Optimal. Leaf size=22 \[ 4-e^x+x-\frac {3 \log (2)}{e (-3+4 x)} \]
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Rubi [A] time = 0.25, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 5, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {12, 27, 6742, 2194, 683} \begin {gather*} x-e^x+\frac {\log (4096)}{4 e (3-4 x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 683
Rule 2194
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{5+x} \left (-9+24 x-16 x^2\right )+e^5 \left (9-24 x+16 x^2\right )+12 e^4 \log (2)}{9-24 x+16 x^2} \, dx}{e^5}\\ &=\frac {\int \frac {e^{5+x} \left (-9+24 x-16 x^2\right )+e^5 \left (9-24 x+16 x^2\right )+12 e^4 \log (2)}{(-3+4 x)^2} \, dx}{e^5}\\ &=\frac {\int \left (-e^{5+x}+\frac {e^4 \left (9 e-24 e x+16 e x^2+\log (4096)\right )}{(-3+4 x)^2}\right ) \, dx}{e^5}\\ &=-\frac {\int e^{5+x} \, dx}{e^5}+\frac {\int \frac {9 e-24 e x+16 e x^2+\log (4096)}{(-3+4 x)^2} \, dx}{e}\\ &=-e^x+\frac {\int \left (e+\frac {\log (4096)}{(-3+4 x)^2}\right ) \, dx}{e}\\ &=-e^x+x+\frac {\log (4096)}{4 e (3-4 x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 25, normalized size = 1.14 \begin {gather*} \frac {-e^{1+x}+e x+\frac {\log (4096)}{12-16 x}}{e} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 40, normalized size = 1.82 \begin {gather*} \frac {{\left ({\left (4 \, x^{2} - 3 \, x\right )} e^{5} - {\left (4 \, x - 3\right )} e^{\left (x + 5\right )} - 3 \, e^{4} \log \relax (2)\right )} e^{\left (-5\right )}}{4 \, x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 42, normalized size = 1.91 \begin {gather*} \frac {{\left (4 \, x^{2} e^{5} - 3 \, x e^{5} - 4 \, x e^{\left (x + 5\right )} - 3 \, e^{4} \log \relax (2) + 3 \, e^{\left (x + 5\right )}\right )} e^{\left (-5\right )}}{4 \, x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 18, normalized size = 0.82
method | result | size |
risch | \(x -\frac {3 \,{\mathrm e}^{-1} \ln \relax (2)}{4 \left (x -\frac {3}{4}\right )}-{\mathrm e}^{x}\) | \(18\) |
norman | \(\frac {4 x^{2}-4 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{x}-\frac {3 \left (4 \,{\mathrm e}^{4} \ln \relax (2)+3 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-5}}{4}}{4 x -3}\) | \(41\) |
default | \({\mathrm e}^{-5} \left (x \,{\mathrm e}^{5}-\frac {3 \,{\mathrm e}^{4} \ln \relax (2)}{4 x -3}-9 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{x}}{16 \left (x -\frac {3}{4}\right )}-\frac {{\mathrm e}^{\frac {3}{4}} \expIntegralEi \left (1, -x +\frac {3}{4}\right )}{16}\right )+24 \,{\mathrm e}^{5} \left (-\frac {3 \,{\mathrm e}^{x}}{64 \left (x -\frac {3}{4}\right )}-\frac {7 \,{\mathrm e}^{\frac {3}{4}} \expIntegralEi \left (1, -x +\frac {3}{4}\right )}{64}\right )-16 \,{\mathrm e}^{5} \left (\frac {{\mathrm e}^{x}}{16}-\frac {9 \,{\mathrm e}^{x}}{256 \left (x -\frac {3}{4}\right )}-\frac {33 \,{\mathrm e}^{\frac {3}{4}} \expIntegralEi \left (1, -x +\frac {3}{4}\right )}{256}\right )\right )\) | \(103\) |
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 {1}{4} \, {\left ({\left (4 \, x - \frac {9}{4 \, x - 3} + 6 \, \log \left (4 \, x - 3\right )\right )} e^{5} + 6 \, {\left (\frac {3}{4 \, x - 3} - \log \left (4 \, x - 3\right )\right )} e^{5} - \frac {32 \, {\left (2 \, x^{2} e^{5} - 3 \, x e^{5}\right )} e^{x}}{16 \, x^{2} - 24 \, x + 9} + \frac {9 \, e^{\frac {23}{4}} E_{2}\left (-x + \frac {3}{4}\right )}{4 \, x - 3} - \frac {12 \, e^{4} \log \relax (2)}{4 \, x - 3} - \frac {9 \, e^{5}}{4 \, x - 3} + 288 \, \int \frac {e^{\left (x + 5\right )}}{64 \, x^{3} - 144 \, x^{2} + 108 \, x - 27}\,{d x}\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 24, normalized size = 1.09 \begin {gather*} x-{\mathrm {e}}^x+\frac {3\,{\mathrm {e}}^4\,\ln \relax (2)}{3\,{\mathrm {e}}^5-4\,x\,{\mathrm {e}}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 20, normalized size = 0.91 \begin {gather*} x - e^{x} - \frac {3 \log {\relax (2 )}}{4 e x - 3 e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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