Optimal. Leaf size=24 \[ -7+e^{6 x}-\frac {1}{x}+\left (1+9 e^{-x} x\right )^2 \]
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Rubi [A] time = 0.37, antiderivative size = 40, normalized size of antiderivative = 1.67, number of steps used = 12, number of rules used = 4, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6688, 2194, 2176, 2196} \begin {gather*} 81 e^{-2 x} x^2+18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2196
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (6 e^{6 x}-18 e^{-x} (-1+x)+\frac {1}{x^2}-162 e^{-2 x} (-1+x) x\right ) \, dx\\ &=-\frac {1}{x}+6 \int e^{6 x} \, dx-18 \int e^{-x} (-1+x) \, dx-162 \int e^{-2 x} (-1+x) x \, dx\\ &=e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}-18 \int e^{-x} \, dx-162 \int \left (-e^{-2 x} x+e^{-2 x} x^2\right ) \, dx\\ &=18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}+162 \int e^{-2 x} x \, dx-162 \int e^{-2 x} x^2 \, dx\\ &=18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}-81 e^{-2 x} x+81 e^{-2 x} x^2+81 \int e^{-2 x} \, dx-162 \int e^{-2 x} x \, dx\\ &=-\frac {81}{2} e^{-2 x}+18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}+81 e^{-2 x} x^2-81 \int e^{-2 x} \, dx\\ &=18 e^{-x}+e^{6 x}-18 e^{-x} (1-x)-\frac {1}{x}+81 e^{-2 x} x^2\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.21 \begin {gather*} e^{6 x}-\frac {1}{x}+18 e^{-x} x+81 e^{-2 x} x^2 \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 33, normalized size = 1.38 \begin {gather*} \frac {{\left (81 \, x^{3} + 18 \, x^{2} e^{x} + x e^{\left (8 \, x\right )} - e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 30, normalized size = 1.25 \begin {gather*} \frac {81 \, x^{3} e^{\left (-2 \, x\right )} + 18 \, x^{2} e^{\left (-x\right )} + x e^{\left (6 \, x\right )} - 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 27, normalized size = 1.12
method | result | size |
default | \(-\frac {1}{x}+{\mathrm e}^{6 x}+18 x \,{\mathrm e}^{-x}+81 x^{2} {\mathrm e}^{-2 x}\) | \(27\) |
risch | \(-\frac {1}{x}+{\mathrm e}^{6 x}+18 x \,{\mathrm e}^{-x}+81 x^{2} {\mathrm e}^{-2 x}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 52, normalized size = 2.17 \begin {gather*} 18 \, {\left (x + 1\right )} e^{\left (-x\right )} + \frac {81}{2} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {81}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} - \frac {1}{x} + e^{\left (6 \, x\right )} - 18 \, e^{\left (-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 26, normalized size = 1.08 \begin {gather*} {\mathrm {e}}^{6\,x}+18\,x\,{\mathrm {e}}^{-x}+81\,x^2\,{\mathrm {e}}^{-2\,x}-\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 24, normalized size = 1.00 \begin {gather*} 81 x^{2} e^{- 2 x} + 18 x e^{- x} + e^{6 x} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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