Integrand size = 92, antiderivative size = 32 \[ \int \frac {1}{3} \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx=4-x-16 \left (-e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )}+\log (4)\right )^4 \]
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Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(32)=64\).
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.81, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {12, 2225} \[ \int \frac {1}{3} \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx=-x+64 \log ^3(4) e^{\frac {1}{3} \left (x+(\log (2)+i \pi )^2\right )}-96 \log ^2(4) e^{\frac {2}{3} \left (x+(\log (2)+i \pi )^2\right )}-16 e^{\frac {4}{3} \left (x+(\log (2)+i \pi )^2\right )}+64 \log (4) e^{x+(\log (2)+i \pi )^2} \]
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Rule 12
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx \\ & = -x-\frac {64}{3} \int e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )} \, dx+(64 \log (4)) \int e^{x+(i \pi +\log (2))^2} \, dx-\left (64 \log ^2(4)\right ) \int e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \, dx+\frac {1}{3} \left (64 \log ^3(4)\right ) \int e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \, dx \\ & = -16 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}-x+64 e^{x+(i \pi +\log (2))^2} \log (4)-96 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(150\) vs. \(2(32)=64\).
Time = 0.41 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.69 \[ \int \frac {1}{3} \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx=\frac {1}{3} \left (-3 2^{4+\frac {8 i \pi }{3}} e^{-\frac {4 \pi ^2}{3}+\frac {4 x}{3}+\frac {4 \log ^2(2)}{3}}-3 x+3\ 2^{6+2 i \pi } e^{-\pi ^2+x+\log ^2(2)} \log (4)-9\ 2^{5+\frac {4 i \pi }{3}} e^{-\frac {2 \pi ^2}{3}+\frac {2 x}{3}+\frac {2 \log ^2(2)}{3}} \log ^2(4)+3\ 2^{6+\frac {2 i \pi }{3}} e^{-\frac {\pi ^2}{3}+\frac {x}{3}+\frac {\log ^2(2)}{3}} \log ^3(4)\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (32 ) = 64\).
Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.91
method | result | size |
default | \(-x -16 \,{\mathrm e}^{\frac {4 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {4 x}{3}}+128 \ln \left (2\right ) {\mathrm e}^{\left (\ln \left (2\right )+i \pi \right )^{2}+x}-384 \ln \left (2\right )^{2} {\mathrm e}^{\frac {2 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {2 x}{3}}+512 \ln \left (2\right )^{3} {\mathrm e}^{\frac {\left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {x}{3}}\) | \(93\) |
norman | \(-x -16 \,{\mathrm e}^{\frac {4 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {4 x}{3}}+128 \ln \left (2\right ) {\mathrm e}^{\left (\ln \left (2\right )+i \pi \right )^{2}+x}-384 \ln \left (2\right )^{2} {\mathrm e}^{\frac {2 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {2 x}{3}}+512 \ln \left (2\right )^{3} {\mathrm e}^{\frac {\left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {x}{3}}\) | \(93\) |
parallelrisch | \(-x -16 \,{\mathrm e}^{\frac {4 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {4 x}{3}}+128 \ln \left (2\right ) {\mathrm e}^{\left (\ln \left (2\right )+i \pi \right )^{2}+x}-384 \ln \left (2\right )^{2} {\mathrm e}^{\frac {2 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {2 x}{3}}+512 \ln \left (2\right )^{3} {\mathrm e}^{\frac {\left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {x}{3}}\) | \(93\) |
parts | \(-x -16 \,{\mathrm e}^{\frac {4 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {4 x}{3}}+128 \ln \left (2\right ) {\mathrm e}^{\left (\ln \left (2\right )+i \pi \right )^{2}+x}-384 \ln \left (2\right )^{2} {\mathrm e}^{\frac {2 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {2 x}{3}}+512 \ln \left (2\right )^{3} {\mathrm e}^{\frac {\left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {x}{3}}\) | \(93\) |
derivativedivides | \(-16 \,{\mathrm e}^{\frac {4 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {4 x}{3}}+128 \ln \left (2\right ) {\mathrm e}^{\left (\ln \left (2\right )+i \pi \right )^{2}+x}-384 \ln \left (2\right )^{2} {\mathrm e}^{\frac {2 \left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {2 x}{3}}+512 \ln \left (2\right )^{3} {\mathrm e}^{\frac {\left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {x}{3}}-3 \ln \left ({\mathrm e}^{\frac {\left (\ln \left (2\right )+i \pi \right )^{2}}{3}+\frac {x}{3}}\right )\) | \(109\) |
risch | \(-x -16 \,2^{\frac {8 i \pi }{3}} {\mathrm e}^{-\frac {4 \pi ^{2}}{3}+\frac {4 \ln \left (2\right )^{2}}{3}+\frac {4 x}{3}}+128 \ln \left (2\right ) 2^{2 i \pi } {\mathrm e}^{-\pi ^{2}+\ln \left (2\right )^{2}+x}-384 \ln \left (2\right )^{2} 2^{\frac {4 i \pi }{3}} {\mathrm e}^{-\frac {2 \pi ^{2}}{3}+\frac {2 \ln \left (2\right )^{2}}{3}+\frac {2 x}{3}}+512 \ln \left (2\right )^{3} 2^{\frac {2 i \pi }{3}} {\mathrm e}^{-\frac {\pi ^{2}}{3}+\frac {\ln \left (2\right )^{2}}{3}+\frac {x}{3}}\) | \(113\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (35) = 70\).
