Optimal. Leaf size=29 \[ e^{1-x}-\frac {-25-\frac {1}{x}+2 x}{4 (4-x)} \]
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Rubi [A] time = 0.34, antiderivative size = 26, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 6, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1594, 27, 12, 6688, 2194, 893} \begin {gather*} e^{1-x}+\frac {1}{16 x}+\frac {69}{16 (4-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 893
Rule 1594
Rule 2194
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (e^x \left (-4+2 x+17 x^2\right )+e \left (-64 x^2+32 x^3-4 x^4\right )\right )}{x^2 \left (64-32 x+4 x^2\right )} \, dx\\ &=\int \frac {e^{-x} \left (e^x \left (-4+2 x+17 x^2\right )+e \left (-64 x^2+32 x^3-4 x^4\right )\right )}{4 (-4+x)^2 x^2} \, dx\\ &=\frac {1}{4} \int \frac {e^{-x} \left (e^x \left (-4+2 x+17 x^2\right )+e \left (-64 x^2+32 x^3-4 x^4\right )\right )}{(-4+x)^2 x^2} \, dx\\ &=\frac {1}{4} \int \left (-4 e^{1-x}+\frac {-4+2 x+17 x^2}{(-4+x)^2 x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-4+2 x+17 x^2}{(-4+x)^2 x^2} \, dx-\int e^{1-x} \, dx\\ &=e^{1-x}+\frac {1}{4} \int \left (\frac {69}{4 (-4+x)^2}-\frac {1}{4 x^2}\right ) \, dx\\ &=e^{1-x}+\frac {69}{16 (4-x)}+\frac {1}{16 x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 24, normalized size = 0.83 \begin {gather*} e^{1-x}-\frac {69}{16 (-4+x)}+\frac {1}{16 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 36, normalized size = 1.24 \begin {gather*} \frac {{\left (4 \, {\left (x^{2} - 4 \, x\right )} e - {\left (17 \, x + 1\right )} e^{x}\right )} e^{\left (-x\right )}}{4 \, {\left (x^{2} - 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 36, normalized size = 1.24 \begin {gather*} \frac {4 \, x^{2} e^{\left (-x + 1\right )} - 16 \, x e^{\left (-x + 1\right )} - 17 \, x - 1}{4 \, {\left (x^{2} - 4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 22, normalized size = 0.76
method | result | size |
risch | \(\frac {-\frac {17 x}{4}-\frac {1}{4}}{\left (x -4\right ) x}+{\mathrm e}^{1-x}\) | \(22\) |
norman | \(\frac {\left (x^{2} {\mathrm e}-\frac {17 \,{\mathrm e}^{x} x}{4}-4 x \,{\mathrm e}-\frac {{\mathrm e}^{x}}{4}\right ) {\mathrm e}^{-x}}{x \left (x -4\right )}\) | \(35\) |
default | \(\frac {1}{16 x}-\frac {69}{16 \left (x -4\right )}-16 \,{\mathrm e} \left (-\frac {{\mathrm e}^{-x}}{x -4}+{\mathrm e}^{-4} \expIntegralEi \left (1, x -4\right )\right )+8 \,{\mathrm e} \left (-\frac {4 \,{\mathrm e}^{-x}}{x -4}+3 \,{\mathrm e}^{-4} \expIntegralEi \left (1, x -4\right )\right )-{\mathrm e} \left (-{\mathrm e}^{-x}-\frac {16 \,{\mathrm e}^{-x}}{x -4}+8 \,{\mathrm e}^{-4} \expIntegralEi \left (1, x -4\right )\right )\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {16 \, e^{\left (-3\right )} E_{2}\left (x - 4\right )}{x - 4} - \frac {17 \, x^{2} - 4 \, {\left (x^{3} e - 8 \, x^{2} e\right )} e^{\left (-x\right )} - 67 \, x - 4}{4 \, {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )}} - 32 \, \int \frac {e^{\left (-x + 1\right )}}{x^{3} - 12 \, x^{2} + 48 \, x - 64}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 33, normalized size = 1.14 \begin {gather*} {\mathrm {e}}^{-x}\,\mathrm {e}+\frac {17\,x}{4\,\left (4\,x-x^2\right )}+\frac {1}{4\,\left (4\,x-x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 20, normalized size = 0.69 \begin {gather*} \frac {- 17 x - 1}{4 x^{2} - 16 x} + e e^{- x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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