Optimal. Leaf size=23 \[ 6 \left (4+\frac {x^3 \left (3+\frac {x}{-4+x^2}\right )}{\log (2)}\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps
used = 8, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {12, 28, 1825,
1814, 1818, 1600, 45} \begin {gather*} \frac {18 x^3}{\log (2)}-\frac {24 x^2}{\left (4-x^2\right ) \log (2)}+\frac {6 x^2}{\log (2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 28
Rule 45
Rule 1600
Rule 1814
Rule 1818
Rule 1825
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {864 x^2-96 x^3-432 x^4+12 x^5+54 x^6}{16-8 x^2+x^4} \, dx}{\log (2)}\\ &=\frac {\int \frac {864 x^2-96 x^3-432 x^4+12 x^5+54 x^6}{\left (-4+x^2\right )^2} \, dx}{\log (2)}\\ &=\frac {\int \frac {x \left (864 x-96 x^2-432 x^3+12 x^4+54 x^5\right )}{\left (-4+x^2\right )^2} \, dx}{\log (2)}\\ &=\frac {\int \frac {x^2 \left (864-96 x-432 x^2+12 x^3+54 x^4\right )}{\left (-4+x^2\right )^2} \, dx}{\log (2)}\\ &=-\frac {24 x^2}{\left (4-x^2\right ) \log (2)}+\frac {\int \frac {x \left (-384-1728 x+96 x^2+432 x^3\right )}{-4+x^2} \, dx}{8 \log (2)}\\ &=-\frac {24 x^2}{\left (4-x^2\right ) \log (2)}+\frac {\int x (96+432 x) \, dx}{8 \log (2)}\\ &=-\frac {24 x^2}{\left (4-x^2\right ) \log (2)}+\frac {\int \left (96 x+432 x^2\right ) \, dx}{8 \log (2)}\\ &=\frac {6 x^2}{\log (2)}+\frac {18 x^3}{\log (2)}-\frac {24 x^2}{\left (4-x^2\right ) \log (2)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.04 \begin {gather*} \frac {6 \left (x^2+3 x^3+\frac {16}{-4+x^2}\right )}{\log (2)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 30, normalized size = 1.30
method | result | size |
gosper | \(\frac {6 x^{3} \left (3 x^{2}+x -12\right )}{\ln \left (2\right ) \left (x^{2}-4\right )}\) | \(25\) |
default | \(\frac {18 x^{3}+6 x^{2}+\frac {24}{x -2}-\frac {24}{2+x}}{\ln \left (2\right )}\) | \(30\) |
risch | \(\frac {18 x^{3}}{\ln \left (2\right )}+\frac {6 x^{2}}{\ln \left (2\right )}+\frac {96}{\ln \left (2\right ) \left (x^{2}-4\right )}\) | \(33\) |
norman | \(\frac {-\frac {72 x^{3}}{\ln \left (2\right )}+\frac {6 x^{4}}{\ln \left (2\right )}+\frac {18 x^{5}}{\ln \left (2\right )}}{x^{2}-4}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 24, normalized size = 1.04 \begin {gather*} \frac {6 \, {\left (3 \, x^{3} + x^{2} + \frac {16}{x^{2} - 4}\right )}}{\log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 33, normalized size = 1.43 \begin {gather*} \frac {6 \, {\left (3 \, x^{5} + x^{4} - 12 \, x^{3} - 4 \, x^{2} + 16\right )}}{{\left (x^{2} - 4\right )} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 29, normalized size = 1.26 \begin {gather*} \frac {18 x^{3}}{\log {\left (2 \right )}} + \frac {6 x^{2}}{\log {\left (2 \right )}} + \frac {96}{x^{2} \log {\left (2 \right )} - 4 \log {\left (2 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 24, normalized size = 1.04 \begin {gather*} \frac {6 \, {\left (3 \, x^{3} + x^{2} + \frac {16}{x^{2} - 4}\right )}}{\log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 33, normalized size = 1.43 \begin {gather*} \frac {6\,\left (3\,x^5+x^4-12\,x^3-4\,x^2+16\right )}{\ln \left (2\right )\,\left (x^2-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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