Optimal. Leaf size=25 \[ x^2 \left (9+e^x-2 \left (3+\frac {1}{4 x^2}+x\right )^2\right )^2 \]
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Rubi [B] time = 0.35, antiderivative size = 116, normalized size of antiderivative = 4.64, number of steps used = 34, number of rules used = 8, integrand size = 116, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {12, 14, 2196, 2176, 2194, 2199, 2177, 2178} \begin {gather*} 4 x^6+\frac {1}{64 x^6}+48 x^5-4 e^x x^4+180 x^4+\frac {3}{4 x^4}-24 e^x x^3+220 x^3+\frac {1}{4 x^3}-18 e^x x^2+e^{2 x} x^2+117 x^2-\frac {e^x}{4 x^2}+\frac {45}{4 x^2}-2 e^x x+90 x-6 e^x+\frac {9}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2196
Rule 2199
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{32} \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{x^7} \, dx\\ &=\frac {1}{32} \int \left (64 e^{2 x} x (1+x)-\frac {8 e^x \left (-2+x+32 x^3+152 x^4+360 x^5+160 x^6+16 x^7\right )}{x^3}+\frac {3 \left (-1-32 x^2-8 x^3-240 x^4-96 x^5+960 x^7+2496 x^8+7040 x^9+7680 x^{10}+2560 x^{11}+256 x^{12}\right )}{x^7}\right ) \, dx\\ &=\frac {3}{32} \int \frac {-1-32 x^2-8 x^3-240 x^4-96 x^5+960 x^7+2496 x^8+7040 x^9+7680 x^{10}+2560 x^{11}+256 x^{12}}{x^7} \, dx-\frac {1}{4} \int \frac {e^x \left (-2+x+32 x^3+152 x^4+360 x^5+160 x^6+16 x^7\right )}{x^3} \, dx+2 \int e^{2 x} x (1+x) \, dx\\ &=\frac {3}{32} \int \left (960-\frac {1}{x^7}-\frac {32}{x^5}-\frac {8}{x^4}-\frac {240}{x^3}-\frac {96}{x^2}+2496 x+7040 x^2+7680 x^3+2560 x^4+256 x^5\right ) \, dx-\frac {1}{4} \int \left (32 e^x-\frac {2 e^x}{x^3}+\frac {e^x}{x^2}+152 e^x x+360 e^x x^2+160 e^x x^3+16 e^x x^4\right ) \, dx+2 \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx\\ &=\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}+\frac {9}{x}+90 x+117 x^2+220 x^3+180 x^4+48 x^5+4 x^6-\frac {1}{4} \int \frac {e^x}{x^2} \, dx+\frac {1}{2} \int \frac {e^x}{x^3} \, dx+2 \int e^{2 x} x \, dx+2 \int e^{2 x} x^2 \, dx-4 \int e^x x^4 \, dx-8 \int e^x \, dx-38 \int e^x x \, dx-40 \int e^x x^3 \, dx-90 \int e^x x^2 \, dx\\ &=-8 e^x+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+\frac {e^x}{4 x}+90 x-38 e^x x+e^{2 x} x+117 x^2-90 e^x x^2+e^{2 x} x^2+220 x^3-40 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6+\frac {1}{4} \int \frac {e^x}{x^2} \, dx-\frac {1}{4} \int \frac {e^x}{x} \, dx-2 \int e^{2 x} x \, dx+16 \int e^x x^3 \, dx+38 \int e^x \, dx+120 \int e^x x^2 \, dx+180 \int e^x x \, dx-\int e^{2 x} \, dx\\ &=30 e^x-\frac {e^{2 x}}{2}+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+90 x+142 e^x x+117 x^2+30 e^x x^2+e^{2 x} x^2+220 x^3-24 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6-\frac {\text {Ei}(x)}{4}+\frac {1}{4} \int \frac {e^x}{x} \, dx-48 \int e^x x^2 \, dx-180 \int e^x \, dx-240 \int e^x x \, dx+\int e^{2 x} \, dx\\ &=-150 e^x+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+90 x-98 e^x x+117 x^2-18 e^x x^2+e^{2 x} x^2+220 x^3-24 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6+96 \int e^x x \, dx+240 \int e^x \, dx\\ &=90 e^x+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+90 x-2 e^x x+117 x^2-18 e^x x^2+e^{2 x} x^2+220 x^3-24 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6-96 \int e^x \, dx\\ &=-6 e^x+\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}-\frac {e^x}{4 x^2}+\frac {9}{x}+90 x-2 e^x x+117 x^2-18 e^x x^2+e^{2 x} x^2+220 x^3-24 e^x x^3+180 x^4-4 e^x x^4+48 x^5+4 