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.19 \[ \int \frac {1}{3} \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx=512 \, e^{\left (-\frac {1}{3} \, \pi ^{2} + \frac {2}{3} i \, \pi \log \left (2\right ) + \frac {1}{3} \, \log \left (2\right )^{2} + \frac {1}{3} \, x\right )} \log \left (2\right )^{3} - 384 \, e^{\left (-\frac {2}{3} \, \pi ^{2} + \frac {4}{3} i \, \pi \log \left (2\right ) + \frac {2}{3} \, \log \left (2\right )^{2} + \frac {2}{3} \, x\right )} \log \left (2\right )^{2} + 128 \, e^{\left (-\pi ^{2} + 2 i \, \pi \log \left (2\right ) + \log \left (2\right )^{2} + x\right )} \log \left (2\right ) - x - 16 \, e^{\left (-\frac {4}{3} \, \pi ^{2} + \frac {8}{3} i \, \pi \log \left (2\right ) + \frac {4}{3} \, \log \left (2\right )^{2} + \frac {4}{3} \, x\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (26) = 52\).
Time = 0.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 5.09 \[ \int \frac {1}{3} \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx=- x + \frac {- 16 e^{2 \pi ^{2}} e^{\frac {4 x}{3}} e^{\frac {8 i \pi \log {\left (2 \right )}}{3}} e^{\frac {4 \log {\left (2 \right )}^{2}}{3}} - 384 e^{\frac {8 \pi ^{2}}{3}} e^{\frac {2 x}{3}} e^{\frac {4 i \pi \log {\left (2 \right )}}{3}} e^{\frac {2 \log {\left (2 \right )}^{2}}{3}} \log {\left (2 \right )}^{2} + 512 e^{3 \pi ^{2}} e^{\frac {x}{3}} e^{\frac {2 i \pi \log {\left (2 \right )}}{3}} e^{\frac {\log {\left (2 \right )}^{2}}{3}} \log {\left (2 \right )}^{3} + 128 e^{\frac {7 \pi ^{2}}{3}} e^{x} e^{2 i \pi \log {\left (2 \right )}} e^{\log {\left (2 \right )}^{2}} \log {\left (2 \right )}}{e^{\frac {10 \pi ^{2}}{3}}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (35) = 70\).
Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.19 \[ \int \frac {1}{3} \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx=512 \, e^{\left (-\frac {1}{3} \, \pi ^{2} + \frac {2}{3} i \, \pi \log \left (2\right ) + \frac {1}{3} \, \log \left (2\right )^{2} + \frac {1}{3} \, x\right )} \log \left (2\right )^{3} - 384 \, e^{\left (-\frac {2}{3} \, \pi ^{2} + \frac {4}{3} i \, \pi \log \left (2\right ) + \frac {2}{3} \, \log \left (2\right )^{2} + \frac {2}{3} \, x\right )} \log \left (2\right )^{2} + 128 \, e^{\left (-\pi ^{2} + 2 i \, \pi \log \left (2\right ) + \log \left (2\right )^{2} + x\right )} \log \left (2\right ) - x - 16 \, e^{\left (-\frac {4}{3} \, \pi ^{2} + \frac {8}{3} i \, \pi \log \left (2\right ) + \frac {4}{3} \, \log \left (2\right )^{2} + \frac {4}{3} \, x\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (35) = 70\).
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.19 \[ \int \frac {1}{3} \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx=512 \, e^{\left (-\frac {1}{3} \, \pi ^{2} + \frac {2}{3} i \, \pi \log \left (2\right ) + \frac {1}{3} \, \log \left (2\right )^{2} + \frac {1}{3} \, x\right )} \log \left (2\right )^{3} - 384 \, e^{\left (-\frac {2}{3} \, \pi ^{2} + \frac {4}{3} i \, \pi \log \left (2\right ) + \frac {2}{3} \, \log \left (2\right )^{2} + \frac {2}{3} \, x\right )} \log \left (2\right )^{2} + 128 \, e^{\left (-\pi ^{2} + 2 i \, \pi \log \left (2\right ) + \log \left (2\right )^{2} + x\right )} \log \left (2\right ) - x - 16 \, e^{\left (-\frac {4}{3} \, \pi ^{2} + \frac {8}{3} i \, \pi \log \left (2\right ) + \frac {4}{3} \, \log \left (2\right )^{2} + \frac {4}{3} \, x\right )} \]
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Time = 9.32 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.31 \[ \int \frac {1}{3} \left (-3-64 e^{\frac {4}{3} \left (x+(i \pi +\log (2))^2\right )}+192 e^{x+(i \pi +\log (2))^2} \log (4)-192 e^{\frac {2}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^2(4)+64 e^{\frac {1}{3} \left (x+(i \pi +\log (2))^2\right )} \log ^3(4)\right ) \, dx=-x-16\,2^{\frac {\Pi \,8{}\mathrm {i}}{3}}\,{\mathrm {e}}^{-\frac {4\,\Pi ^2}{3}+\frac {4\,x}{3}+\frac {4\,{\ln \left (2\right )}^2}{3}}+512\,2^{\frac {\Pi \,2{}\mathrm {i}}{3}}\,{\mathrm {e}}^{-\frac {\Pi ^2}{3}+\frac {x}{3}+\frac {{\ln \left (2\right )}^2}{3}}\,{\ln \left (2\right )}^3-384\,2^{\frac {\Pi \,4{}\mathrm {i}}{3}}\,{\mathrm {e}}^{-\frac {2\,\Pi ^2}{3}+\frac {2\,x}{3}+\frac {2\,{\ln \left (2\right )}^2}{3}}\,{\ln \left (2\right )}^2+128\,2^{\Pi \,2{}\mathrm {i}}\,{\mathrm {e}}^{-\Pi ^2+x+{\ln \left (2\right )}^2}\,\ln \left (2\right ) \]
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