Optimal. Leaf size=32 \[ \frac {4 \left (\frac {e}{4}-\left (2-x^2\right )^2\right )^2}{\left (-\frac {e^2}{8}+x\right )^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(32)=64\).
time = 0.22, antiderivative size = 153, normalized size of antiderivative = 4.78, number of steps
used = 2, number of rules used = 1, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2099}
\begin {gather*} 4 x^6+e^2 x^5-\frac {1}{16} \left (512-3 e^4\right ) x^4-\frac {1}{32} e^2 \left (256-e^4\right ) x^3+\frac {\left (98304-2048 e-1536 e^4+5 e^8\right ) x^2}{1024}+\frac {e^2 \left (98304-2048 e-1024 e^4+3 e^8\right ) x}{4096}+\frac {e^2 \left (128-e^4\right ) \left (16384-1024 e-256 e^4+e^8\right )}{8192 \left (e^2-8 x\right )}+\frac {\left (16384-1024 e-256 e^4+e^8\right )^2}{65536 \left (e^2-8 x\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^2 \left (98304-2048 e-1024 e^4+3 e^8\right )}{4096}+\frac {\left (16384-1024 e-256 e^4+e^8\right )^2}{4096 \left (e^2-8 x\right )^3}+\frac {e^2 \left (128-e^4\right ) \left (16384-1024 e-256 e^4+e^8\right )}{1024 \left (e^2-8 x\right )^2}+\frac {1}{512} \left (98304-2048 e-1536 e^4+5 e^8\right ) x+\frac {3}{32} e^2 \left (-256+e^4\right ) x^2+\frac {1}{4} \left (-512+3 e^4\right ) x^3+5 e^2 x^4+24 x^5\right ) \, dx\\ &=\frac {\left (16384-1024 e-256 e^4+e^8\right )^2}{65536 \left (e^2-8 x\right )^2}+\frac {e^2 \left (128-e^4\right ) \left (16384-1024 e-256 e^4+e^8\right )}{8192 \left (e^2-8 x\right )}+\frac {e^2 \left (98304-2048 e-1024 e^4+3 e^8\right ) x}{4096}+\frac {\left (98304-2048 e-1536 e^4+5 e^8\right ) x^2}{1024}-\frac {1}{32} e^2 \left (256-e^4\right ) x^3-\frac {1}{16} \left (512-3 e^4\right ) x^4+e^2 x^5+4 x^6\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(137\) vs. \(2(32)=64\).
time = 0.04, size = 137, normalized size = 4.28 \begin {gather*} -\frac {-3072 e^9+7 e^{16}-2097152 e^3 x-2359296 e^6 x+49152 e^7 x+30720 e^{10} x-112 e^{14} x+262144 e^2 (-1+128 x)-65536 e^5 \left (-2+3 x^2\right )-24576 e^8 \left (-6+5 x^2\right )+64 e^{12} \left (-30+7 x^2\right )+1048576 e^4 \left (-2+9 x^2\right )+2097152 e \left (4+x^4\right )-4194304 \left (16+24 x^4-8 x^6+x^8\right )}{16384 \left (e^2-8 x\right )^2} \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 = 0.38, size = 314, normalized size = 9.81
method | result | size |
norman | \(\frac {\left (6144-128 \,{\mathrm e}\right ) x^{4}+\left (128 \,{\mathrm e} \,{\mathrm e}^{2}-2048 \,{\mathrm e}^{2}\right ) x -2048 x^{6}+256 x^{8}+4096-8 \,{\mathrm e} \,{\mathrm e}^{4}+128 \,{\mathrm e}^{4}+16 \,{\mathrm e}^{2}-512 \,{\mathrm e}}{\left ({\mathrm e}^{2}-8 x \right )^{2}}\) | \(69\) |
