Optimal. Leaf size=16 \[ e^x-\frac {3 e^x}{1-x} \]
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Rubi [A] time = 0.07, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2199, 2194, 2177, 2178} \[ e^x-\frac {3 e^x}{1-x} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {align*} \int \frac {e^x \left (-5+x+x^2\right )}{(-1+x)^2} \, dx &=\int \left (e^x-\frac {3 e^x}{(-1+x)^2}+\frac {3 e^x}{-1+x}\right ) \, dx\\ &=-\left (3 \int \frac {e^x}{(-1+x)^2} \, dx\right )+3 \int \frac {e^x}{-1+x} \, dx+\int e^x \, dx\\ &=e^x-\frac {3 e^x}{1-x}+3 e \text {Ei}(-1+x)-3 \int \frac {e^x}{-1+x} \, dx\\ &=e^x-\frac {3 e^x}{1-x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 12, normalized size = 0.75 \[ \frac {e^x (x+2)}{x-1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 11, normalized size = 0.69 \[ \frac {{\left (x + 2\right )} e^{x}}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 54, normalized size = 3.38 \[ \frac {{\left (x - 1\right )} {\left (\frac {1}{x - 1} + 1\right )} e^{\left ({\left (x - 1\right )} {\left (\frac {1}{x - 1} + 1\right )}\right )} + 2 \, e^{\left ({\left (x - 1\right )} {\left (\frac {1}{x - 1} + 1\right )}\right )}}{{\left (x - 1\right )} {\left (\frac {1}{x - 1} + 1\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 12, normalized size = 0.75 \[ \frac {\left (x +2\right ) {\mathrm e}^{x}}{x -1} \]
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 \[ \frac {{\left (x^{2} + x\right )} e^{x}}{x^{2} - 2 \, x + 1} + \frac {5 \, e E_{2}\left (-x + 1\right )}{x - 1} + \int \frac {{\left (3 \, x + 1\right )} e^{x}}{x^{3} - 3 \, x^{2} + 3 \, x - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 11, normalized size = 0.69 \[ \frac {{\mathrm {e}}^x\,\left (x+2\right )}{x-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 8, normalized size = 0.50 \[ \frac {\left (x + 2\right ) e^{x}}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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