Integrand size = 15, antiderivative size = 43 \[ \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx=3 x-\frac {a^{-12 x}}{12 \log (a)}+\frac {a^{-6 x}}{2 \log (a)}-\frac {a^{6 x}}{6 \log (a)} \]
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Time = 0.02 (sec) , antiderivative size = 43, 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 = {2320, 272, 45} \[ \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx=-\frac {a^{-12 x}}{12 \log (a)}+\frac {a^{-6 x}}{2 \log (a)}-\frac {a^{6 x}}{6 \log (a)}+3 x \]
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Rule 45
Rule 272
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-x^3\right )^3}{x^7} \, dx,x,a^{2 x}\right )}{2 \log (a)} \\ & = \frac {\text {Subst}\left (\int \frac {(1-x)^3}{x^3} \, dx,x,a^{6 x}\right )}{6 \log (a)} \\ & = \frac {\text {Subst}\left (\int \left (-1+\frac {1}{x^3}-\frac {3}{x^2}+\frac {3}{x}\right ) \, dx,x,a^{6 x}\right )}{6 \log (a)} \\ & = 3 x-\frac {a^{-12 x}}{12 \log (a)}+\frac {a^{-6 x}}{2 \log (a)}-\frac {a^{6 x}}{6 \log (a)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx=-\frac {a^{-12 x}-6 a^{-6 x}+2 a^{6 x}-36 \log \left (a^x\right )}{12 \log (a)} \]
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Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.02
method | result | size |
risch | \(3 x -\frac {a^{6 x}}{6 \ln \left (a \right )}+\frac {a^{-6 x}}{2 \ln \left (a \right )}-\frac {a^{-12 x}}{12 \ln \left (a \right )}\) | \(44\) |
norman | \(\left (-\frac {1}{12 \ln \left (a \right )}+3 x \,{\mathrm e}^{12 x \ln \left (a \right )}+\frac {{\mathrm e}^{6 x \ln \left (a \right )}}{2 \ln \left (a \right )}-\frac {{\mathrm e}^{18 x \ln \left (a \right )}}{6 \ln \left (a \right )}\right ) {\mathrm e}^{-12 x \ln \left (a \right )}\) | \(56\) |
parallelrisch | \(-\frac {\left (1-36 a^{8 x} a^{4 x} x \ln \left (a \right )+2 a^{12 x} a^{6 x}-6 a^{2 x} a^{4 x}\right ) a^{-12 x}}{12 \ln \left (a \right )}\) | \(63\) |
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Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx=\frac {36 \, a^{12 \, x} x \log \left (a\right ) - 2 \, a^{18 \, x} + 6 \, a^{6 \, x} - 1}{12 \, a^{12 \, x} \log \left (a\right )} \]
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Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.23 \[ \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx=3 x + \begin {cases} \frac {- 24 a^{6 x} \log {\left (a \right )}^{2} + 72 a^{- 6 x} \log {\left (a \right )}^{2} - 12 a^{- 12 x} \log {\left (a \right )}^{2}}{144 \log {\left (a \right )}^{3}} & \text {for}\: \log {\left (a \right )}^{3} \neq 0 \\- 3 x & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx=3 \, x - \frac {a^{6 \, x}}{6 \, \log \left (a\right )} - \frac {1}{12 \, a^{12 \, x} \log \left (a\right )} + \frac {1}{2 \, a^{6 \, x} \log \left (a\right )} \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.07 \[ \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx=-\frac {2 \, a^{6 \, x} + \frac {9 \, a^{12 \, x} - 6 \, a^{6 \, x} + 1}{a^{12 \, x}} - 6 \, \log \left (a^{6 \, x}\right )}{12 \, \log \left (a\right )} \]
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Time = 0.50 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx=3\,x+\frac {1}{2\,a^{6\,x}\,\ln \left (a\right )}-\frac {a^{6\,x}}{6\,\ln \left (a\right )}-\frac {1}{12\,a^{12\,x}\,\ln \left (a\right )} \]
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