Optimal. Leaf size=24 \[ \left (e^x-\frac {1}{x}+5 x+3 \left (-8-x+x^2\right )\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).
time = 0.15, antiderivative size = 59, normalized size of antiderivative = 2.46, number of steps
used = 16, number of rules used = 6, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.090, Rules used = {14, 2225,
2230, 2208, 2209, 2207} \begin {gather*} 9 x^4+12 x^3+6 e^x x^2-140 x^2+\frac {1}{x^2}+4 e^x x-102 x-48 e^x+e^{2 x}-\frac {2 e^x}{x}+\frac {48}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 e^{2 x}+\frac {2 e^x \left (1-x-22 x^2+8 x^3+3 x^4\right )}{x^2}+\frac {2 \left (-1-24 x-51 x^3-140 x^4+18 x^5+18 x^6\right )}{x^3}\right ) \, dx\\ &=2 \int e^{2 x} \, dx+2 \int \frac {e^x \left (1-x-22 x^2+8 x^3+3 x^4\right )}{x^2} \, dx+2 \int \frac {-1-24 x-51 x^3-140 x^4+18 x^5+18 x^6}{x^3} \, dx\\ &=e^{2 x}+2 \int \left (-22 e^x+\frac {e^x}{x^2}-\frac {e^x}{x}+8 e^x x+3 e^x x^2\right ) \, dx+2 \int \left (-51-\frac {1}{x^3}-\frac {24}{x^2}-140 x+18 x^2+18 x^3\right ) \, dx\\ &=e^{2 x}+\frac {1}{x^2}+\frac {48}{x}-102 x-140 x^2+12 x^3+9 x^4+2 \int \frac {e^x}{x^2} \, dx-2 \int \frac {e^x}{x} \, dx+6 \int e^x x^2 \, dx+16 \int e^x x \, dx-44 \int e^x \, dx\\ &=-44 e^x+e^{2 x}+\frac {1}{x^2}+\frac {48}{x}-\frac {2 e^x}{x}-102 x+16 e^x x-140 x^2+6 e^x x^2+12 x^3+9 x^4-2 \text {Ei}(x)+2 \int \frac {e^x}{x} \, dx-12 \int e^x x \, dx-16 \int e^x \, dx\\ &=-60 e^x+e^{2 x}+\frac {1}{x^2}+\frac {48}{x}-\frac {2 e^x}{x}-102 x+4 e^x x-140 x^2+6 e^x x^2+12 x^3+9 x^4+12 \int e^x \, dx\\ &=-48 e^x+e^{2 x}+\frac {1}{x^2}+\frac {48}{x}-\frac {2 e^x}{x}-102 x+4 e^x x-140 x^2+6 e^x x^2+12 x^3+9 x^4\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(24)=48\).
time = 3.73, size = 51, normalized size = 2.12 \begin {gather*} e^{2 x}+\frac {1}{x^2}+\frac {48}{x}-102 x-140 x^2+12 x^3+9 x^4+e^x \left (-48-\frac {2}{x}+4 x+6 x^2\right ) \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. \(54\) vs.
\(2(19)=38\).
time = 0.15, size = 55, normalized size = 2.29
method | result | size |
default | \(-140 x^{2}-102 x +\frac {1}{x^{2}}+\frac {48}{x}+12 x^{3}+9 x^{4}+{\mathrm e}^{2 x}-\frac {2 \,{\mathrm e}^{x}}{x}+4 \,{\mathrm e}^{x} x -48 \,{\mathrm e}^{x}+6 \,{\mathrm e}^{x} x^{2}\) | \(55\) |
risch | \(9 x^{4}+12 x^{3}-140 x^{2}-102 x +\frac {48 x +1}{x^{2}}+{\mathrm e}^{2 x}+\frac {2 \left (3 x^{3}+2 x^{2}-24 x -1\right ) {\mathrm e}^{x}}{x}\) | \(55\) |
norman | \(\frac {1+{\mathrm e}^{2 x} x^{2}+48 x -102 x^{3}-140 x^{4}+12 x^{5}+9 x^{6}-2 \,{\mathrm e}^{x} x -48 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{x} x^{4}}{x^{2}}\) | \(64\) |
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 = 0.30, size = 65, normalized size = 2.71 \begin {gather*} 9 \, x^{4} + 12 \, x^{3} - 140 \, x^{2} + 6 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + 16 \, {\left (x - 1\right )} e^{x} - 102 \, x + \frac {48}{x} + \frac {1}{x^{2}} - 2 \, {\rm Ei}\left (x\right ) + e^{\left (2 \, x\right )} - 44 \, e^{x} + 2 \, \Gamma \left (-1, -x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (19) = 38\).
time = 0.41, size = 60, normalized size = 2.50 \begin {gather*} \frac {9 \, x^{6} + 12 \, x^{5} - 140 \, x^{4} - 102 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 2 \, {\left (3 \, x^{4} + 2 \, x^{3} - 24 \, x^{2} - x\right )} e^{x} + 48 \, x + 1}{x^{2}} \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. 53 vs.
\(2 (17) = 34\).
time = 0.06, size = 53, normalized size = 2.21 \begin {gather*} 9 x^{4} + 12 x^{3} - 140 x^{2} - 102 x + \frac {x e^{2 x} + \left (6 x^{3} + 4 x^{2} - 48 x - 2\right ) e^{x}}{x} + \frac {48 x + 1}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (19) = 38\).
time = 0.39, size = 63, normalized size = 2.62 \begin {gather*} \frac {9 \, x^{6} + 12 \, x^{5} + 6 \, x^{4} e^{x} - 140 \, x^{4} + 4 \, x^{3} e^{x} - 102 \, x^{3} + x^{2} e^{\left (2 \, x\right )} - 48 \, x^{2} e^{x} - 2 \, x e^{x} + 48 \, x + 1}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 52, normalized size = 2.17 \begin {gather*} {\mathrm {e}}^{2\,x}-48\,{\mathrm {e}}^x-\frac {x\,\left (2\,{\mathrm {e}}^x-48\right )-1}{x^2}+x\,\left (4\,{\mathrm {e}}^x-102\right )+x^2\,\left (6\,{\mathrm {e}}^x-140\right )+12\,x^3+9\,x^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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