Optimal. Leaf size=43 \[ -\frac {a^{-12 x}}{12 \log (a)}+\frac {a^{-6 x}}{2 \log (a)}-\frac {a^{6 x}}{6 \log (a)}+3 x \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2282, 266, 43} \[ -\frac {a^{-12 x}}{12 \log (a)}+\frac {a^{-6 x}}{2 \log (a)}-\frac {a^{6 x}}{6 \log (a)}+3 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2282
Rubi steps
\begin {align*} \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^3\right )^3}{x^7} \, dx,x,a^{2 x}\right )}{2 \log (a)}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(1-x)^3}{x^3} \, dx,x,a^{6 x}\right )}{6 \log (a)}\\ &=\frac {\operatorname {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*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 0.77 \[ -\frac {a^{-12 x}-6 a^{-6 x}+2 a^{6 x}-36 x \log (a)}{12 \log (a)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^{-4 x}-a^{2 x}\right )^3 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.14, size = 39, normalized size = 0.91 \[ \frac {36 \, a^{12 \, x} x \log \relax (a) - 2 \, a^{18 \, x} + 6 \, a^{6 \, x} - 1}{12 \, a^{12 \, x} \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 46, normalized size = 1.07 \[ -\frac {2 \, a^{6 \, x} + \frac {9 \, a^{12 \, x} - 6 \, a^{6 \, x} + 1}{a^{12 \, x}} - 6 \, \log \left (a^{6 \, x}\right )}{12 \, \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 44, normalized size = 1.02
method | result | size |
risch | \(3 x -\frac {a^{6 x}}{6 \ln \relax (a )}+\frac {a^{-6 x}}{2 \ln \relax (a )}-\frac {a^{-12 x}}{12 \ln \relax (a )}\) | \(44\) |
norman | \(\left (-\frac {1}{12 \ln \relax (a )}+3 x \,{\mathrm e}^{12 x \ln \relax (a )}+\frac {{\mathrm e}^{6 x \ln \relax (a )}}{2 \ln \relax (a )}-\frac {{\mathrm e}^{18 x \ln \relax (a )}}{6 \ln \relax (a )}\right ) {\mathrm e}^{-12 x \ln \relax (a )}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 41, normalized size = 0.95 \[ 3 \, x - \frac {a^{6 \, x}}{6 \, \log \relax (a)} - \frac {1}{12 \, a^{12 \, x} \log \relax (a)} + \frac {1}{2 \, a^{6 \, x} \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 41, normalized size = 0.95 \[ 3\,x+\frac {1}{2\,a^{6\,x}\,\ln \relax (a)}-\frac {a^{6\,x}}{6\,\ln \relax (a)}-\frac {1}{12\,a^{12\,x}\,\ln \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 54, normalized size = 1.26 \[ 3 x + \begin {cases} \frac {- 24 a^{6 x} \log {\relax (a )}^{2} + 72 a^{- 6 x} \log {\relax (a )}^{2} - 12 a^{- 12 x} \log {\relax (a )}^{2}}{144 \log {\relax (a )}^{3}} & \text {for}\: 144 \log {\relax (a )}^{3} \neq 0 \\- 3 x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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