Integrand size = 15, antiderivative size = 55 \[ \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx=-140 e^{x^4}+490 e^{2 x^4}-\frac {3430 e^{3 x^4}}{3}+\frac {12005 e^{4 x^4}}{8}-\frac {16807 e^{5 x^4}}{20}+8 x^4 \]
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Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6847, 2320, 45} \[ \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx=8 x^4-140 e^{x^4}+490 e^{2 x^4}-\frac {3430 e^{3 x^4}}{3}+\frac {12005 e^{4 x^4}}{8}-\frac {16807 e^{5 x^4}}{20} \]
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Rule 45
Rule 2320
Rule 6847
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \left (2-7 e^x\right )^5 \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {(2-7 x)^5}{x} \, dx,x,e^{x^4}\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-560+\frac {32}{x}+3920 x-13720 x^2+24010 x^3-16807 x^4\right ) \, dx,x,e^{x^4}\right ) \\ & = -140 e^{x^4}+490 e^{2 x^4}-\frac {3430 e^{3 x^4}}{3}+\frac {12005 e^{4 x^4}}{8}-\frac {16807 e^{5 x^4}}{20}+8 x^4 \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98 \[ \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx=-\frac {7}{120} e^{x^4} \left (2400-8400 e^{x^4}+19600 e^{2 x^4}-25725 e^{3 x^4}+14406 e^{4 x^4}\right )+8 \log \left (e^{x^4}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82
method | result | size |
norman | \(-140 \,{\mathrm e}^{x^{4}}+490 \,{\mathrm e}^{2 x^{4}}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+8 x^{4}\) | \(45\) |
risch | \(-140 \,{\mathrm e}^{x^{4}}+490 \,{\mathrm e}^{2 x^{4}}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+8 x^{4}\) | \(45\) |
parallelrisch | \(-140 \,{\mathrm e}^{x^{4}}+490 \,{\mathrm e}^{2 x^{4}}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+8 x^{4}\) | \(45\) |
parts | \(-140 \,{\mathrm e}^{x^{4}}+490 \,{\mathrm e}^{2 x^{4}}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+8 x^{4}\) | \(45\) |
derivativedivides | \(-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+490 \,{\mathrm e}^{2 x^{4}}-140 \,{\mathrm e}^{x^{4}}+8 \ln \left ({\mathrm e}^{x^{4}}\right )\) | \(47\) |
default | \(-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+490 \,{\mathrm e}^{2 x^{4}}-140 \,{\mathrm e}^{x^{4}}+8 \ln \left ({\mathrm e}^{x^{4}}\right )\) | \(47\) |
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Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx=8 \, x^{4} - \frac {16807}{20} \, e^{\left (5 \, x^{4}\right )} + \frac {12005}{8} \, e^{\left (4 \, x^{4}\right )} - \frac {3430}{3} \, e^{\left (3 \, x^{4}\right )} + 490 \, e^{\left (2 \, x^{4}\right )} - 140 \, e^{\left (x^{4}\right )} \]
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Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx=8 x^{4} - \frac {16807 e^{5 x^{4}}}{20} + \frac {12005 e^{4 x^{4}}}{8} - \frac {3430 e^{3 x^{4}}}{3} + 490 e^{2 x^{4}} - 140 e^{x^{4}} \]
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Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx=8 \, x^{4} - \frac {16807}{20} \, e^{\left (5 \, x^{4}\right )} + \frac {12005}{8} \, e^{\left (4 \, x^{4}\right )} - \frac {3430}{3} \, e^{\left (3 \, x^{4}\right )} + 490 \, e^{\left (2 \, x^{4}\right )} - 140 \, e^{\left (x^{4}\right )} \]
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Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx=8 \, x^{4} - \frac {16807}{20} \, e^{\left (5 \, x^{4}\right )} + \frac {12005}{8} \, e^{\left (4 \, x^{4}\right )} - \frac {3430}{3} \, e^{\left (3 \, x^{4}\right )} + 490 \, e^{\left (2 \, x^{4}\right )} - 140 \, e^{\left (x^{4}\right )} \]
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Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.80 \[ \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx=490\,{\mathrm {e}}^{2\,x^4}-140\,{\mathrm {e}}^{x^4}-\frac {3430\,{\mathrm {e}}^{3\,x^4}}{3}+\frac {12005\,{\mathrm {e}}^{4\,x^4}}{8}-\frac {16807\,{\mathrm {e}}^{5\,x^4}}{20}+8\,x^4 \]
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