gosper | \(-\frac {8 \left (-32 x^{8}+256 x^{6}+16 x^{4} {\mathrm e}-768 x^{4}+{\mathrm e} \,{\mathrm e}^{4}-16 x \,{\mathrm e} \,{\mathrm e}^{2}-16 \,{\mathrm e}^{4}+256 \,{\mathrm e}^{2} x -2 \,{\mathrm e}^{2}+64 \,{\mathrm e}-512\right )}{{\mathrm e}^{4}-16 \,{\mathrm e}^{2} x +64 x^{2}}\) | \(79\) |
risch | \(4 x^{6}+{\mathrm e}^{2} x^{5}+\frac {3 x^{4} {\mathrm e}^{4}}{16}-32 x^{4}+\frac {{\mathrm e}^{2} {\mathrm e}^{4} x^{3}}{48}-8 x^{3} {\mathrm e}^{2}+\frac {x^{3} {\mathrm e}^{6}}{96}-\frac {3 x^{2} {\mathrm e}^{4}}{2}+\frac {x^{2} {\mathrm e}^{2} {\mathrm e}^{6}}{512}+\frac {3 x^{2} {\mathrm e}^{8}}{1024}-2 x^{2} {\mathrm e}+96 x^{2}-\frac {{\mathrm e}^{2} {\mathrm e}^{4} x}{4}-\frac {x \,{\mathrm e} \,{\mathrm e}^{2}}{2}+\frac {3 \,{\mathrm e}^{2} x \,{\mathrm e}^{8}}{4096}+24 \,{\mathrm e}^{2} x +\frac {\frac {\left (4 \,{\mathrm e}^{14}-1536 \,{\mathrm e}^{10}-4096 \,{\mathrm e}^{7}+196608 \,{\mathrm e}^{6}+524288 \,{\mathrm e}^{3}-8388608 \,{\mathrm e}^{2}\right ) x}{4096}-\frac {7 \,{\mathrm e}^{16}}{65536}+\frac {5 \,{\mathrm e}^{12}}{128}+\frac {3 \,{\mathrm e}^{9}}{32}-\frac {9 \,{\mathrm e}^{8}}{2}-8 \,{\mathrm e}^{5}+128 \,{\mathrm e}^{4}+16 \,{\mathrm e}^{2}-512 \,{\mathrm e}+4096}{{\mathrm e}^{4}-16 \,{\mathrm e}^{2} x +64 x^{2}}\) | \(196\) |
default | \(-\frac {3 x \,{\mathrm e} \,{\mathrm e}^{2}}{2}+\frac {405 x \,{\mathrm e}^{10}}{4096}+\frac {81 x^{2} {\mathrm e}^{8}}{512}+\frac {x^{3} {\mathrm e}^{6}}{4}+\frac {27 \,{\mathrm e}^{2} {\mathrm e}^{4} x}{8}-\frac {171 x^{2} \left ({\mathrm e}^{4}\right )^{2}}{1024}-\frac {3 x^{2} {\mathrm e}^{4}}{2}-\frac {29 x \,{\mathrm e}^{6}}{8}-8 x^{3} {\mathrm e}^{2}+x \,{\mathrm e}^{3}-2 x^{2} {\mathrm e}+\frac {3 x^{4} {\mathrm e}^{4}}{16}+{\mathrm e}^{2} x^{5}+24 \,{\mathrm e}^{2} x +4 x^{6}-32 x^{4}+96 x^{2}-\frac {7 \,{\mathrm e}^{2} {\mathrm e}^{4} x^{3}}{32}-\frac {609 \,{\mathrm e}^{4} {\mathrm e}^{6} x}{4096}+\frac {7 x^{2} {\mathrm e}^{2} {\mathrm e}^{6}}{512}+\frac {207 \,{\mathrm e}^{2} x \,{\mathrm e}^{8}}{4096}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (-{\mathrm e}^{6}+24 \textit {\_Z} \,{\mathrm e}^{4}-192 \textit {\_Z}^{2} {\mathrm e}^{2}+512 \textit {\_Z}^{3}\right )}{\sum }\frac {\left (-67108864 \,{\mathrm e}^{2} \textit {\_R} +4194304 \textit {\_R} \,{\mathrm e}^{3}+32 \textit {\_R} \,{\mathrm e}^{14}-12288 \textit {\_R} \,{\mathrm e}^{10}-32768 \textit {\_R} \,{\mathrm e}^{7}+1572864 \textit {\_R} \,{\mathrm e}^{6}+1048576 \,{\mathrm e}^{2}-33554432 \,{\mathrm e}+1024 \,{\mathrm e}^{12}+2048 \,{\mathrm e}^{9}-98304 \,{\mathrm e}^{8}-3 \,{\mathrm e}^{16}+268435456\right ) \ln \left (x -\textit {\_R} \right )}{{\mathrm e}^{4}-16 \,{\mathrm e}^{2} \textit {\_R} +64 \textit {\_R}^{2}}\right )}{98304}\) | \(314\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs.
