Optimal. Leaf size=30 \[ -5+\left (-2-x+\frac {x}{8 \left (1-e^{2 x} (5-x)\right )^2}\right )^2 \]
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Rubi [F]
time = 2.54, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {112+49 x+e^{2 x} \left (-2960-1061 x+211 x^2+28 x^3\right )+e^{4 x} \left (30800+5360 x-4464 x^2+112 x^3+64 x^4\right )+e^{6 x} \left (-158000+9600 x+28720 x^2-6784 x^3+224 x^4+32 x^5\right )+e^{8 x} \left (400000-120000 x-64000 x^2+35200 x^3-5760 x^4+320 x^5\right )+e^{10 x} \left (-400000+200000 x+40000 x^2-48000 x^3+12800 x^4-1472 x^5+64 x^6\right )}{32+e^{2 x} (-800+160 x)+e^{4 x} \left (8000-3200 x+320 x^2\right )+e^{6 x} \left (-40000+24000 x-4800 x^2+320 x^3\right )+e^{8 x} \left (100000-80000 x+24000 x^2-3200 x^3+160 x^4\right )+e^{10 x} \left (-100000+100000 x-40000 x^2+8000 x^3-800 x^4+32 x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {320 e^{8 x} (-5+x)^4 (2+x)+64 e^{10 x} (-5+x)^5 (2+x)+7 (16+7 x)+16 e^{6 x} (-5+x)^2 \left (-395-134 x+34 x^2+2 x^3\right )+e^{2 x} \left (-2960-1061 x+211 x^2+28 x^3\right )+16 e^{4 x} \left (1925+335 x-279 x^2+7 x^3+4 x^4\right )}{32 \left (1+e^{2 x} (-5+x)\right )^5} \, dx\\ &=\frac {1}{32} \int \frac {320 e^{8 x} (-5+x)^4 (2+x)+64 e^{10 x} (-5+x)^5 (2+x)+7 (16+7 x)+16 e^{6 x} (-5+x)^2 \left (-395-134 x+34 x^2+2 x^3\right )+e^{2 x} \left (-2960-1061 x+211 x^2+28 x^3\right )+16 e^{4 x} \left (1925+335 x-279 x^2+7 x^3+4 x^4\right )}{\left (1+e^{2 x} (-5+x)\right )^5} \, dx\\ &=\frac {1}{32} \int \left (64 (2+x)+\frac {2 x^2 (-9+2 x)}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^5}-\frac {16 x \left (-18-5 x+2 x^2\right )}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^3}-\frac {x \left (5-19 x+4 x^2\right )}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^4}+\frac {16 \left (5-14 x-6 x^2+2 x^3\right )}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^2}\right ) \, dx\\ &=(2+x)^2-\frac {1}{32} \int \frac {x \left (5-19 x+4 x^2\right )}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^4} \, dx+\frac {1}{16} \int \frac {x^2 (-9+2 x)}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^5} \, dx-\frac {1}{2} \int \frac {x \left (-18-5 x+2 x^2\right )}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^3} \, dx+\frac {1}{2} \int \frac {5-14 x-6 x^2+2 x^3}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^2} \, dx\\ &=(2+x)^2-\frac {1}{32} \int \left (\frac {10}{\left (1-5 e^{2 x}+e^{2 x} x\right )^4}+\frac {50}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^4}+\frac {x}{\left (1-5 e^{2 x}+e^{2 x} x\right )^4}+\frac {4 x^2}{\left (1-5 e^{2 x}+e^{2 x} x\right )^4}\right ) \, dx+\frac {1}{16} \int \left (\frac {5}{\left (1-5 e^{2 x}+e^{2 x} x\right )^5}+\frac {25}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^5}+\frac {x}{\left (1-5 e^{2 x}+e^{2 x} x\right )^5}+\frac {2 x^2}{\left (1-5 e^{2 x}+e^{2 x} x\right )^5}\right ) \, dx-\frac {1}{2} \int \left (\frac {7}{\left (1-5 e^{2 x}+e^{2 x} x\right )^3}+\frac {35}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^3}+\frac {5 x}{\left (1-5 e^{2 x}+e^{2 x} x\right )^3}+\frac {2 x^2}{\left (1-5 e^{2 x}+e^{2 x} x\right )^3}\right ) \, dx+\frac {1}{2} \int \left (\frac {6}{\left (1-5 e^{2 x}+e^{2 x} x\right )^2}+\frac {35}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^2}+\frac {4 x}{\left (1-5 e^{2 x}+e^{2 x} x\right )^2}+\frac {2 x^2}{\left (1-5 e^{2 x}+e^{2 x} x\right )^2}\right ) \, dx\\ &=(2+x)^2-\frac {1}{32} \int \frac {x}{\left (1-5 e^{2 x}+e^{2 x} x\right )^4} \, dx+\frac {1}{16} \int \frac {x}{\left (1-5 e^{2 x}+e^{2 x} x\right )^5} \, dx+\frac {1}{8} \int \frac {x^2}{\left (1-5 e^{2 x}+e^{2 x} x\right )^5} \, dx-\frac {1}{8} \int \frac {x^2}{\left (1-5 e^{2 x}+e^{2 x} x\right )^4} \, dx+\frac {5}{16} \int \frac {1}{\left (1-5 e^{2 x}+e^{2 x} x\right )^5} \, dx-\frac {5}{16} \int \frac {1}{\left (1-5 e^{2 x}+e^{2 x} x\right )^4} \, dx+\frac {25}{16} \int \frac {1}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^5} \, dx-\frac {25}{16} \int \frac {1}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^4} \, dx+2 \int \frac {x}{\left (1-5 e^{2 x}+e^{2 x} x\right )^2} \, dx-\frac {5}{2} \int \frac {x}{\left (1-5 e^{2 x}+e^{2 x} x\right )^3} \, dx+3 \int \frac {1}{\left (1-5 e^{2 x}+e^{2 x} x\right )^2} \, dx-\frac {7}{2} \int \frac {1}{\left (1-5 e^{2 x}+e^{2 x} x\right )^3} \, dx-\frac {35}{2} \int \frac {1}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^3} \, dx+\frac {35}{2} \int \frac {1}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^2} \, dx-\int \frac {x^2}{\left (1-5 e^{2 x}+e^{2 x} x\right )^3} \, dx+\int \frac {x^2}{\left (1-5 e^{2 x}+e^{2 x} x\right )^2} \, dx\\ &=(2+x)^2-\frac {1}{32} \int \frac {x}{\left (1+e^{2 x} (-5+x)\right )^4} \, dx+\frac {1}{16} \int \frac {x}{\left (1+e^{2 x} (-5+x)\right )^5} \, dx+\frac {1}{8} \int \frac {x^2}{\left (1+e^{2 x} (-5+x)\right )^5} \, dx-\frac {1}{8} \int \frac {x^2}{\left (1+e^{2 x} (-5+x)\right )^4} \, dx+\frac {5}{16} \int \frac {1}{\left (1+e^{2 x} (-5+x)\right )^5} \, dx-\frac {5}{16} \int \frac {1}{\left (1+e^{2 x} (-5+x)\right )^4} \, dx-\frac {25}{16} \int \frac {1}{\left (1+e^{2 x} (-5+x)\right )^4 (-5+x)} \, dx+\frac {25}{16} \int \frac {1}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^5} \, dx+2 \int \frac {x}{\left (1+e^{2 x} (-5+x)\right )^2} \, dx-\frac {5}{2} \int \frac {x}{\left (1+e^{2 x} (-5+x)\right )^3} \, dx+3 \int \frac {1}{\left (1+e^{2 x} (-5+x)\right )^2} \, dx-\frac {7}{2} \int \frac {1}{\left (1+e^{2 x} (-5+x)\right )^3} \, dx+\frac {35}{2} \int \frac {1}{\left (1+e^{2 x} (-5+x)\right )^2 (-5+x)} \, dx-\frac {35}{2} \int \frac {1}{(-5+x) \left (1-5 e^{2 x}+e^{2 x} x\right )^3} \, dx-\int \frac {x^2}{\left (1+e^{2 x} (-5+x)\right )^3} \, dx+\int \frac {x^2}{\left (1+e^{2 x} (-5+x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(30)=60\).
time = 18.77, size = 63, normalized size = 2.10 \begin {gather*} \frac {1}{32} \left (-1440+\left (128-\frac {16}{\left (1+e^{2 x} (-5+x)\right )^2}\right ) x+\left (32+\frac {1}{2 \left (1+e^{2 x} (-5+x)\right )^4}-\frac {8}{\left (1+e^{2 x} (-5+x)\right )^2}\right ) x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs.
