Optimal. Leaf size=32 \[ -5-x-\left (4+\frac {2}{x}+\frac {x}{4}\right ) \left (-1+\left (e^x-x+\log (2)\right )^2\right ) \]
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Rubi [C] time = 0.37, antiderivative size = 161, normalized size of antiderivative = 5.03, number of steps used = 24, number of rules used = 7, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {12, 14, 2199, 2194, 2177, 2178, 2176} \begin {gather*} \frac {1}{2} \log (256) \text {Ei}(x)-4 \log (2) \text {Ei}(x)-\frac {x^3}{4}+\frac {e^x x^2}{2}-\frac {1}{2} x^2 (8-\log (2))-e^x x-\frac {1}{4} e^{2 x} x+e^x-4 e^{2 x}-\frac {2 e^{2 x}}{x}-\frac {1}{4} x \left (11+\log ^2(2)-32 \log (2)\right )+\frac {2 \left (1-\log ^2(2)\right )}{x}+\frac {1}{2} e^x x (18-\log (2))-\frac {1}{2} e^x (18-\log (2))+\frac {1}{2} e^x (24-17 \log (2))-\frac {e^x \log (256)}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {-8-11 x^2-32 x^3-3 x^4+e^{2 x} \left (8-16 x-33 x^2-2 x^3\right )+\left (32 x^2+4 x^3\right ) \log (2)+\left (8-x^2\right ) \log ^2(2)+e^x \left (48 x^2+36 x^3+2 x^4+\left (16-16 x-34 x^2-2 x^3\right ) \log (2)\right )}{x^2} \, dx\\ &=\frac {1}{4} \int \left (-\frac {e^{2 x} \left (-8+16 x+33 x^2+2 x^3\right )}{x^2}+\frac {-3 x^4-4 x^3 (8-\log (2))-8 \left (1-\log ^2(2)\right )-x^2 \left (11-32 \log (2)+\log ^2(2)\right )}{x^2}+\frac {2 e^x \left (x^4+x^2 (24-17 \log (2))+x^3 (18-\log (2))-8 x \log (2)+\log (256)\right )}{x^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {e^{2 x} \left (-8+16 x+33 x^2+2 x^3\right )}{x^2} \, dx\right )+\frac {1}{4} \int \frac {-3 x^4-4 x^3 (8-\log (2))-8 \left (1-\log ^2(2)\right )-x^2 \left (11-32 \log (2)+\log ^2(2)\right )}{x^2} \, dx+\frac {1}{2} \int \frac {e^x \left (x^4+x^2 (24-17 \log (2))+x^3 (18-\log (2))-8 x \log (2)+\log (256)\right )}{x^2} \, dx\\ &=-\left (\frac {1}{4} \int \left (33 e^{2 x}-\frac {8 e^{2 x}}{x^2}+\frac {16 e^{2 x}}{x}+2 e^{2 x} x\right ) \, dx\right )+\frac {1}{4} \int \left (-3 x^2+4 x (-8+\log (2))-11 \left (1+\frac {1}{11} (-32+\log (2)) \log (2)\right )+\frac {8 \left (-1+\log ^2(2)\right )}{x^2}\right ) \, dx+\frac {1}{2} \int \left (e^x x^2+24 e^x \left (1-\frac {17 \log (2)}{24}\right )-e^x x (-18+\log (2))-\frac {8 e^x \log (2)}{x}+\frac {e^x \log (256)}{x^2}\right ) \, dx\\ &=-\frac {x^3}{4}-\frac {1}{2} x^2 (8-\log (2))+\frac {2 \left (1-\log ^2(2)\right )}{x}-\frac {1}{4} x \left (11-32 \log (2)+\log ^2(2)\right )-\frac {1}{2} \int e^{2 x} x \, dx+\frac {1}{2} \int e^x x^2 \, dx+2 \int \frac {e^{2 x}}{x^2} \, dx-4 \int \frac {e^{2 x}}{x} \, dx-\frac {33}{4} \int e^{2 x} \, dx+\frac {1}{2} (24-17 \log (2)) \int e^x \, dx+\frac {1}{2} (18-\log (2)) \int e^x x \, dx-(4 \log (2)) \int \frac {e^x}{x} \, dx+\frac {1}{2} \log (256) \int \frac {e^x}{x^2} \, dx\\ &=-\frac {33 e^{2 x}}{8}-\frac {2 e^{2 x}}{x}-\frac {1}{4} e^{2 x} x+\frac {e^x x^2}{2}-\frac {x^3}{4}-4 \text {Ei}(2 x)+\frac {1}{2} e^x (24-17 \log (2))-\frac {1}{2} x^2 (8-\log (2))+\frac {1}{2} e^x x (18-\log (2))-4 \text {Ei}(x) \log (2)+\frac {2 \left (1-\log ^2(2)\right )}{x}-\frac {1}{4} x \left (11-32 \log (2)+\log ^2(2)\right )-\frac {e^x \log (256)}{2 x}+\frac {1}{4} \int e^{2 x} \, dx+4 \int \frac {e^{2 x}}{x} \, dx+\frac {1}{2} (-18+\log (2)) \int e^x \, dx+\frac {1}{2} \log (256) \int \frac {e^x}{x} \, dx-\int e^x x \, dx\\ &=-4 e^{2 x}-\frac {2 e^{2 x}}{x}-e^x x-\frac {1}{4} e^{2 x} x+\frac {e^x x^2}{2}-\frac {x^3}{4}+\frac {1}{2} e^x (24-17 \log (2))-\frac {1}{2} x^2 (8-\log (2))-\frac {1}{2} e^x (18-\log (2))+\frac {1}{2} e^x x (18-\log (2))-4 \text {Ei}(x) \log (2)+\frac {2 \left (1-\log ^2(2)\right )}{x}-\frac {1}{4} x \left (11-32 \log (2)+\log ^2(2)\right )-\frac {e^x \log (256)}{2 x}+\frac {1}{2} \text {Ei}(x) \log (256)+\int e^x \, dx\\ &=e^x-4 e^{2 x}-\frac {2 e^{2 x}}{x}-e^x x-\frac {1}{4} e^{2 x} x+\frac {e^x x^2}{2}-\frac {x^3}{4}+\frac {1}{2} e^x (24-17 \log (2))-\frac {1}{2} x^2 (8-\log (2))-\frac {1}{2} e^x (18-\log (2))+\frac {1}{2} e^x x (18-\log (2))-4 \text {Ei}(x) \log (2)+\frac {2 \left (1-\log ^2(2)\right )}{x}-\frac {1}{4} x \left (11-32 \log (2)+\log ^2(2)\right )-\frac {e^x \log (256)}{2 x}+\frac {1}{2} \text {Ei}(x) \log (256)\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.14, size = 82, normalized size = 2.56 \begin {gather*} -\frac {x^4+e^{2 x} \left (8+16 x+x^2\right )+8 \left (-1+\log ^2(2)\right )-x^3 (-16+\log (4))+e^x \left (-2 x^3+x^2 (-32+\log (4))+\log (65536)+x (-16+\log (4294967296))\right )+x^2 \left (11+\log ^2(2)-\log (4294967296)\right )}{4 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 84, normalized size = 2.62 \begin {gather*} -\frac {x^{4} + 16 \, x^{3} + {\left (x^{2} + 8\right )} \log \relax (2)^{2} + 11 \, x^{2} + {\left (x^{2} + 16 \, x + 8\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} + 16 \, x^{2} - {\left (x^{2} + 16 \, x + 8\right )} \log \relax (2) + 8 \, x\right )} e^{x} - 2 \, {\left (x^{3} + 16 \, x^{2}\right )} \log \relax (2) - 8}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 110, normalized size = 3.44 \begin {gather*} -\frac {x^{4} - 2 \, x^{3} e^{x} - 2 \, x^{3} \log \relax (2) + 2 \, x^{2} e^{x} \log \relax (2) + x^{2} \log \relax (2)^{2} + 16 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 32 \, x^{2} e^{x} - 32 \, x^{2} \log \relax (2) + 32 \, x e^{x} \log \relax (2) + 11 \, x^{2} + 16 \, x e^{\left (2 \, x\right )} - 16 \, x e^{x} + 16 \, e^{x} \log \relax (2) + 8 \, \log \relax (2)^{2} + 8 \, e^{\left (2 \, x\right )} - 8}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 101, normalized size = 3.16
method | result | size |
norman | \(\frac {\left (\frac {\ln \relax (2)}{2}-4\right ) x^{3}+\left (-\frac {\ln \relax (2)^{2}}{4}+8 \ln \relax (2)-\frac {11}{4}\right ) x^{2}+\left (4-8 \ln \relax (2)\right ) x \,{\mathrm e}^{x}+\left (8-\frac {\ln \relax (2)}{2}\right ) x^{2} {\mathrm e}^{x}-\frac {x^{4}}{4}-2 \,{\mathrm e}^{2 x}-4 x \,{\mathrm e}^{2 x}+\frac {{\mathrm e}^{x} x^{3}}{2}-4 \,{\mathrm e}^{x} \ln \relax (2)-\frac {{\mathrm e}^{2 x} x^{2}}{4}-2 \ln \relax (2)^{2}+2}{x}\) | \(101\) |