\(2 (28) = 56\).
time = 0.27, size = 151, normalized size = 4.72 \begin {gather*} 4 \, x^{6} + x^{5} e^{2} + \frac {1}{16} \, x^{4} {\left (3 \, e^{4} - 512\right )} + \frac {1}{32} \, x^{3} {\left (e^{6} - 256 \, e^{2}\right )} + \frac {1}{1024} \, x^{2} {\left (5 \, e^{8} - 1536 \, e^{4} - 2048 \, e + 98304\right )} + \frac {1}{4096} \, x {\left (3 \, e^{10} - 1024 \, e^{6} - 2048 \, e^{3} + 98304 \, e^{2}\right )} + \frac {64 \, x {\left (e^{14} - 384 \, e^{10} - 1024 \, e^{7} + 49152 \, e^{6} + 131072 \, e^{3} - 2097152 \, e^{2}\right )} - 7 \, e^{16} + 2560 \, e^{12} + 6144 \, e^{9} - 294912 \, e^{8} - 524288 \, e^{5} + 8388608 \, e^{4} + 1048576 \, e^{2} - 33554432 \, e + 268435456}{65536 \, {\left (64 \, x^{2} - 16 \, x e^{2} + e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (28) = 56\).
time = 0.36, size = 129, normalized size = 4.03 \begin {gather*} \frac {16777216 \, x^{8} - 134217728 \, x^{6} + 402653184 \, x^{4} + 112 \, x e^{14} - 64 \, {\left (7 \, x^{2} - 40\right )} e^{12} - 40960 \, x e^{10} + 32768 \, {\left (5 \, x^{2} - 9\right )} e^{8} - 98304 \, x e^{7} + 4718592 \, x e^{6} + 131072 \, {\left (3 \, x^{2} - 4\right )} e^{5} - 2097152 \, {\left (9 \, x^{2} - 4\right )} e^{4} + 8388608 \, x e^{3} - 1048576 \, {\left (128 \, x - 1\right )} e^{2} - 8388608 \, {\left (x^{4} + 4\right )} e - 7 \, e^{16} + 6144 \, e^{9} + 268435456}{65536 \, {\left (64 \, x^{2} - 16 \, x e^{2} + e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 175 vs.
\(2 (22) = 44\).
time = 1.06, size = 175, normalized size = 5.47 \begin {gather*} 4 x^{6} + x^{5} e^{2} + x^{4} \left (-32 + \frac {3 e^{4}}{16}\right ) + x^{3} \left (- 8 e^{2} + \frac {e^{6}}{32}\right ) + x^{2} \left (- \frac {3 e^{4}}{2} - 2 e + \frac {5 e^{8}}{1024} + 96\right ) + x \left (- \frac {e^{6}}{4} - \frac {e^{3}}{2} + \frac {3 e^{10}}{4096} + 24 e^{2}\right ) + \frac {x \left (- 134217728 e^{2} - 24576 e^{10} - 65536 e^{7} + 64 e^{14} + 8388608 e^{3} + 3145728 e^{6}\right ) - 294912 e^{8} - 33554432 e - 524288 e^{5} - 7 e^{16} + 1048576 e^{2} + 6144 e^{9} + 268435456 + 2560 e^{12} + 8388608 e^{4}}{4194304 x^{2} - 1048576 x e^{2} + 65536 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs.
