Optimal. Leaf size=21 \[ e^{4-4 (-2+x)} \left (3+\frac {5}{2 x^3}\right )+x \]
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Rubi [A] time = 0.40, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {12, 6688, 2199, 2194, 2177, 2178} \begin {gather*} \frac {5 e^{12-4 x}}{2 x^3}+x+3 e^{12-4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {e^{8-4 x} \left (2 e^{-8+4 x} x^4+e^4 \left (-15-20 x-24 x^4\right )\right )}{x^4} \, dx\\ &=\frac {1}{2} \int \left (2-\frac {e^{12-4 x} \left (15+20 x+24 x^4\right )}{x^4}\right ) \, dx\\ &=x-\frac {1}{2} \int \frac {e^{12-4 x} \left (15+20 x+24 x^4\right )}{x^4} \, dx\\ &=x-\frac {1}{2} \int \left (24 e^{12-4 x}+\frac {15 e^{12-4 x}}{x^4}+\frac {20 e^{12-4 x}}{x^3}\right ) \, dx\\ &=x-\frac {15}{2} \int \frac {e^{12-4 x}}{x^4} \, dx-10 \int \frac {e^{12-4 x}}{x^3} \, dx-12 \int e^{12-4 x} \, dx\\ &=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+\frac {5 e^{12-4 x}}{x^2}+x+10 \int \frac {e^{12-4 x}}{x^3} \, dx+20 \int \frac {e^{12-4 x}}{x^2} \, dx\\ &=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}-\frac {20 e^{12-4 x}}{x}+x-20 \int \frac {e^{12-4 x}}{x^2} \, dx-80 \int \frac {e^{12-4 x}}{x} \, dx\\ &=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x-80 e^{12} \text {Ei}(-4 x)+80 \int \frac {e^{12-4 x}}{x} \, dx\\ &=3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 25, normalized size = 1.19 \begin {gather*} 3 e^{12-4 x}+\frac {5 e^{12-4 x}}{2 x^3}+x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 33, normalized size = 1.57 \begin {gather*} \frac {{\left (2 \, x^{4} e^{\left (4 \, x - 8\right )} + {\left (6 \, x^{3} + 5\right )} e^{4}\right )} e^{\left (-4 \, x + 8\right )}}{2 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 30, normalized size = 1.43 \begin {gather*} \frac {2 \, x^{4} + 6 \, x^{3} e^{\left (-4 \, x + 12\right )} + 5 \, e^{\left (-4 \, x + 12\right )}}{2 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 21, normalized size = 1.00
method | result | size |
risch | \(x +\frac {\left (6 x^{3}+5\right ) {\mathrm e}^{-4 x +12}}{2 x^{3}}\) | \(21\) |
norman | \(\frac {\left (x^{4} {\mathrm e}^{4 x -8}+3 x^{3} {\mathrm e}^{4}+\frac {5 \,{\mathrm e}^{4}}{2}\right ) {\mathrm e}^{-4 x +8}}{x^{3}}\) | \(35\) |
derivativedivides | \(x -2-14048 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{384 x^{3}}+\frac {{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{6}\right )-6304 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{384 x^{3}}-\frac {11 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{6}\right )-1152 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{192 x^{3}}+\frac {59 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )-96 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{24 x^{3}}-\frac {619 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )-3 \,{\mathrm e}^{4} \left (-{\mathrm e}^{-4 x +8}-\frac {2 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{3 x^{3}}+\frac {6368 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )\) | \(205\) |
default | \(x -2-14048 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (\left (4 x -8\right )^{2}+60 x -62\right )}{384 x^{3}}+\frac {{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{6}\right )-6304 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (11 \left (4 x -8\right )^{2}+660 x -688\right )}{384 x^{3}}-\frac {11 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{6}\right )-1152 \,{\mathrm e}^{4} \left (-\frac {{\mathrm e}^{-4 x +8} \left (59 \left (4 x -8\right )^{2}+3552 x -3712\right )}{192 x^{3}}+\frac {59 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )-96 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{-4 x +8} \left (77 \left (4 x -8\right )^{2}+4656 x -4864\right )}{24 x^{3}}-\frac {619 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )-3 \,{\mathrm e}^{4} \left (-{\mathrm e}^{-4 x +8}-\frac {2 \,{\mathrm e}^{-4 x +8} \left (49 \left (4 x -8\right )^{2}+2976 x -3104\right )}{3 x^{3}}+\frac {6368 \,{\mathrm e}^{8} \expIntegralEi \left (1, 4 x \right )}{3}\right )\) | \(205\) |
meijerg | \(-\frac {{\mathrm e}^{-4 x +4 x \,{\mathrm e}^{8}} \left (1-{\mathrm e}^{4 x \left (-{\mathrm e}^{8}+1\right )}\right )}{4 \left (-{\mathrm e}^{8}+1\right )}-3 \,{\mathrm e}^{-4 x +4+4 x \,{\mathrm e}^{8}} \left (1-{\mathrm e}^{-4 x \,{\mathrm e}^{8}}\right )-160 \,{\mathrm e}^{-4 x +28+4 x \,{\mathrm e}^{8}} \left (-\frac {{\mathrm e}^{-16}}{32 x^{2}}+\frac {{\mathrm e}^{-8}}{4 x}+\frac {13}{4}+\frac {\ln \relax (x )}{2}+\ln \relax (2)+\frac {{\mathrm e}^{-16} \left (144 x^{2} {\mathrm e}^{16}-48 x \,{\mathrm e}^{8}+6\right )}{192 x^{2}}-\frac {{\mathrm e}^{-16-4 x \,{\mathrm e}^{8}} \left (-12 x \,{\mathrm e}^{8}+3\right )}{96 x^{2}}-\frac {\ln \left (4 x \,{\mathrm e}^{8}\right )}{2}-\frac {\expIntegralEi \left (1, 4 x \,{\mathrm e}^{8}\right )}{2}\right )-480 \,{\mathrm e}^{4 x \,{\mathrm e}^{8}+36-4 x} \left (-\frac {{\mathrm e}^{-24}}{192 x^{3}}+\frac {{\mathrm e}^{-16}}{32 x^{2}}-\frac {{\mathrm e}^{-8}}{8 x}-\frac {37}{36}-\frac {\ln \relax (x )}{6}-\frac {\ln \relax (2)}{3}+\frac {{\mathrm e}^{-24} \left (-1408 x^{3} {\mathrm e}^{24}+576 x^{2} {\mathrm e}^{16}-144 x \,{\mathrm e}^{8}+24\right )}{4608 x^{3}}-\frac {{\mathrm e}^{-24-4 x \,{\mathrm e}^{8}} \left (64 x^{2} {\mathrm e}^{16}-16 x \,{\mathrm e}^{8}+8\right )}{1536 x^{3}}+\frac {\ln \left (4 x \,{\mathrm e}^{8}\right )}{6}+\frac {\expIntegralEi \left (1, 4 x \,{\mathrm e}^{8}\right )}{6}\right )\) | \(268\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 28, normalized size = 1.33 \begin {gather*} 160 \, e^{12} \Gamma \left (-2, 4 \, x\right ) + 480 \, e^{12} \Gamma \left (-3, 4 \, x\right ) + x + 3 \, e^{\left (-4 \, x + 12\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.24, size = 21, normalized size = 1.00 \begin {gather*} x+3\,{\mathrm {e}}^{12-4\,x}+\frac {5\,{\mathrm {e}}^{12-4\,x}}{2\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 26, normalized size = 1.24 \begin {gather*} x + \frac {\left (6 x^{3} e^{4} + 5 e^{4}\right ) e^{8 - 4 x}}{2 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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