Optimal. Leaf size=53 \[ -\frac{2 a^{x (k+l)}}{\log (a) (k+l)}+\frac{a^{2 k x}}{2 k \log (a)}+\frac{a^{2 l x}}{2 l \log (a)} \]
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Rubi [A] time = 0.127586, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 a^{x (k+l)}}{\log (a) (k+l)}+\frac{a^{2 k x}}{2 k \log (a)}+\frac{a^{2 l x}}{2 l \log (a)} \]
Antiderivative was successfully verified.
[In] Int[(a^(k*x) - a^(l*x))^2,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a^{k x} - a^{l x}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a**(k*x)-a**(l*x))**2,x)
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Mathematica [A] time = 0.0573637, size = 55, normalized size = 1.04 \[ -\frac{2 a^{k x+l x}}{\log (a) (k+l)}+\frac{a^{2 k x}}{2 k \log (a)}+\frac{a^{2 l x}}{2 l \log (a)} \]
Antiderivative was successfully verified.
[In] Integrate[(a^(k*x) - a^(l*x))^2,x]
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Maple [A] time = 0.023, size = 59, normalized size = 1.1 \[{\frac{ \left ({{\rm e}^{kx\ln \left ( a \right ) }} \right ) ^{2}}{2\,k\ln \left ( a \right ) }}+{\frac{ \left ({{\rm e}^{lx\ln \left ( a \right ) }} \right ) ^{2}}{2\,l\ln \left ( a \right ) }}-2\,{\frac{{{\rm e}^{kx\ln \left ( a \right ) }}{{\rm e}^{lx\ln \left ( a \right ) }}}{\ln \left ( a \right ) \left ( k+l \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a^(k*x)-a^(l*x))^2,x)
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Maxima [A] time = 1.39032, size = 69, normalized size = 1.3 \[ -\frac{2 \, a^{k x + l x}}{{\left (k + l\right )} \log \left (a\right )} + \frac{a^{2 \, k x}}{2 \, k \log \left (a\right )} + \frac{a^{2 \, l x}}{2 \, l \log \left (a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a^(k*x) - a^(l*x))^2,x, algorithm="maxima")
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Fricas [A] time = 0.218455, size = 86, normalized size = 1.62 \[ -\frac{4 \, a^{k x} a^{l x} k l -{\left (k l + l^{2}\right )} a^{2 \, k x} -{\left (k^{2} + k l\right )} a^{2 \, l x}}{2 \,{\left (k^{2} l + k l^{2}\right )} \log \left (a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a^(k*x) - a^(l*x))^2,x, algorithm="fricas")
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Sympy [A] time = 3.47077, size = 248, normalized size = 4.68 \[ \begin{cases} 0 & \text{for}\: a = 1 \wedge \left (a = 1 \vee k = 0\right ) \wedge \left (a = 1 \vee l = 0\right ) \\\frac{a^{2 l x}}{2 l \log{\left (a \right )}} - \frac{2 a^{l x}}{l \log{\left (a \right )}} + x & \text{for}\: k = 0 \\\frac{a^{2 l x}}{2 l \log{\left (a \right )}} - 2 x - \frac{a^{- 2 l x}}{2 l \log{\left (a \right )}} & \text{for}\: k = - l \\\frac{a^{2 k x}}{2 k \log{\left (a \right )}} - \frac{2 a^{k x}}{k \log{\left (a \right )}} + x & \text{for}\: l = 0 \\\frac{a^{2 k x} k l}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} + \frac{a^{2 k x} l^{2}}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} - \frac{4 a^{k x} a^{l x} k l}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} + \frac{a^{2 l x} k^{2}}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} + \frac{a^{2 l x} k l}{2 k^{2} l \log{\left (a \right )} + 2 k l^{2} \log{\left (a \right )}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a**(k*x)-a**(l*x))**2,x)
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GIAC/XCAS [A] time = 0.256413, size = 938, normalized size = 17.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a^(k*x) - a^(l*x))^2,x, algorithm="giac")
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