Optimal. Leaf size=32 \[ x-16 x^3 \left (3+\frac {1}{5} \left (-1-e^4+\frac {5 (4+x)}{4+x^2}\right )\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 37, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 3, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {28, 1814, 1586} \begin {gather*} -\frac {16}{5} \left (14-e^4\right ) x^3-16 x^2-\frac {256 (1-x)}{x^2+4}-63 x \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 1586
Rule 1814
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=5 \int \frac {80-14552 x^2-1280 x^3-5691 x^4-160 x^5-672 x^6+e^4 \left (768 x^2+384 x^4+48 x^6\right )}{\left (20+5 x^2\right )^2} \, dx\\ &=-\frac {256 (1-x)}{4+x^2}-\frac {1}{8} \int \frac {10080+5120 x+24 \left (1001-64 e^4\right ) x^2+1280 x^3+384 \left (14-e^4\right ) x^4}{20+5 x^2} \, dx\\ &=-\frac {256 (1-x)}{4+x^2}-\frac {1}{8} \int \left (504+256 x+\left (\frac {5376}{5}-\frac {384 e^4}{5}\right ) x^2\right ) \, dx\\ &=-63 x-16 x^2-\frac {16}{5} \left (14-e^4\right ) x^3-\frac {256 (1-x)}{4+x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 1.09 \begin {gather*} \frac {1}{5} \left (-315 x-80 x^2+16 \left (-14+e^4\right ) x^3+\frac {1280 (-1+x)}{4+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 47, normalized size = 1.47 \begin {gather*} -\frac {224 \, x^{5} + 80 \, x^{4} + 1211 \, x^{3} + 320 \, x^{2} - 16 \, {\left (x^{5} + 4 \, x^{3}\right )} e^{4} - 20 \, x + 1280}{5 \, {\left (x^{2} + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 33, normalized size = 1.03 \begin {gather*} \frac {16}{5} \, x^{3} e^{4} - \frac {224}{5} \, x^{3} - 16 \, x^{2} - 63 \, x + \frac {256 \, {\left (x - 1\right )}}{x^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 1.09
method | result | size |
risch | \(\frac {16 x^{3} {\mathrm e}^{4}}{5}-\frac {224 x^{3}}{5}-16 x^{2}-63 x +\frac {256 x -256}{x^{2}+4}\) | \(35\) |
default | \(\frac {16 x^{3} {\mathrm e}^{4}}{5}-\frac {224 x^{3}}{5}-16 x^{2}-63 x -\frac {256 \left (1-x \right )}{x^{2}+4}\) | \(36\) |
norman | \(\frac {\left (\frac {16 \,{\mathrm e}^{4}}{5}-\frac {224}{5}\right ) x^{5}+\left (\frac {64 \,{\mathrm e}^{4}}{5}-\frac {1211}{5}\right ) x^{3}+4 x -16 x^{4}}{x^{2}+4}\) | \(38\) |
gosper | \(\frac {x \left (16 x^{4} {\mathrm e}^{4}-224 x^{4}+64 x^{2} {\mathrm e}^{4}-80 x^{3}-1211 x^{2}+20\right )}{5 x^{2}+20}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 30, normalized size = 0.94 \begin {gather*} \frac {16}{5} \, x^{3} {\left (e^{4} - 14\right )} - 16 \, x^{2} - 63 \, x + \frac {256 \, {\left (x - 1\right )}}{x^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 32, normalized size = 1.00 \begin {gather*} \frac {256\,x-256}{x^2+4}-63\,x+x^3\,\left (\frac {16\,{\mathrm {e}}^4}{5}-\frac {224}{5}\right )-16\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 32, normalized size = 1.00 \begin {gather*} - x^{3} \left (\frac {224}{5} - \frac {16 e^{4}}{5}\right ) - 16 x^{2} - 63 x - \frac {256 - 256 x}{x^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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