Optimal. Leaf size=33 \[ \left (e^x+e^{2 x}-x^2\right ) \left (2+\frac {5 \left (e^x+2 x-\log (2)\right )}{x}\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.20, antiderivative size = 87, normalized size of antiderivative = 2.64, number of steps
used = 17, number of rules used = 7, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {14, 2228,
2230, 2225, 2209, 2208, 2207} \begin {gather*} -2 (5-\log (32)) \text {Ei}(2 x)+10 (1-\log (2)) \text {Ei}(2 x)-12 x^2-5 e^x x+12 e^x+12 e^{2 x}+\frac {5 e^{3 x}}{x}+x \log (32)-\frac {e^x \log (32)}{x}+\frac {e^{2 x} (5-\log (32))}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2228
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-24 x+\frac {5 e^{3 x} (-1+3 x)}{x^2}+\log (32)+\frac {e^{2 x} \left (-5+24 x^2+10 x (1-\log (2))+\log (32)\right )}{x^2}-\frac {e^x \left (-7 x^2+5 x^3-\log (32)+x \log (32)\right )}{x^2}\right ) \, dx\\ &=-12 x^2+x \log (32)+5 \int \frac {e^{3 x} (-1+3 x)}{x^2} \, dx+\int \frac {e^{2 x} \left (-5+24 x^2+10 x (1-\log (2))+\log (32)\right )}{x^2} \, dx-\int \frac {e^x \left (-7 x^2+5 x^3-\log (32)+x \log (32)\right )}{x^2} \, dx\\ &=\frac {5 e^{3 x}}{x}-12 x^2+x \log (32)+\int \left (24 e^{2 x}-\frac {10 e^{2 x} (-1+\log (2))}{x}+\frac {e^{2 x} (-5+\log (32))}{x^2}\right ) \, dx-\int \left (-7 e^x+5 e^x x-\frac {e^x \log (32)}{x^2}+\frac {e^x \log (32)}{x}\right ) \, dx\\ &=\frac {5 e^{3 x}}{x}-12 x^2+x \log (32)-5 \int e^x x \, dx+7 \int e^x \, dx+24 \int e^{2 x} \, dx+(10 (1-\log (2))) \int \frac {e^{2 x}}{x} \, dx+(-5+\log (32)) \int \frac {e^{2 x}}{x^2} \, dx+\log (32) \int \frac {e^x}{x^2} \, dx-\log (32) \int \frac {e^x}{x} \, dx\\ &=7 e^x+12 e^{2 x}+\frac {5 e^{3 x}}{x}-5 e^x x-12 x^2+10 \text {Ei}(2 x) (1-\log (2))+\frac {e^{2 x} (5-\log (32))}{x}-\frac {e^x \log (32)}{x}+x \log (32)-\text {Ei}(x) \log (32)+5 \int e^x \, dx-(2 (5-\log (32))) \int \frac {e^{2 x}}{x} \, dx+\log (32) \int \frac {e^x}{x} \, dx\\ &=12 e^x+12 e^{2 x}+\frac {5 e^{3 x}}{x}-5 e^x x-12 x^2+10 \text {Ei}(2 x) (1-\log (2))+\frac {e^{2 x} (5-\log (32))}{x}-2 \text {Ei}(2 x) (5-\log (32))-\frac {e^x \log (32)}{x}+x \log (32)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 31, normalized size = 0.94 \begin {gather*} \frac {\left (e^x+e^{2 x}-x^2\right ) \left (5 e^x+12 x-\log (32)\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs.
\(2(30)=60\).
time = 0.38, size = 65, normalized size = 1.97
method | result | size |
risch | \(5 x \ln \left (2\right )-12 x^{2}+\frac {5 \,{\mathrm e}^{3 x}}{x}-\frac {\left (-12 x +5 \ln \left (2\right )-5\right ) {\mathrm e}^{2 x}}{x}-\frac {\left (5 x^{2}+5 \ln \left (2\right )-12 x \right ) {\mathrm e}^{x}}{x}\) | \(59\) |
norman | \(\frac {\left (-5 \ln \left (2\right )+5\right ) {\mathrm e}^{2 x}-12 x^{3}+5 \,{\mathrm e}^{3 x}+12 x \,{\mathrm e}^{2 x}+5 x^{2} \ln \left (2\right )+12 \,{\mathrm e}^{x} x -5 \,{\mathrm e}^{x} x^{2}-5 \,{\mathrm e}^{x} \ln \left (2\right )}{x}\) | \(60\) |
default | \(-12 x^{2}+12 \,{\mathrm e}^{2 x}+\frac {5 \,{\mathrm e}^{3 x}}{x}-5 \,{\mathrm e}^{x} x +12 \,{\mathrm e}^{x}+\frac {5 \,{\mathrm e}^{2 x}}{x}-\frac {5 \ln \left (2\right ) {\mathrm e}^{x}}{x}-\frac {5 \ln \left (2\right ) {\mathrm e}^{2 x}}{x}+5 x \ln \left (2\right )\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.32, size = 86, normalized size = 2.61 \begin {gather*} -12 \, x^{2} - 5 \, {\left (x - 1\right )} e^{x} + 5 \, x \log \left (2\right ) - 10 \, {\rm Ei}\left (2 \, x\right ) \log \left (2\right ) - 5 \, {\rm Ei}\left (x\right ) \log \left (2\right ) + 5 \, \Gamma \left (-1, -x\right ) \log \left (2\right ) + 10 \, \Gamma \left (-1, -2 \, x\right ) \log \left (2\right ) + 15 \, {\rm Ei}\left (3 \, x\right ) + 10 \, {\rm Ei}\left (2 \, x\right ) + 12 \, e^{\left (2 \, x\right )} + 7 \, e^{x} - 10 \, \Gamma \left (-1, -2 \, x\right ) - 15 \, \Gamma \left (-1, -3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 55, normalized size = 1.67 \begin {gather*} -\frac {12 \, x^{3} - 5 \, x^{2} \log \left (2\right ) - {\left (12 \, x - 5 \, \log \left (2\right ) + 5\right )} e^{\left (2 \, x\right )} + {\left (5 \, x^{2} - 12 \, x + 5 \, \log \left (2\right )\right )} e^{x} - 5 \, e^{\left (3 \, x\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (27) = 54\).
time = 0.09, size = 70, normalized size = 2.12 \begin {gather*} - 12 x^{2} + 5 x \log {\left (2 \right )} + \frac {5 x^{2} e^{3 x} + \left (12 x^{3} - 5 x^{2} \log {\left (2 \right )} + 5 x^{2}\right ) e^{2 x} + \left (- 5 x^{4} + 12 x^{3} - 5 x^{2} \log {\left (2 \right )}\right ) e^{x}}{x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 63, normalized size = 1.91 \begin {gather*} -\frac {12 \, x^{3} + 5 \, x^{2} e^{x} - 5 \, x^{2} \log \left (2\right ) - 12 \, x e^{\left (2 \, x\right )} - 12 \, x e^{x} + 5 \, e^{\left (2 \, x\right )} \log \left (2\right ) + 5 \, e^{x} \log \left (2\right ) - 5 \, e^{\left (3 \, x\right )} - 5 \, e^{\left (2 \, x\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.48, size = 28, normalized size = 0.85 \begin {gather*} \frac {\left (12\,x-\ln \left (32\right )+5\,{\mathrm {e}}^x\right )\,\left ({\mathrm {e}}^{2\,x}+{\mathrm {e}}^x-x^2\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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