Optimal. Leaf size=20 \[ e^3 \left (3+4 e^x-\frac {3 e}{x}+x\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(20)=40\).
time = 0.16, antiderivative size = 56, normalized size of antiderivative = 2.80, number of steps
used = 12, number of rules used = 7, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {14, 2225,
1604, 2230, 2208, 2209, 2207} \begin {gather*} \frac {e^3 \left (-x^2-3 x+3 e\right )^2}{x^2}+24 e^{x+3}+16 e^{2 x+3}+8 e^{x+3} x-\frac {24 e^{x+4}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1604
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (32 e^{3+2 x}-\frac {2 e^3 \left (3 e-3 x-x^2\right ) \left (3 e+x^2\right )}{x^3}-\frac {8 e^{3+x} \left (-3 e+3 e x-4 x^2-x^3\right )}{x^2}\right ) \, dx\\ &=-\left (8 \int \frac {e^{3+x} \left (-3 e+3 e x-4 x^2-x^3\right )}{x^2} \, dx\right )+32 \int e^{3+2 x} \, dx-\left (2 e^3\right ) \int \frac {\left (3 e-3 x-x^2\right ) \left (3 e+x^2\right )}{x^3} \, dx\\ &=16 e^{3+2 x}+\frac {e^3 \left (3 e-3 x-x^2\right )^2}{x^2}-8 \int \left (-4 e^{3+x}-\frac {3 e^{4+x}}{x^2}+\frac {3 e^{4+x}}{x}-e^{3+x} x\right ) \, dx\\ &=16 e^{3+2 x}+\frac {e^3 \left (3 e-3 x-x^2\right )^2}{x^2}+8 \int e^{3+x} x \, dx+24 \int \frac {e^{4+x}}{x^2} \, dx-24 \int \frac {e^{4+x}}{x} \, dx+32 \int e^{3+x} \, dx\\ &=32 e^{3+x}+16 e^{3+2 x}-\frac {24 e^{4+x}}{x}+8 e^{3+x} x+\frac {e^3 \left (3 e-3 x-x^2\right )^2}{x^2}-24 e^4 \text {Ei}(x)-8 \int e^{3+x} \, dx+24 \int \frac {e^{4+x}}{x} \, dx\\ &=24 e^{3+x}+16 e^{3+2 x}-\frac {24 e^{4+x}}{x}+8 e^{3+x} x+\frac {e^3 \left (3 e-3 x-x^2\right )^2}{x^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(20)=40\).
time = 2.13, size = 54, normalized size = 2.70 \begin {gather*} -2 e^3 \left (-8 e^{2 x}+e^x \left (-12+\frac {12 e}{x}-4 x\right )-\frac {9 e^2}{2 x^2}+\frac {9 e}{x}-3 x-\frac {x^2}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.87, size = 92, normalized size = 4.60
method | result | size |
risch | \(x^{2} {\mathrm e}^{3}+6 x \,{\mathrm e}^{3}+\frac {9 \,{\mathrm e}^{5}-18 x \,{\mathrm e}^{4}}{x^{2}}+16 \,{\mathrm e}^{2 x +3}-\frac {8 \left (-x^{2}+3 \,{\mathrm e}-3 x \right ) {\mathrm e}^{3+x}}{x}\) | \(57\) |
norman | \(\frac {x^{4} {\mathrm e}^{3}+6 x^{3} {\mathrm e}^{3}+9 \,{\mathrm e}^{2} {\mathrm e}^{3}-18 x \,{\mathrm e} \,{\mathrm e}^{3}+24 x^{2} {\mathrm e}^{3} {\mathrm e}^{x}+16 x^{2} {\mathrm e}^{3} {\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} {\mathrm e}^{3} x^{3}-24 \,{\mathrm e}^{x} {\mathrm e}^{3} x \,{\mathrm e}}{x^{2}}\) | \(72\) |