\(2 (28) = 56\).
time = 0.39, size = 160, normalized size = 5.00 \begin {gather*} 4 \, x^{6} + x^{5} e^{2} + \frac {3}{16} \, x^{4} e^{4} - 32 \, x^{4} + \frac {1}{32} \, x^{3} e^{6} - 8 \, x^{3} e^{2} + \frac {5}{1024} \, x^{2} e^{8} - \frac {3}{2} \, x^{2} e^{4} - 2 \, x^{2} e + 96 \, x^{2} + \frac {3}{4096} \, x e^{10} - \frac {1}{4} \, x e^{6} - \frac {1}{2} \, x e^{3} + 24 \, x e^{2} + \frac {64 \, x e^{14} - 24576 \, x e^{10} - 65536 \, x e^{7} + 3145728 \, x e^{6} + 8388608 \, x e^{3} - 134217728 \, x e^{2} - 7 \, e^{16} + 2560 \, e^{12} + 6144 \, e^{9} - 294912 \, e^{8} - 524288 \, e^{5} + 8388608 \, e^{4} + 1048576 \, e^{2} - 33554432 \, e + 268435456}{65536 \, {\left (8 \, x - e^{2}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.19, size = 262, normalized size = 8.19 \begin {gather*} x^3\,\left (8\,{\mathrm {e}}^2-\frac {{\mathrm {e}}^6}{16}+\frac {{\mathrm {e}}^2\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{8}\right )+x^4\,\left (\frac {3\,{\mathrm {e}}^4}{16}-32\right )+x\,\left (\frac {3\,{\mathrm {e}}^2\,\left (\frac {5\,{\mathrm {e}}^8}{512}-4\,\mathrm {e}+\frac {3\,{\mathrm {e}}^2\,\left (24\,{\mathrm {e}}^2-\frac {3\,{\mathrm {e}}^6}{16}+\frac {3\,{\mathrm {e}}^2\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{8}\right )}{8}-\frac {3\,{\mathrm {e}}^4\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{64}+192\right )}{8}+{\mathrm {e}}^2\,\left (\mathrm {e}-48\right )-\frac {3\,{\mathrm {e}}^4\,\left (24\,{\mathrm {e}}^2-\frac {3\,{\mathrm {e}}^6}{16}+\frac {3\,{\mathrm {e}}^2\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{8}\right )}{64}+\frac {{\mathrm {e}}^6\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{512}\right )+x^5\,{\mathrm {e}}^2+x^2\,\left (\frac {5\,{\mathrm {e}}^8}{1024}-2\,\mathrm {e}+\frac {3\,{\mathrm {e}}^2\,\left (24\,{\mathrm {e}}^2-\frac {3\,{\mathrm {e}}^6}{16}+\frac {3\,{\mathrm {e}}^2\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{8}\right )}{16}-\frac {3\,{\mathrm {e}}^4\,\left (\frac {3\,{\mathrm {e}}^4}{4}-128\right )}{128}+96\right )-\frac {2097152\,\mathrm {e}-65536\,{\mathrm {e}}^2-524288\,{\mathrm {e}}^4+32768\,{\mathrm {e}}^5+18432\,{\mathrm {e}}^8-384\,{\mathrm {e}}^9-160\,{\mathrm {e}}^{12}+\frac {7\,{\mathrm {e}}^{16}}{16}+x\,\left (8388608\,{\mathrm {e}}^2-524288\,{\mathrm {e}}^3-196608\,{\mathrm {e}}^6+4096\,{\mathrm {e}}^7+1536\,{\mathrm {e}}^{10}-4\,{\mathrm {e}}^{14}\right )-16777216}{262144\,x^2-65536\,{\mathrm {e}}^2\,x+4096\,{\mathrm {e}}^4}+4\,x^6 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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