\(2(27)=54\).
time = 0.11, size = 85, normalized size = 2.83
method | result | size |
risch | \(x^{2}+4 x -\frac {x \left (16 x^{3} {\mathrm e}^{4 x}-128 x^{2} {\mathrm e}^{4 x}+80 x \,{\mathrm e}^{4 x}+32 \,{\mathrm e}^{2 x} x^{2}+800 \,{\mathrm e}^{4 x}-96 x \,{\mathrm e}^{2 x}-320 \,{\mathrm e}^{2 x}+15 x +32\right )}{64 \left (x \,{\mathrm e}^{2 x}-5 \,{\mathrm e}^{2 x}+1\right )^{4}}\) | \(85\) |
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. 186 vs.
\(2 (26) = 52\).
time = 0.48, size = 186, normalized size = 6.20 \begin {gather*} \frac {49 \, x^{2} + 64 \, {\left (x^{6} - 16 \, x^{5} + 70 \, x^{4} + 100 \, x^{3} - 1375 \, x^{2} + 2500 \, x\right )} e^{\left (8 \, x\right )} + 256 \, {\left (x^{5} - 11 \, x^{4} + 15 \, x^{3} + 175 \, x^{2} - 500 \, x\right )} e^{\left (6 \, x\right )} + 16 \, {\left (23 \, x^{4} - 136 \, x^{3} - 365 \, x^{2} + 2350 \, x\right )} e^{\left (4 \, x\right )} + 32 \, {\left (7 \, x^{3} - 5 \, x^{2} - 150 \, x\right )} e^{\left (2 \, x\right )} + 224 \, x}{64 \, {\left ({\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )} e^{\left (8 \, x\right )} + 4 \, {\left (x^{3} - 15 \, x^{2} + 75 \, x - 125\right )} e^{\left (6 \, x\right )} + 6 \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (4 \, x\right )} + 4 \, {\left (x - 5\right )} e^{\left (2 \, x\right )} + 1\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. 186 vs.
\(2 (26) = 52\).
time = 0.32, size = 186, normalized size = 6.20 \begin {gather*} \frac {49 \, x^{2} + 64 \, {\left (x^{6} - 16 \, x^{5} + 70 \, x^{4} + 100 \, x^{3} - 1375 \, x^{2} + 2500 \, x\right )} e^{\left (8 \, x\right )} + 256 \, {\left (x^{5} - 11 \, x^{4} + 15 \, x^{3} + 175 \, x^{2} - 500 \, x\right )} e^{\left (6 \, x\right )} + 16 \, {\left (23 \, x^{4} - 136 \, x^{3} - 365 \, x^{2} + 2350 \, x\right )} e^{\left (4 \, x\right )} + 32 \, {\left (7 \, x^{3} - 5 \, x^{2} - 150 \, x\right )} e^{\left (2 \, x\right )} + 224 \, x}{64 \, {\left ({\left (x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625\right )} e^{\left (8 \, x\right )} + 4 \, {\left (x^{3} - 15 \, x^{2} + 75 \, x - 125\right )} e^{\left (6 \, x\right )} + 6 \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (4 \, x\right )} + 4 \, {\left (x - 5\right )} e^{\left (2 \, x\right )} + 1\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. 129 vs.