x^6\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.23, size = 101, normalized size = 4.04 \begin {gather*} \frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}+\frac {9}{x}+90 x+117 x^2+e^{2 x} x^2+220 x^3+180 x^4+48 x^5+4 x^6-\frac {1}{4} e^x \left (24+\frac {1}{x^2}+8 x+72 x^2+96 x^3+16 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 99, normalized size = 3.96 \begin {gather*} \frac {256 \, x^{12} + 3072 \, x^{11} + 11520 \, x^{10} + 14080 \, x^{9} + 64 \, x^{8} e^{\left (2 \, x\right )} + 7488 \, x^{8} + 5760 \, x^{7} + 576 \, x^{5} + 720 \, x^{4} + 16 \, x^{3} + 48 \, x^{2} - 16 \, {\left (16 \, x^{10} + 96 \, x^{9} + 72 \, x^{8} + 8 \, x^{7} + 24 \, x^{6} + x^{4}\right )} e^{x} + 1}{64 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 108, normalized size = 4.32 \begin {gather*} \frac {256 \, x^{12} + 3072 \, x^{11} - 256 \, x^{10} e^{x} + 11520 \, x^{10} - 1536 \, x^{9} e^{x} + 14080 \, x^{9} + 64 \, x^{8} e^{\left (2 \, x\right )} - 1152 \, x^{8} e^{x} + 7488 \, x^{8} - 128 \, x^{7} e^{x} + 5760 \, x^{7} - 384 \, x^{6} e^{x} + 576 \, x^{5} - 16 \, x^{4} e^{x} + 720 \, x^{4} + 16 \, x^{3} + 48 \, x^{2} + 1}{64 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 99, normalized size = 3.96
method | result | size |
risch | \(4 x^{6}+48 x^{5}+180 x^{4}+220 x^{3}+117 x^{2}+90 x +\frac {288 x^{5}+360 x^{4}+8 x^{3}+24 x^{2}+\frac {1}{2}}{32 x^{6}}+{\mathrm e}^{2 x} x^{2}-\frac {\left (16 x^{6}+96 x^{5}+72 x^{4}+8 x^{3}+24 x^{2}+1\right ) {\mathrm e}^{x}}{4 x^{2}}\) | \(99\) |
default | \(117 x^{2}+90 x +\frac {1}{64 x^{6}}+\frac {3}{4 x^{4}}+\frac {1}{4 x^{3}}+\frac {45}{4 x^{2}}+\frac {9}{x}+220 x^{3}+180 x^{4}+48 x^{5}+4 x^{6}-\frac {{\mathrm e}^{x}}{4 x^{2}}-2 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}-18 \,{\mathrm e}^{x} x^{2}-24 \,{\mathrm e}^{x} x^{3}-4 \,{\mathrm e}^{x} x^{4}+{\mathrm e}^{2 x} x^{2}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.40, size = 157, normalized size = 6.28 \begin {gather*} 4 \, x^{6} + 48 \, x^{5} + 180 \, x^{4} + 220 \, x^{3} + 117 \, x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} - 40 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} - 90 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 38 \, {\left (x - 1\right )} e^{x} + 90 \, x + \frac {9}{x} + \frac {45}{4 \, x^{2}} + \frac {1}{4 \, x^{3}} + \frac {3}{4 \, x^{4}} + \frac {1}{64 \, x^{6}} - 8 \, e^{x} - \frac {1}{4} \, \Gamma \left (-1, -x\right ) - \frac {1}{2} \, \Gamma \left (-2, -x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.34, size = 92, normalized size = 3.68 \begin {gather*} \frac {\frac {3\,x^2}{4}-x^4\,\left (\frac {{\mathrm {e}}^x}{4}-\frac {45}{4}\right )+\frac {x^3}{4}+9\,x^5+\frac {1}{64}}{x^6}-6\,{\mathrm {e}}^x+x^2\,\left ({\mathrm {e}}^{2\,x}-18\,{\mathrm {e}}^x+117\right )-x\,\left (2\,{\mathrm {e}}^x-90\right )-x^4\,\left (4\,{\mathrm {e}}^x-180\right )-x^3\,\left (24\,{\mathrm {e}}^x-220\right )+48\,x^5+4\,x^6 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.21, size = 102, normalized size = 4.08 \begin {gather*} 4 x^{6} + 48 x^{5} + 180 x^{4} + 220 x^{3} + 117 x^{2} + 90 x + \frac {4 x^{4} e^{2 x} + \left (- 16 x^{6} - 96 x^{5} - 72 x^{4} - 8 x^{3} - 24 x^{2} - 1\right ) e^{x}}{4 x^{2}} + \frac {576 x^{5} + 720 x^{4} + 16 x^{3} + 48 x^{2} + 1}{64 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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