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 = 0.0272945, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 2282
Rule 266
Rule 43
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*}
Mathematica [A] time = 0.0315887, 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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Maple [A] time = 0.019, size = 56, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ({{\rm e}^{2\,x\ln \left ( a \right ) }} \right ) ^{6}} \left ( -{\frac{1}{12\,\ln \left ( a \right ) }}+3\,x \left ({{\rm e}^{2\,x\ln \left ( a \right ) }} \right ) ^{6}+{\frac{ \left ({{\rm e}^{2\,x\ln \left ( a \right ) }} \right ) ^{3}}{2\,\ln \left ( a \right ) }}-{\frac{ \left ({{\rm e}^{2\,x\ln \left ( a \right ) }} \right ) ^{9}}{6\,\ln \left ( a \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.925228, size = 55, normalized size = 1.28 \begin{align*} 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91736, size = 103, normalized size = 2.4 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.21758, size = 54, normalized size = 1.26 \begin{align*} 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}\: 144 \log{\left (a \right )}^{3} \neq 0 \\- 3 x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06233, size = 62, normalized size = 1.44 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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