default | \(x^{2} {\mathrm e}^{3}+16 \,{\mathrm e}^{3} {\mathrm e}^{2 x}+32 \,{\mathrm e}^{x} {\mathrm e}^{3}+\frac {9 \,{\mathrm e}^{2} {\mathrm e}^{3}}{x^{2}}-\frac {18 \,{\mathrm e} \,{\mathrm e}^{3}}{x}+8 \,{\mathrm e}^{3} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+24 \,{\mathrm e} \,{\mathrm e}^{3} \left (-\frac {{\mathrm e}^{x}}{x}-\expIntegral \left (1, -x \right )\right )+24 \,{\mathrm e} \,{\mathrm e}^{3} \expIntegral \left (1, -x \right )+6 x \,{\mathrm e}^{3}\) | \(92\) |
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 = 0.29, size = 68, normalized size = 3.40 \begin {gather*} x^{2} e^{3} - 24 \, {\rm Ei}\left (x\right ) e^{4} + 6 \, x e^{3} + 8 \, {\left (x e^{3} - e^{3}\right )} e^{x} + 24 \, e^{4} \Gamma \left (-1, -x\right ) - \frac {18 \, e^{4}}{x} + \frac {9 \, e^{5}}{x^{2}} + 16 \, e^{\left (2 \, x + 3\right )} + 32 \, e^{\left (x + 3\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. 64 vs.
\(2 (19) = 38\).
time = 0.37, size = 64, normalized size = 3.20 \begin {gather*} \frac {{\left (16 \, x^{2} e^{\left (2 \, x + 6\right )} - 18 \, x e^{7} + {\left (x^{4} + 6 \, x^{3}\right )} e^{6} - 8 \, {\left (3 \, x e^{4} - {\left (x^{3} + 3 \, x^{2}\right )} e^{3}\right )} e^{\left (x + 3\right )} + 9 \, e^{8}\right )} e^{\left (-3\right )}}{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. 65 vs.
\(2 (19) = 38\).
time = 0.12, size = 65, normalized size = 3.25 \begin {gather*} x^{2} e^{3} + 6 x e^{3} + \frac {16 x e^{3} e^{2 x} + \left (8 x^{2} e^{3} + 24 x e^{3} - 24 e^{4}\right ) e^{x}}{x} + \frac {- 18 x e^{4} + 9 e^{5}}{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. 143 vs.
\(2 (19) = 38\).
time = 0.45, size = 143, normalized size = 7.15 \begin {gather*} \frac {{\left (x + 3\right )}^{4} e^{6} - 6 \, {\left (x + 3\right )}^{3} e^{6} + 8 \, {\left (x + 3\right )}^{3} e^{\left (x + 6\right )} + 9 \, {\left (x + 3\right )}^{2} e^{6} + 16 \, {\left (x + 3\right )}^{2} e^{\left (2 \, x + 6\right )} - 48 \, {\left (x + 3\right )}^{2} e^{\left (x + 6\right )} - 18 \, {\left (x + 3\right )} e^{7} - 96 \, {\left (x + 3\right )} e^{\left (2 \, x + 6\right )} - 24 \, {\left (x + 3\right )} e^{\left (x + 7\right )} + 72 \, {\left (x + 3\right )} e^{\left (x + 6\right )} + 9 \, e^{8} + 54 \, e^{7} + 144 \, e^{\left (2 \, x + 6\right )} + 72 \, e^{\left (x + 7\right )}}{{\left (x + 3\right )}^{2} e^{3} - 6 \, {\left (x + 3\right )} e^{3} + 9 \, e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 56, normalized size = 2.80 \begin {gather*} \frac {9\,{\mathrm {e}}^5-x\,{\mathrm {e}}^3\,\left (24\,{\mathrm {e}}^{x+1}+18\,\mathrm {e}\right )}{x^2}+x^2\,{\mathrm {e}}^3+{\mathrm {e}}^3\,\left (16\,{\mathrm {e}}^{2\,x}+24\,{\mathrm {e}}^x\right )+x\,{\mathrm {e}}^3\,\left (8\,{\mathrm {e}}^x+6\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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