Optimal. Leaf size=19 \[ \frac {16}{1+e^5+\frac {81}{256 x^2}+x^2} \]
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Rubi [A]
time = 0.11, antiderivative size = 24, normalized size of antiderivative = 1.26, number of steps
used = 5, number of rules used = 5, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6, 1607, 6820,
12, 1602} \begin {gather*} \frac {4096 x^2}{256 x^4+256 \left (1+e^5\right ) x^2+81} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 1602
Rule 1607
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {663552 x-2097152 x^5}{6561+41472 x^2+\left (107008+65536 e^{10}\right ) x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx\\ &=\int \frac {x \left (663552-2097152 x^4\right )}{6561+41472 x^2+\left (107008+65536 e^{10}\right ) x^4+131072 x^6+65536 x^8+e^5 \left (41472 x^2+131072 x^4+131072 x^6\right )} \, dx\\ &=\int \frac {8192 x \left (81-256 x^4\right )}{\left (81+256 \left (1+e^5\right ) x^2+256 x^4\right )^2} \, dx\\ &=8192 \int \frac {x \left (81-256 x^4\right )}{\left (81+256 \left (1+e^5\right ) x^2+256 x^4\right )^2} \, dx\\ &=\frac {4096 x^2}{81+256 \left (1+e^5\right ) x^2+256 x^4}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.26 \begin {gather*} \frac {8192 x^2}{162+512 \left (1+e^5\right ) x^2+512 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 6.80, size = 100, normalized size = 5.26
method | result | size |
risch | \(\frac {16 x^{2}}{x^{4}+x^{2} {\mathrm e}^{5}+x^{2}+\frac {81}{256}}\) | \(22\) |
gosper | \(\frac {4096 x^{2}}{256 x^{4}+256 x^{2} {\mathrm e}^{5}+256 x^{2}+81}\) | \(27\) |
norman | \(\frac {4096 x^{2}}{256 x^{4}+256 x^{2} {\mathrm e}^{5}+256 x^{2}+81}\) | \(27\) |
default | \(8 \left (\munderset {\textit {\_R} =\RootOf \left (6561+65536 \textit {\_Z}^{4}+\left (131072 \,{\mathrm e}^{5}+131072\right ) \textit {\_Z}^{3}+\left (65536 \,{\mathrm e}^{10}+131072 \,{\mathrm e}^{5}+107008\right ) \textit {\_Z}^{2}+\left (41472 \,{\mathrm e}^{5}+41472\right ) \textit {\_Z} \right )}{\sum }\frac {\left (-256 \textit {\_R}^{2}+81\right ) \ln \left (x^{2}-\textit {\_R} \right )}{81+512 \textit {\_R}^{3}+768 \textit {\_R}^{2} {\mathrm e}^{5}+256 \textit {\_R} \,{\mathrm e}^{10}+768 \textit {\_R}^{2}+512 \textit {\_R} \,{\mathrm e}^{5}+418 \textit {\_R} +81 \,{\mathrm e}^{5}}\right )\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 23, normalized size = 1.21 \begin {gather*} \frac {4096 \, x^{2}}{256 \, x^{4} + 256 \, x^{2} {\left (e^{5} + 1\right )} + 81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 26, normalized size = 1.37 \begin {gather*} \frac {4096 \, x^{2}}{256 \, x^{4} + 256 \, x^{2} e^{5} + 256 \, x^{2} + 81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.62, size = 20, normalized size = 1.05 \begin {gather*} \frac {4096 x^{2}}{256 x^{4} + x^{2} \cdot \left (256 + 256 e^{5}\right ) + 81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 26, normalized size = 1.37 \begin {gather*} \frac {4096 \, x^{2}}{256 \, x^{4} + 256 \, x^{2} e^{5} + 256 \, x^{2} + 81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.49, size = 24, normalized size = 1.26 \begin {gather*} \frac {4096\,x^2}{256\,x^4+\left (256\,{\mathrm {e}}^5+256\right )\,x^2+81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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