Optimal. Leaf size=53 \[ \frac {a^{2 k x}}{2 k \log (a)}+\frac {a^{2 l x}}{2 l \log (a)}+\frac {2 a^{(k+l) x}}{(k+l) \log (a)} \]
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Rubi [A]
time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6874, 2225}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 6874
Rubi steps
\begin {align*} \int \left (a^{k x}+a^{l x}\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (e^{k x}+e^{l x}\right )^2 \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\text {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 {\text {Subst}\left (\int e^{2 k x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {\text {Subst}\left (\int e^{2 l x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {2 \text {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 = 0.04, size = 53, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.70, size = 277, normalized size = 5.23 \begin {gather*} \text {Piecewise}\left [\left \{\left \{4 x,a\text {==}1\text {\&\&}a\text {==}1\text {$\vert $$\vert $}k\text {==}0\text {\&\&}a\text {==}1\text {$\vert $$\vert $}l\text {==}0\right \},\left \{\frac {l x \text {Log}\left [a\right ]+\frac {a^{2 l x}}{2}+2 a^{l x}}{l \text {Log}\left [a\right ]},k\text {==}0\right \},\left \{-\frac {a^{-2 l x}}{2 l \text {Log}\left [a\right ]}+\frac {a^{2 l x}}{2 l \text {Log}\left [a\right ]}+2 x,k\text {==}-l\right \},\left \{\frac {k x \text {Log}\left [a\right ]+\frac {a^{2 k x}}{2}+2 a^{k x}}{k \text {Log}\left [a\right ]},l\text {==}0\right \}\right \},\frac {k^2 a^{2 l x}}{2 k^2 l \text {Log}\left [a\right ]+2 k l^2 \text {Log}\left [a\right ]}+\frac {4 k l a^{k x} a^{l x}}{2 k^2 l \text {Log}\left [a\right ]+2 k l^2 \text {Log}\left [a\right ]}+\frac {k l a^{2 k x}}{2 k^2 l \text {Log}\left [a\right ]+2 k l^2 \text {Log}\left [a\right ]}+\frac {k l a^{2 l x}}{2 k^2 l \text {Log}\left [a\right ]+2 k l^2 \text {Log}\left [a\right ]}+\frac {l^2 a^{2 k x}}{2 k^2 l \text {Log}\left [a\right ]+2 k l^2 \text {Log}\left [a\right ]}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.04, size = 55, normalized size = 1.04
method | result | size |
risch | \(\frac {a^{2 k x}}{2 k \ln \left (a \right )}+\frac {a^{2 l x}}{2 l \ln \left (a \right )}+\frac {2 a^{k x} a^{l x}}{\ln \left (a \right ) \left (k +l \right )}\) | \(55\) |
norman | \(\frac {{\mathrm e}^{2 k x \ln \left (a \right )}}{2 k \ln \left (a \right )}+\frac {{\mathrm e}^{2 l x \ln \left (a \right )}}{2 l \ln \left (a \right )}+\frac {2 \,{\mathrm e}^{k x \ln \left (a \right )} {\mathrm e}^{l x \ln \left (a \right )}}{\ln \left (a \right ) \left (k +l \right )}\) | \(59\) |
meijerg | \(-\frac {1-{\mathrm e}^{2 k x \ln \left (a \right )}}{2 k \ln \left (a \right )}-\frac {2 \left (1-{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}-\frac {1-{\mathrm e}^{2 l x \ln \left (a \right )}}{2 l \ln \left (a \right )}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.24, size = 51, normalized size = 0.96 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 62, normalized size = 1.17 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.52, size = 250, normalized size = 4.72 \begin {gather*} \begin {cases} 4 x & \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.02, size = 677, normalized size = 12.77
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 68, normalized size = 1.28 \begin {gather*} \frac {a^{2\,k\,x}}{2\,k\,\ln \left (a\right )}+\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 \left (a\right )\,\left (k+l\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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