\(2 (20) = 40\).
time = 0.41, size = 129, normalized size = 4.30 \begin {gather*} x^{2} + 4 x + \frac {- 15 x^{2} - 32 x + \left (- 32 x^{3} + 96 x^{2} + 320 x\right ) e^{2 x} + \left (- 16 x^{4} + 128 x^{3} - 80 x^{2} - 800 x\right ) e^{4 x}}{\left (256 x - 1280\right ) e^{2 x} + \left (384 x^{2} - 3840 x + 9600\right ) e^{4 x} + \left (256 x^{3} - 3840 x^{2} + 19200 x - 32000\right ) e^{6 x} + \left (64 x^{4} - 1280 x^{3} + 9600 x^{2} - 32000 x + 40000\right ) e^{8 x} + 64} \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. 274 vs.
\(2 (26) = 52\).
time = 0.52, size = 274, normalized size = 9.13 \begin {gather*} \frac {64 \, x^{6} e^{\left (8 \, x\right )} - 1024 \, x^{5} e^{\left (8 \, x\right )} + 256 \, x^{5} e^{\left (6 \, x\right )} + 4480 \, x^{4} e^{\left (8 \, x\right )} - 2816 \, x^{4} e^{\left (6 \, x\right )} + 368 \, x^{4} e^{\left (4 \, x\right )} + 6400 \, x^{3} e^{\left (8 \, x\right )} + 3840 \, x^{3} e^{\left (6 \, x\right )} - 2176 \, x^{3} e^{\left (4 \, x\right )} + 224 \, x^{3} e^{\left (2 \, x\right )} - 88000 \, x^{2} e^{\left (8 \, x\right )} + 44800 \, x^{2} e^{\left (6 \, x\right )} - 5840 \, x^{2} e^{\left (4 \, x\right )} - 160 \, x^{2} e^{\left (2 \, x\right )} + 49 \, x^{2} + 160000 \, x e^{\left (8 \, x\right )} - 128000 \, x e^{\left (6 \, x\right )} + 37600 \, x e^{\left (4 \, x\right )} - 4800 \, x e^{\left (2 \, x\right )} + 224 \, x}{64 \, {\left (x^{4} e^{\left (8 \, x\right )} - 20 \, x^{3} e^{\left (8 \, x\right )} + 4 \, x^{3} e^{\left (6 \, x\right )} + 150 \, x^{2} e^{\left (8 \, x\right )} - 60 \, x^{2} e^{\left (6 \, x\right )} + 6 \, x^{2} e^{\left (4 \, x\right )} - 500 \, x e^{\left (8 \, x\right )} + 300 \, x e^{\left (6 \, x\right )} - 60 \, x e^{\left (4 \, x\right )} + 4 \, x e^{\left (2 \, x\right )} + 625 \, e^{\left (8 \, x\right )} - 500 \, e^{\left (6 \, x\right )} + 150 \, e^{\left (4 \, x\right )} - 20 \, e^{\left (2 \, x\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {49\,x-{\mathrm {e}}^{2\,x}\,\left (-28\,x^3-211\,x^2+1061\,x+2960\right )+{\mathrm {e}}^{4\,x}\,\left (64\,x^4+112\,x^3-4464\,x^2+5360\,x+30800\right )+{\mathrm {e}}^{6\,x}\,\left (32\,x^5+224\,x^4-6784\,x^3+28720\,x^2+9600\,x-158000\right )-{\mathrm {e}}^{8\,x}\,\left (-320\,x^5+5760\,x^4-35200\,x^3+64000\,x^2+120000\,x-400000\right )+{\mathrm {e}}^{10\,x}\,\left (64\,x^6-1472\,x^5+12800\,x^4-48000\,x^3+40000\,x^2+200000\,x-400000\right )+112}{{\mathrm {e}}^{4\,x}\,\left (320\,x^2-3200\,x+8000\right )+{\mathrm {e}}^{6\,x}\,\left (320\,x^3-4800\,x^2+24000\,x-40000\right )+{\mathrm {e}}^{8\,x}\,\left (160\,x^4-3200\,x^3+24000\,x^2-80000\,x+100000\right )+{\mathrm {e}}^{10\,x}\,\left (32\,x^5-800\,x^4+8000\,x^3-40000\,x^2+100000\,x-100000\right )+{\mathrm {e}}^{2\,x}\,\left (160\,x-800\right )+32} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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