Optimal. Leaf size=27 \[ -x+\left (4+2 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} x\right )^4 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(27)=54\).
time = 13.92, antiderivative size = 128, normalized size of antiderivative = 4.74, number of steps
used = 65, number of rules used = 9, integrand size = 177, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {12, 6873,
6874, 2276, 2273, 2272, 2266, 2236, 2326} \begin {gather*} \frac {256 \left (3 x^2+x\right ) x^2 e^{-\frac {3 x (3 x+2)}{4 \log ^2(4)}}}{6 x+2}+\frac {768 \left (3 x^2+x\right ) x e^{-\frac {x (3 x+2)}{2 \log ^2(4)}}}{6 x+2}+\frac {1024 \left (3 x^2+x\right ) e^{-\frac {x (3 x+2)}{4 \log ^2(4)}}}{6 x+2}+16 x^4 e^{-\frac {3 x^2}{\log ^2(4)}-\frac {2 x}{\log ^2(4)}}-x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2236
Rule 2266
Rule 2272
Rule 2273
Rule 2276
Rule 2326
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int e^{-\frac {2 x+3 x^2}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right ) \, dx}{\log ^2(4)}\\ &=\frac {\int e^{-\frac {x (2+3 x)}{\log ^2(4)}} \left (-32 x^4-96 x^5-e^{\frac {2 x+3 x^2}{\log ^2(4)}} \log ^2(4)+64 x^3 \log ^2(4)+e^{\frac {3 \left (2 x+3 x^2\right )}{4 \log ^2(4)}} \left (-256 x-768 x^2+512 \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{2 \log ^2(4)}} \left (-384 x^2-1152 x^3+768 x \log ^2(4)\right )+e^{\frac {2 x+3 x^2}{4 \log ^2(4)}} \left (-192 x^3-576 x^4+384 x^2 \log ^2(4)\right )\right ) \, dx}{\log ^2(4)}\\ &=\frac {\int \left (-32 e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^4-96 e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^5-\log ^2(4)+64 e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^3 \log ^2(4)-256 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2-2 \log ^2(4)\right )-384 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2-2 \log ^2(4)\right )-192 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2-2 \log ^2(4)\right )\right ) \, dx}{\log ^2(4)}\\ &=-x+64 \int e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^3 \, dx-\frac {32 \int e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^4 \, dx}{\log ^2(4)}-\frac {96 \int e^{-\frac {x (2+3 x)}{\log ^2(4)}} x^5 \, dx}{\log ^2(4)}-\frac {192 \int e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2-2 \log ^2(4)\right ) \, dx}{\log ^2(4)}-\frac {256 \int e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2-2 \log ^2(4)\right ) \, dx}{\log ^2(4)}-\frac {384 \int e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2-2 \log ^2(4)\right ) \, dx}{\log ^2(4)}\\ &=-x+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}+64 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3 \, dx-\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4 \, dx}{\log ^2(4)}-\frac {96 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^5 \, dx}{\log ^2(4)}\\ &=-x+\frac {16}{3} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3+16 e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}-\frac {32}{3} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \log ^2(4)-16 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx-\frac {64}{3} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx-64 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3 \, dx+\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3 \, dx}{3 \log ^2(4)}+\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4 \, dx}{\log ^2(4)}+\frac {1}{3} \left (64 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx\\ &=-x-\frac {16}{9} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2+16 e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^4+\frac {1024 e^{-\frac {x (2+3 x)}{4 \log ^2(4)}} \left (x+3 x^2\right )}{2+6 x}+\frac {768 e^{-\frac {x (2+3 x)}{2 \log ^2(4)}} x \left (x+3 x^2\right )}{2+6 x}+\frac {256 e^{-\frac {3 x (2+3 x)}{4 \log ^2(4)}} x^2 \left (x+3 x^2\right )}{2+6 x}+\frac {56}{9} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \log ^2(4)-\frac {32}{9} e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \log ^4(4)+\frac {32}{9} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx+\frac {16}{3} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx+\frac {64}{9} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx+16 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx+\frac {64}{3} \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx-\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^2 \, dx}{9 \log ^2(4)}-\frac {32 \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x^3 \, dx}{3 \log ^2(4)}-\frac {1}{3} \left (8 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {1}{9} \left (32 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {1}{9} \left (64 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} \, dx-\frac {1}{3} \left (64 \log ^2(4)\right ) \int e^{-\frac {2 x}{\log ^2(4)}-\frac {3 x^2}{\log ^2(4)}} x \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(27)=54\).
time = 0.18, size = 94, normalized size = 3.48 \begin {gather*} -e^{-\frac {x (2+3 x)}{\log ^2(4)}} x \left (-512 e^{\frac {3 x (2+3 x)}{4 \log ^2(4)}}+e^{\frac {x (2+3 x)}{\log ^2(4)}}-384 e^{\frac {x (2+3 x)}{2 \log ^2(4)}} x-128 e^{\frac {x (2+3 x)}{4 \log ^2(4)}} x^2-16 x^3\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. \(120\) vs.
