Integrand size = 11, antiderivative size = 27 \[ \int \left (a^{k x}+a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)} \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2225} \[ \int \left (a^{k x}+a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)} \]
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Rule 2225
Rubi steps \begin{align*} \text {integral}& = \int a^{k x} \, dx+\int a^{l x} \, dx \\ & = \frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \left (a^{k x}+a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)} \]
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Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {a^{k x} l +a^{l x} k}{k \ln \left (a \right ) l}\) | \(27\) |
default | \(\frac {a^{k x}}{k \ln \left (a \right )}+\frac {a^{l x}}{l \ln \left (a \right )}\) | \(28\) |
risch | \(\frac {a^{k x}}{k \ln \left (a \right )}+\frac {a^{l x}}{l \ln \left (a \right )}\) | \(28\) |
parts | \(\frac {a^{k x}}{k \ln \left (a \right )}+\frac {a^{l x}}{l \ln \left (a \right )}\) | \(28\) |
norman | \(\frac {{\mathrm e}^{k x \ln \left (a \right )}}{k \ln \left (a \right )}+\frac {{\mathrm e}^{l x \ln \left (a \right )}}{l \ln \left (a \right )}\) | \(30\) |
meijerg | \(-\frac {1-{\mathrm e}^{k x \ln \left (a \right )}}{k \ln \left (a \right )}-\frac {1-{\mathrm e}^{l x \ln \left (a \right )}}{l \ln \left (a \right )}\) | \(40\) |
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none
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \left (a^{k x}+a^{l x}\right ) \, dx=\frac {a^{l x} k + a^{k x} l}{k l \log \left (a\right )} \]
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Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \left (a^{k x}+a^{l x}\right ) \, dx=\begin {cases} \frac {a^{k x}}{k \log {\left (a \right )}} & \text {for}\: k \log {\left (a \right )} \neq 0 \\x & \text {otherwise} \end {cases} + \begin {cases} \frac {a^{l x}}{l \log {\left (a \right )}} & \text {for}\: l \log {\left (a \right )} \neq 0 \\x & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \left (a^{k x}+a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log \left (a\right )} + \frac {a^{l x}}{l \log \left (a\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \left (a^{k x}+a^{l x}\right ) \, dx=\frac {a^{k x}}{k \log \left (a\right )} + \frac {a^{l x}}{l \log \left (a\right )} \]
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Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \left (a^{k x}+a^{l x}\right ) \, dx=\frac {a^{k\,x}\,l+a^{l\,x}\,k}{k\,l\,\ln \left (a\right )} \]
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