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 = 0.07, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6742, 2194} \[ -\frac {2 a^{x (k+l)}}{\log (a) (k+l)}+\frac {a^{2 k x}}{2 k \log (a)}+\frac {a^{2 l x}}{2 l \log (a)} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 6742
Rubi steps
\begin {align*} \int \left (a^{k x}-a^{l x}\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (e^{k x}-e^{l x}\right )^2 \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\operatorname {Subst}\left (\int \left (e^{2 k x}+e^{2 l x}-2 e^{(k+l) x}\right ) \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\operatorname {Subst}\left (\int e^{2 k x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {\operatorname {Subst}\left (\int e^{2 l x} \, dx,x,x \log (a)\right )}{\log (a)}-\frac {2 \operatorname {Subst}\left (\int e^{(k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {a^{2 k x}}{2 k \log (a)}+\frac {a^{2 l x}}{2 l \log (a)}-\frac {2 a^{(k+l) x}}{(k+l) \log (a)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 1.00 \[ -\frac {2 a^{x (k+l)}}{\log (a) (k+l)}+\frac {a^{2 k x}}{2 k \log (a)}+\frac {a^{2 l x}}{2 l \log (a)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^{k x}-a^{l x}\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.38, size = 64, normalized size = 1.21 \[ -\frac {4 \, a^{k x} a^{l x} k l - {\left (k l + l^{2}\right )} a^{2 \, k x} - {\left (k^{2} + k l\right )} a^{2 \, l x}}{2 \, {\left (k^{2} l + k l^{2}\right )} \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.75, size = 691, normalized size = 13.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 55, normalized size = 1.04
method | result | size |
risch | \(\frac {a^{2 k x}}{2 k \ln \relax (a )}+\frac {a^{2 l x}}{2 l \ln \relax (a )}-\frac {2 a^{k x} a^{l x}}{\ln \relax (a ) \left (k +l \right )}\) | \(55\) |
norman | \(\frac {{\mathrm e}^{2 k x \ln \relax (a )}}{2 k \ln \relax (a )}+\frac {{\mathrm e}^{2 l x \ln \relax (a )}}{2 l \ln \relax (a )}-\frac {2 \,{\mathrm e}^{k x \ln \relax (a )} {\mathrm e}^{l x \ln \relax (a )}}{\ln \relax (a ) \left (k +l \right )}\) | \(59\) |
meijerg | error in int/gbinthm/express: improper op or subscript selector\ | N/A |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 51, normalized size = 0.96 \[ -\frac {2 \, a^{k x + l x}}{{\left (k + l\right )} \log \relax (a)} + \frac {a^{2 \, k x}}{2 \, k \log \relax (a)} + \frac {a^{2 \, l x}}{2 \, l \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 69, normalized size = 1.30 \[ \frac {a^{2\,k\,x}}{2\,k\,\ln \relax (a)}+\frac {\frac {a^{2\,l\,x}\,k^2}{2}-l\,\left (2\,a^{k\,x+l\,x}\,k-\frac {a^{2\,l\,x}\,k}{2}\right )}{k\,l\,\ln \relax (a)\,\left (k+l\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.98, size = 248, normalized size = 4.68 \[ \begin {cases} 0 & \text {for}\: a = 1 \wedge \left (a = 1 \vee k = 0\right ) \wedge \left (a = 1 \vee l = 0\right ) \\\frac {a^{2 l x}}{2 l \log {\relax (a )}} - \frac {2 a^{l x}}{l \log {\relax (a )}} + x & \text {for}\: k = 0 \\\frac {a^{2 l x}}{2 l \log {\relax (a )}} - 2 x - \frac {a^{- 2 l x}}{2 l \log {\relax (a )}} & \text {for}\: k = - l \\\frac {a^{2 k x}}{2 k \log {\relax (a )}} - \frac {2 a^{k x}}{k \log {\relax (a )}} + x & \text {for}\: l = 0 \\\frac {a^{2 k x} k l}{2 k^{2} l \log {\relax (a )} + 2 k l^{2} \log {\relax (a )}} + \frac {a^{2 k x} l^{2}}{2 k^{2} l \log {\relax (a )} + 2 k l^{2} \log {\relax (a )}} - \frac {4 a^{k x} a^{l x} k l}{2 k^{2} l \log {\relax (a )} + 2 k l^{2} \log {\relax (a )}} + \frac {a^{2 l x} k^{2}}{2 k^{2} l \log {\relax (a )} + 2 k l^{2} \log {\relax (a )}} + \frac {a^{2 l x} k l}{2 k^{2} l \log {\relax (a )} + 2 k l^{2} \log {\relax (a )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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