x^2\right )}{8 \left (1+x^2\right )^2}+\frac {3}{8} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 0.87 \begin {gather*} \frac {5 x-3 x^3+3 x^5-5 x^7+3 \left (1+x^2\right )^4 \tan ^{-1}(x)}{8 \left (1+x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 33, normalized size = 0.70
method | result | size |
default | \(\frac {-\frac {5}{8} x^{7}+\frac {3}{8} x^{5}-\frac {3}{8} x^{3}+\frac {5}{8} x}{\left (x^{2}+1\right )^{4}}+\frac {3 \arctan \left (x \right )}{8}\) | \(33\) |
risch | \(\frac {-\frac {5}{8} x^{7}+\frac {3}{8} x^{5}-\frac {3}{8} x^{3}+\frac {5}{8} x}{\left (x^{2}+1\right )^{4}}+\frac {3 \arctan \left (x \right )}{8}\) | \(33\) |
meijerg | \(\frac {x \left (105 x^{6}+385 x^{4}+511 x^{2}+279\right )}{384 \left (x^{2}+1\right )^{4}}+\frac {3 \arctan \left (x \right )}{8}-\frac {x \left (837 x^{6}+1533 x^{4}+1155 x^{2}+315\right )}{1152 \left (x^{2}+1\right )^{4}}+\frac {x \left (-105 x^{6}+511 x^{4}+385 x^{2}+105\right )}{672 \left (x^{2}+1\right )^{4}}-\frac {3 x \left (-15 x^{6}-55 x^{4}+55 x^{2}+15\right )}{320 \left (x^{2}+1\right )^{4}}+\frac {x \left (-15 x^{6}-55 x^{4}-73 x^{2}+15\right )}{96 \left (x^{2}+1\right )^{4}}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 48, normalized size = 1.02 \begin {gather*} -\frac {5 \, x^{7} - 3 \, x^{5} + 3 \, x^{3} - 5 \, x}{8 \, {\left (x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1\right )}} + \frac {3}{8} \, \arctan \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 67, normalized size = 1.43 \begin {gather*} -\frac {5 \, x^{7} - 3 \, x^{5} + 3 \, x^{3} - 3 \, {\left (x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1\right )} \arctan \left (x\right ) - 5 \, x}{8 \, {\left (x^{8} + 4 \, x^{6} + 6 \, x^{4} + 4 \, x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 46, normalized size = 0.98 \begin {gather*} \frac {- 5 x^{7} + 3 x^{5} - 3 x^{3} + 5 x}{8 x^{8} + 32 x^{6} + 48 x^{4} + 32 x^{2} + 8} + \frac {3 \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 = 0.91, size = 54, normalized size = 1.15 \begin {gather*} \frac {3}{32} \, \pi \mathrm {sgn}\left (x\right ) - \frac {5 \, {\left (x - \frac {1}{x}\right )}^{3} + 12 \, x - \frac {12}{x}}{8 \, {\left ({\left (x - \frac {1}{x}\right )}^{2} + 4\right )}^{2}} + \frac {3}{16} \, \arctan \left (\frac {x^{2} - 1}{2 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 47, normalized size = 1.00 \begin {gather*} \frac {3\,\mathrm {atan}\left (x\right )}{8}+\frac {-\frac {5\,x^7}{8}+\frac {3\,x^5}{8}-\frac {3\,x^3}{8}+\frac {5\,x}{8}}{x^8+4\,x^6+6\,x^4+4\,x^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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