Optimal. Leaf size=28 \[ e^{3+x}+\frac {e^{2 x} (-1+x)^2}{81 (-4+x)^2 x^2} \]
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Rubi [B] time = 0.49, antiderivative size = 62, normalized size of antiderivative = 2.21, number of steps used = 17, number of rules used = 5, integrand size = 78, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {6688, 2194, 6742, 2177, 2178} \begin {gather*} \frac {e^{2 x}}{1296 x^2}+e^{x+3}-\frac {e^{2 x}}{864 x}-\frac {e^{2 x}}{864 (4-x)}+\frac {e^{2 x}}{144 (4-x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2194
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^{3+x}+\frac {2 e^{2 x} \left (4-10 x+12 x^2-7 x^3+x^4\right )}{81 (-4+x)^3 x^3}\right ) \, dx\\ &=\frac {2}{81} \int \frac {e^{2 x} \left (4-10 x+12 x^2-7 x^3+x^4\right )}{(-4+x)^3 x^3} \, dx+\int e^{3+x} \, dx\\ &=e^{3+x}+\frac {2}{81} \int \left (-\frac {9 e^{2 x}}{16 (-4+x)^3}+\frac {33 e^{2 x}}{64 (-4+x)^2}+\frac {3 e^{2 x}}{32 (-4+x)}-\frac {e^{2 x}}{16 x^3}+\frac {7 e^{2 x}}{64 x^2}-\frac {3 e^{2 x}}{32 x}\right ) \, dx\\ &=e^{3+x}-\frac {1}{648} \int \frac {e^{2 x}}{x^3} \, dx+\frac {1}{432} \int \frac {e^{2 x}}{-4+x} \, dx-\frac {1}{432} \int \frac {e^{2 x}}{x} \, dx+\frac {7 \int \frac {e^{2 x}}{x^2} \, dx}{2592}+\frac {11}{864} \int \frac {e^{2 x}}{(-4+x)^2} \, dx-\frac {1}{72} \int \frac {e^{2 x}}{(-4+x)^3} \, dx\\ &=e^{3+x}+\frac {e^{2 x}}{144 (4-x)^2}+\frac {11 e^{2 x}}{864 (4-x)}+\frac {e^{2 x}}{1296 x^2}-\frac {7 e^{2 x}}{2592 x}+\frac {1}{432} e^8 \text {Ei}(-2 (4-x))-\frac {\text {Ei}(2 x)}{432}-\frac {1}{648} \int \frac {e^{2 x}}{x^2} \, dx+\frac {7 \int \frac {e^{2 x}}{x} \, dx}{1296}-\frac {1}{72} \int \frac {e^{2 x}}{(-4+x)^2} \, dx+\frac {11}{432} \int \frac {e^{2 x}}{-4+x} \, dx\\ &=e^{3+x}+\frac {e^{2 x}}{144 (4-x)^2}-\frac {e^{2 x}}{864 (4-x)}+\frac {e^{2 x}}{1296 x^2}-\frac {e^{2 x}}{864 x}+\frac {1}{36} e^8 \text {Ei}(-2 (4-x))+\frac {\text {Ei}(2 x)}{324}-\frac {1}{324} \int \frac {e^{2 x}}{x} \, dx-\frac {1}{36} \int \frac {e^{2 x}}{-4+x} \, dx\\ &=e^{3+x}+\frac {e^{2 x}}{144 (4-x)^2}-\frac {e^{2 x}}{864 (4-x)}+\frac {e^{2 x}}{1296 x^2}-\frac {e^{2 x}}{864 x}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 38, normalized size = 1.36 \begin {gather*} \frac {e^x \left (e^x (-1+x)^2+81 e^3 (-4+x)^2 x^2\right )}{81 (-4+x)^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 56, normalized size = 2.00 \begin {gather*} \frac {{\left ({\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x + 6\right )} + 81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} e^{\left (x + 9\right )}\right )} e^{\left (-6\right )}}{81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 65, normalized size = 2.32 \begin {gather*} \frac {81 \, x^{4} e^{\left (x + 3\right )} - 648 \, x^{3} e^{\left (x + 3\right )} + x^{2} e^{\left (2 \, x\right )} + 1296 \, x^{2} e^{\left (x + 3\right )} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )}}{81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 28, normalized size = 1.00
method | result | size |
risch | \(\frac {\left (x^{2}-2 x +1\right ) {\mathrm e}^{2 x}}{81 \left (x -4\right )^{2} x^{2}}+{\mathrm e}^{3+x}\) | \(28\) |
norman | \(\frac {{\mathrm e}^{x} {\mathrm e}^{3} x^{4}+\frac {{\mathrm e}^{2 x}}{81}-\frac {2 x \,{\mathrm e}^{2 x}}{81}+\frac {{\mathrm e}^{2 x} x^{2}}{81}+16 x^{2} {\mathrm e}^{3} {\mathrm e}^{x}-8 \,{\mathrm e}^{x} {\mathrm e}^{3} x^{3}}{\left (x -4\right )^{2} x^{2}}\) | \(59\) |
default | \(\frac {{\mathrm e}^{2 x}}{144 \left (x -4\right )^{2}}+\frac {{\mathrm e}^{2 x}}{864 x -3456}-\frac {{\mathrm e}^{2 x}}{864 x}+\frac {{\mathrm e}^{2 x}}{1296 x^{2}}-64 \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{x}}{2 \left (x -4\right )^{2}}-\frac {{\mathrm e}^{x}}{2 \left (x -4\right )}-\frac {{\mathrm e}^{4} \expIntegralEi \left (1, -x +4\right )}{2}\right )+48 \,{\mathrm e}^{3} \left (-\frac {2 \,{\mathrm e}^{x}}{\left (x -4\right )^{2}}-\frac {3 \,{\mathrm e}^{x}}{x -4}-3 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x +4\right )\right )-12 \,{\mathrm e}^{3} \left (-\frac {8 \,{\mathrm e}^{x}}{\left (x -4\right )^{2}}-\frac {16 \,{\mathrm e}^{x}}{x -4}-17 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x +4\right )\right )+{\mathrm e}^{3} \left ({\mathrm e}^{x}-\frac {32 \,{\mathrm e}^{x}}{\left (x -4\right )^{2}}-\frac {80 \,{\mathrm e}^{x}}{x -4}-92 \,{\mathrm e}^{4} \expIntegralEi \left (1, -x +4\right )\right )\) | \(179\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 57, normalized size = 2.04 \begin {gather*} \frac {{\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + 81 \, {\left (x^{4} e^{3} - 8 \, x^{3} e^{3} + 16 \, x^{2} e^{3}\right )} e^{x}}{81 \, {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.21, size = 55, normalized size = 1.96 \begin {gather*} \frac {{\mathrm {e}}^{x-3}\,\left ({\mathrm {e}}^{x+3}-2\,x\,{\mathrm {e}}^{x+3}+x^2\,{\mathrm {e}}^{x+3}+1296\,x^2\,{\mathrm {e}}^6-648\,x^3\,{\mathrm {e}}^6+81\,x^4\,{\mathrm {e}}^6\right )}{81\,x^2\,{\left (x-4\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 61, normalized size = 2.18 \begin {gather*} \frac {\left (x^{2} - 2 x + 1\right ) e^{2 x} + \left (81 x^{4} e^{3} - 648 x^{3} e^{3} + 1296 x^{2} e^{3}\right ) \sqrt {e^{2 x}}}{81 x^{4} - 648 x^{3} + 1296 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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