risch | \(-\frac {x \ln \relax (2)^{2}}{4}+\frac {x^{2} \ln \relax (2)}{2}-\frac {x^{3}}{4}+8 x \ln \relax (2)-4 x^{2}-\frac {11 x}{4}-\frac {2 \ln \relax (2)^{2}}{x}+\frac {2}{x}-\frac {\left (x^{2}+16 x +8\right ) {\mathrm e}^{2 x}}{4 x}-\frac {\left (x^{2} \ln \relax (2)-x^{3}+16 x \ln \relax (2)-16 x^{2}+8 \ln \relax (2)-8 x \right ) {\mathrm e}^{x}}{2 x}\) | \(101\) |
default | \(-4 x^{2}-\frac {11 x}{4}+\frac {2}{x}-\frac {x^{3}}{4}-4 \,{\mathrm e}^{2 x}-\frac {x \ln \relax (2)^{2}}{4}-\frac {x \,{\mathrm e}^{2 x}}{4}+\frac {x^{2} \ln \relax (2)}{2}+8 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x}+\frac {{\mathrm e}^{x} x^{2}}{2}-8 \,{\mathrm e}^{x} \ln \relax (2)-\frac {2 \,{\mathrm e}^{2 x}}{x}-\frac {2 \ln \relax (2)^{2}}{x}-\frac {4 \ln \relax (2) {\mathrm e}^{x}}{x}-\frac {x \ln \relax (2) {\mathrm e}^{x}}{2}+8 x \ln \relax (2)\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.39, size = 130, normalized size = 4.06 \begin {gather*} -\frac {1}{4} \, x^{3} + \frac {1}{2} \, x^{2} \log \relax (2) - \frac {1}{2} \, {\left (x - 1\right )} e^{x} \log \relax (2) - \frac {1}{4} \, x \log \relax (2)^{2} - 4 \, x^{2} - \frac {1}{8} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + 9 \, {\left (x - 1\right )} e^{x} + 8 \, x \log \relax (2) - 4 \, {\rm Ei}\relax (x) \log \relax (2) - \frac {17}{2} \, e^{x} \log \relax (2) + 4 \, \Gamma \left (-1, -x\right ) \log \relax (2) - \frac {11}{4} \, x - \frac {2 \, \log \relax (2)^{2}}{x} + \frac {2}{x} - 4 \, {\rm Ei}\left (2 \, x\right ) - \frac {33}{8} \, e^{\left (2 \, x\right )} + 12 \, e^{x} + 4 \, \Gamma \left (-1, -2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.22, size = 90, normalized size = 2.81 \begin {gather*} x^2\,\left (\frac {\ln \relax (4)}{4}+\frac {{\mathrm {e}}^x}{2}-4\right )-\frac {2\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^x\,\ln \relax (2)+2\,{\ln \relax (2)}^2-2}{x}-{\mathrm {e}}^x\,\left (8\,\ln \relax (2)-4\right )-4\,{\mathrm {e}}^{2\,x}-x\,\left (\frac {{\mathrm {e}}^{2\,x}}{4}-8\,\ln \relax (2)+\frac {{\mathrm {e}}^x\,\left (\ln \relax (4)-32\right )}{4}+\frac {{\ln \relax (2)}^2}{4}+\frac {11}{4}\right )-\frac {x^3}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.26, size = 110, normalized size = 3.44 \begin {gather*} - \frac {x^{3}}{4} - \frac {x^{2} \left (16 - 2 \log {\relax (2 )}\right )}{4} - \frac {x \left (- 32 \log {\relax (2 )} + \log {\relax (2 )}^{2} + 11\right )}{4} - \frac {-8 + 8 \log {\relax (2 )}^{2}}{4 x} + \frac {\left (- 2 x^{3} - 32 x^{2} - 16 x\right ) e^{2 x} + \left (4 x^{4} - 4 x^{3} \log {\relax (2 )} + 64 x^{3} - 64 x^{2} \log {\relax (2 )} + 32 x^{2} - 32 x \log {\relax (2 )}\right ) e^{x}}{8 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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