Optimal. Leaf size=25 \[ \left (-2+e^4+\left (-1-e^{2 x}+\frac {3}{4 x^2}\right ) x\right )^2 \]
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Rubi [B] time = 0.20, antiderivative size = 104, normalized size of antiderivative = 4.16, number of steps used = 19, number of rules used = 6, integrand size = 89, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {12, 14, 2196, 2176, 2194, 1590} \begin {gather*} 2 e^{2 x} x^2+e^{4 x} x^2+\frac {\left (-4 x^2-4 \left (2-e^4\right ) x+3\right )^2}{16 x^2}-2 e^{2 x} x+2 \left (3-e^4\right ) e^{2 x} x+e^{2 x}-\left (3-e^4\right ) e^{2 x}+\frac {1}{2} \left (1-2 e^4\right ) e^{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 1590
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \frac {-9+24 x+32 x^3+16 x^4+e^4 \left (-12 x-16 x^3\right )+e^{4 x} \left (16 x^4+32 x^5\right )+e^{2 x} \left (8 x^3+96 x^4+32 x^5+e^4 \left (-16 x^3-32 x^4\right )\right )}{x^3} \, dx\\ &=\frac {1}{8} \int \left (16 e^{4 x} x (1+2 x)+\frac {\left (-3-4 x^2\right ) \left (3-4 \left (2-e^4\right ) x-4 x^2\right )}{x^3}+8 e^{2 x} \left (1-2 e^4+4 \left (3-e^4\right ) x+4 x^2\right )\right ) \, dx\\ &=\frac {1}{8} \int \frac {\left (-3-4 x^2\right ) \left (3-4 \left (2-e^4\right ) x-4 x^2\right )}{x^3} \, dx+2 \int e^{4 x} x (1+2 x) \, dx+\int e^{2 x} \left (1-2 e^4+4 \left (3-e^4\right ) x+4 x^2\right ) \, dx\\ &=\frac {\left (3-4 \left (2-e^4\right ) x-4 x^2\right )^2}{16 x^2}+2 \int \left (e^{4 x} x+2 e^{4 x} x^2\right ) \, dx+\int \left (e^{2 x} \left (1-2 e^4\right )-4 e^{2 x} \left (-3+e^4\right ) x+4 e^{2 x} x^2\right ) \, dx\\ &=\frac {\left (3-4 \left (2-e^4\right ) x-4 x^2\right )^2}{16 x^2}+2 \int e^{4 x} x \, dx+4 \int e^{2 x} x^2 \, dx+4 \int e^{4 x} x^2 \, dx+\left (1-2 e^4\right ) \int e^{2 x} \, dx+\left (4 \left (3-e^4\right )\right ) \int e^{2 x} x \, dx\\ &=\frac {1}{2} e^{2 x} \left (1-2 e^4\right )+\frac {1}{2} e^{4 x} x+2 e^{2 x} \left (3-e^4\right ) x+2 e^{2 x} x^2+e^{4 x} x^2+\frac {\left (3-4 \left (2-e^4\right ) x-4 x^2\right )^2}{16 x^2}-\frac {1}{2} \int e^{4 x} \, dx-2 \int e^{4 x} x \, dx-4 \int e^{2 x} x \, dx-\left (2 \left (3-e^4\right )\right ) \int e^{2 x} \, dx\\ &=-\frac {e^{4 x}}{8}+\frac {1}{2} e^{2 x} \left (1-2 e^4\right )-e^{2 x} \left (3-e^4\right )-2 e^{2 x} x+2 e^{2 x} \left (3-e^4\right ) x+2 e^{2 x} x^2+e^{4 x} x^2+\frac {\left (3-4 \left (2-e^4\right ) x-4 x^2\right )^2}{16 x^2}+\frac {1}{2} \int e^{4 x} \, dx+2 \int e^{2 x} \, dx\\ &=e^{2 x}+\frac {1}{2} e^{2 x} \left (1-2 e^4\right )-e^{2 x} \left (3-e^4\right )-2 e^{2 x} x+2 e^{2 x} \left (3-e^4\right ) x+2 e^{2 x} x^2+e^{4 x} x^2+\frac {\left (3-4 \left (2-e^4\right ) x-4 x^2\right )^2}{16 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.08, size = 72, normalized size = 2.88 \begin {gather*} \frac {9}{16 x^2}-\frac {3}{x}+\frac {3 e^4}{2 x}+4 x-2 e^4 x-2 e^{4+2 x} x+x^2+e^{4 x} x^2+e^{2 x} \left (-\frac {3}{2}+4 x+2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 71, normalized size = 2.84 \begin {gather*} \frac {16 \, x^{4} e^{\left (4 \, x\right )} + 16 \, x^{4} + 64 \, x^{3} - 8 \, {\left (4 \, x^{3} - 3 \, x\right )} e^{4} + 8 \, {\left (4 \, x^{4} - 4 \, x^{3} e^{4} + 8 \, x^{3} - 3 \, x^{2}\right )} e^{\left (2 \, x\right )} - 48 \, x + 9}{16 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 79, normalized size = 3.16 \begin {gather*} \frac {16 \, x^{4} e^{\left (4 \, x\right )} + 32 \, x^{4} e^{\left (2 \, x\right )} + 16 \, x^{4} - 32 \, x^{3} e^{4} + 64 \, x^{3} e^{\left (2 \, x\right )} - 32 \, x^{3} e^{\left (2 \, x + 4\right )} + 64 \, x^{3} - 24 \, x^{2} e^{\left (2 \, x\right )} + 24 \, x e^{4} - 48 \, x + 9}{16 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 57, normalized size = 2.28
method | result | size |
risch | \(-2 x \,{\mathrm e}^{4}+x^{2}+4 x +\frac {\left (12 \,{\mathrm e}^{4}-24\right ) x +\frac {9}{2}}{8 x^{2}}+x^{2} {\mathrm e}^{4 x}+\frac {\left (-16 x \,{\mathrm e}^{4}+16 x^{2}+32 x -12\right ) {\mathrm e}^{2 x}}{8}\) | \(57\) |
norman | \(\frac {\frac {9}{16}+x^{4}+x^{4} {\mathrm e}^{4 x}+\left (-2 \,{\mathrm e}^{4}+4\right ) x^{3}+\left (\frac {3 \,{\mathrm e}^{4}}{2}-3\right ) x +\left (-2 \,{\mathrm e}^{4}+4\right ) x^{3} {\mathrm e}^{2 x}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{2}+2 \,{\mathrm e}^{2 x} x^{4}}{x^{2}}\) | \(68\) |
default | \(x^{2}+4 x +\frac {9}{16 x^{2}}-\frac {3}{x}-\frac {3 \,{\mathrm e}^{2 x}}{2}+4 x \,{\mathrm e}^{2 x}+x^{2} {\mathrm e}^{4 x}+\frac {3 \,{\mathrm e}^{4}}{2 x}-{\mathrm e}^{4} {\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x} x^{2}-4 \,{\mathrm e}^{4} \left (\frac {x \,{\mathrm e}^{2 x}}{2}-\frac {{\mathrm e}^{2 x}}{4}\right )-2 x \,{\mathrm e}^{4}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 112, normalized size = 4.48 \begin {gather*} x^{2} - 2 \, x e^{4} + \frac {1}{8} \, {\left (8 \, x^{2} - 4 \, x + 1\right )} e^{\left (4 \, x\right )} + \frac {1}{8} \, {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} + {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - {\left (2 \, x e^{4} - e^{4}\right )} e^{\left (2 \, x\right )} + 3 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 4 \, x + \frac {3 \, e^{4}}{2 \, x} - \frac {3}{x} + \frac {9}{16 \, x^{2}} + \frac {1}{2} \, e^{\left (2 \, x\right )} - e^{\left (2 \, x + 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.55, size = 58, normalized size = 2.32 \begin {gather*} x^2\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )-\frac {3\,{\mathrm {e}}^{2\,x}}{2}+\frac {x\,\left (\frac {3\,{\mathrm {e}}^4}{2}-3\right )+\frac {9}{16}}{x^2}-x\,\left (2\,{\mathrm {e}}^4+\frac {{\mathrm {e}}^{2\,x}\,\left (16\,{\mathrm {e}}^4-32\right )}{8}-4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.25, size = 60, normalized size = 2.40 \begin {gather*} x^{2} e^{4 x} + x^{2} + \frac {x \left (32 - 16 e^{4}\right )}{8} + \frac {\left (4 x^{2} - 4 x e^{4} + 8 x - 3\right ) e^{2 x}}{2} + \frac {x \left (-48 + 24 e^{4}\right ) + 9}{16 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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