\(2(26)=52\).
time = 0.14, size = 121, normalized size = 4.48
method | result | size |
risch | \(-x +512 x \,{\mathrm e}^{-\frac {x \left (3 x +2\right )}{16 \ln \left (2\right )^{2}}}+384 x^{2} {\mathrm e}^{-\frac {x \left (3 x +2\right )}{8 \ln \left (2\right )^{2}}}+128 x^{3} {\mathrm e}^{-\frac {3 x \left (3 x +2\right )}{16 \ln \left (2\right )^{2}}}+16 x^{4} {\mathrm e}^{-\frac {x \left (3 x +2\right )}{4 \ln \left (2\right )^{2}}}\) | \(75\) |
default | \(\frac {-4 x \ln \left (2\right )^{2}+64 \ln \left (2\right )^{2} x^{4} {\mathrm e}^{-\frac {3 x^{2}}{4 \ln \left (2\right )^{2}}-\frac {x}{2 \ln \left (2\right )^{2}}}+512 \ln \left (2\right )^{2} x^{3} {\mathrm e}^{-\frac {9 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {3 x}{8 \ln \left (2\right )^{2}}}+1536 \ln \left (2\right )^{2} x^{2} {\mathrm e}^{-\frac {3 x^{2}}{8 \ln \left (2\right )^{2}}-\frac {x}{4 \ln \left (2\right )^{2}}}+2048 \ln \left (2\right )^{2} x \,{\mathrm e}^{-\frac {3 x^{2}}{16 \ln \left (2\right )^{2}}-\frac {x}{8 \ln \left (2\right )^{2}}}}{4 \ln \left (2\right )^{2}}\) | \(121\) |
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 = 0.68, size = 2622, normalized size = 97.11 \begin {gather*} \text {Too large to display} \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. 103 vs.
\(2 (25) = 50\).
time = 0.39, size = 103, normalized size = 3.81 \begin {gather*} {\left (16 \, x^{4} + 128 \, x^{3} e^{\left (\frac {3 \, x^{2} + 2 \, x}{16 \, \log \left (2\right )^{2}}\right )} + 384 \, x^{2} e^{\left (\frac {3 \, x^{2} + 2 \, x}{8 \, \log \left (2\right )^{2}}\right )} - x e^{\left (\frac {3 \, x^{2} + 2 \, x}{4 \, \log \left (2\right )^{2}}\right )} + 512 \, x e^{\left (\frac {3 \, {\left (3 \, x^{2} + 2 \, x\right )}}{16 \, \log \left (2\right )^{2}}\right )}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{4 \, \log \left (2\right )^{2}}\right )} \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. 88 vs.
\(2 (22) = 44\).
time = 0.19, size = 88, normalized size = 3.26 \begin {gather*} 16 x^{4} e^{- \frac {4 \cdot \left (\frac {3 x^{2}}{16} + \frac {x}{8}\right )}{\log {\left (2 \right )}^{2}}} + 128 x^{3} e^{- \frac {3 \cdot \left (\frac {3 x^{2}}{16} + \frac {x}{8}\right )}{\log {\left (2 \right )}^{2}}} + 384 x^{2} e^{- \frac {2 \cdot \left (\frac {3 x^{2}}{16} + \frac {x}{8}\right )}{\log {\left (2 \right )}^{2}}} - x + 512 x e^{- \frac {\frac {3 x^{2}}{16} + \frac {x}{8}}{\log {\left (2 \right )}^{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. 228 vs.
\(2 (25) = 50\).
time = 0.46, size = 228, normalized size = 8.44 \begin {gather*} -\frac {81 \, x \log \left (2\right )^{2} - 13824 \, {\left ({\left (3 \, x + 1\right )} \log \left (2\right )^{2} - \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{16 \, \log \left (2\right )^{2}}\right )} - 3456 \, {\left ({\left (3 \, x + 1\right )}^{2} \log \left (2\right )^{2} - 2 \, {\left (3 \, x + 1\right )} \log \left (2\right )^{2} + \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{8 \, \log \left (2\right )^{2}}\right )} - 384 \, {\left ({\left (3 \, x + 1\right )}^{3} \log \left (2\right )^{2} - 3 \, {\left (3 \, x + 1\right )}^{2} \log \left (2\right )^{2} + 3 \, {\left (3 \, x + 1\right )} \log \left (2\right )^{2} - \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, {\left (3 \, x^{2} + 2 \, x\right )}}{16 \, \log \left (2\right )^{2}}\right )} - 16 \, {\left ({\left (3 \, x + 1\right )}^{4} \log \left (2\right )^{2} - 4 \, {\left (3 \, x + 1\right )}^{3} \log \left (2\right )^{2} + 6 \, {\left (3 \, x + 1\right )}^{2} \log \left (2\right )^{2} - 4 \, {\left (3 \, x + 1\right )} \log \left (2\right )^{2} + \log \left (2\right )^{2}\right )} e^{\left (-\frac {3 \, x^{2} + 2 \, x}{4 \, \log \left (2\right )^{2}}\right )}}{81 \, \log \left (2\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 94, normalized size = 3.48 \begin {gather*} 512\,x\,{\mathrm {e}}^{-\frac {3\,x^2}{16\,{\ln \left (2\right )}^2}-\frac {x}{8\,{\ln \left (2\right )}^2}}-x+16\,x^4\,{\mathrm {e}}^{-\frac {3\,x^2}{4\,{\ln \left (2\right )}^2}-\frac {x}{2\,{\ln \left (2\right )}^2}}+384\,x^2\,{\mathrm {e}}^{-\frac {3\,x^2}{8\,{\ln \left (2\right )}^2}-\frac {x}{4\,{\ln \left (2\right )}^2}}+128\,x^3\,{\mathrm {e}}^{-\frac {9\,x^2}{16\,{\ln \left (2\right )}^2}-\frac {3\,x}{8\,{\ln \left (2\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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