Optimal. Leaf size=58 \[ \frac {35 x}{16}-\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}-\frac {35 x^3}{48 \left (4+x^2\right )}-\frac {35}{8} \tan ^{-1}\left (\frac {x}{2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {294, 327, 209}
\begin {gather*} -\frac {35}{8} \text {ArcTan}\left (\frac {x}{2}\right )-\frac {x^7}{6 \left (x^2+4\right )^3}-\frac {7 x^5}{24 \left (x^2+4\right )^2}-\frac {35 x^3}{48 \left (x^2+4\right )}+\frac {35 x}{16} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 294
Rule 327
Rubi steps
\begin {align*} \int \frac {x^8}{\left (4+x^2\right )^4} \, dx &=-\frac {x^7}{6 \left (4+x^2\right )^3}+\frac {7}{6} \int \frac {x^6}{\left (4+x^2\right )^3} \, dx\\ &=-\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}+\frac {35}{24} \int \frac {x^4}{\left (4+x^2\right )^2} \, dx\\ &=-\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}-\frac {35 x^3}{48 \left (4+x^2\right )}+\frac {35}{16} \int \frac {x^2}{4+x^2} \, dx\\ &=\frac {35 x}{16}-\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}-\frac {35 x^3}{48 \left (4+x^2\right )}-\frac {35}{4} \int \frac {1}{4+x^2} \, dx\\ &=\frac {35 x}{16}-\frac {x^7}{6 \left (4+x^2\right )^3}-\frac {7 x^5}{24 \left (4+x^2\right )^2}-\frac {35 x^3}{48 \left (4+x^2\right )}-\frac {35}{8} \tan ^{-1}\left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 40, normalized size = 0.69 \begin {gather*} \frac {x \left (1680+1120 x^2+231 x^4+12 x^6\right )}{12 \left (4+x^2\right )^3}-\frac {35}{8} \tan ^{-1}\left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 32, normalized size = 0.55
method | result | size |
risch | \(x +\frac {\frac {29}{4} x^{5}+\frac {136}{3} x^{3}+76 x}{\left (x^{2}+4\right )^{3}}-\frac {35 \arctan \left (\frac {x}{2}\right )}{8}\) | \(31\) |
default | \(x -\frac {16 \left (-\frac {29}{64} x^{5}-\frac {17}{6} x^{3}-\frac {19}{4} x \right )}{\left (x^{2}+4\right )^{3}}-\frac {35 \arctan \left (\frac {x}{2}\right )}{8}\) | \(32\) |
meijerg | \(\frac {x \left (\frac {9}{4} x^{6}+\frac {693}{16} x^{4}+210 x^{2}+315\right )}{144 \left (1+\frac {x^{2}}{4}\right )^{3}}-\frac {35 \arctan \left (\frac {x}{2}\right )}{8}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.93, size = 41, normalized size = 0.71 \begin {gather*} x + \frac {87 \, x^{5} + 544 \, x^{3} + 912 \, x}{12 \, {\left (x^{6} + 12 \, x^{4} + 48 \, x^{2} + 64\right )}} - \frac {35}{8} \, \arctan \left (\frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 59, normalized size = 1.02 \begin {gather*} \frac {24 \, x^{7} + 462 \, x^{5} + 2240 \, x^{3} - 105 \, {\left (x^{6} + 12 \, x^{4} + 48 \, x^{2} + 64\right )} \arctan \left (\frac {1}{2} \, x\right ) + 3360 \, x}{24 \, {\left (x^{6} + 12 \, x^{4} + 48 \, x^{2} + 64\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 39, normalized size = 0.67 \begin {gather*} x + \frac {87 x^{5} + 544 x^{3} + 912 x}{12 x^{6} + 144 x^{4} + 576 x^{2} + 768} - \frac {35 \operatorname {atan}{\left (\frac {x}{2} \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 31, normalized size = 0.53 \begin {gather*} x + \frac {87 \, x^{5} + 544 \, x^{3} + 912 \, x}{12 \, {\left (x^{2} + 4\right )}^{3}} - \frac {35}{8} \, \arctan \left (\frac {1}{2} \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 40, normalized size = 0.69 \begin {gather*} x-\frac {35\,\mathrm {atan}\left (\frac {x}{2}\right )}{8}+\frac {\frac {29\,x^5}{4}+\frac {136\,x^3}{3}+76\,x}{x^6+12\,x^4+48\,x^2+64} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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