Integrand size = 9, antiderivative size = 33 \[ \int 3^{2^x} 4^x \, dx=-\frac {3^{2^x}}{\log (2) \log ^2(3)}+\frac {2^x 3^{2^x}}{\log (2) \log (3)} \]
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Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2320, 2207, 2225} \[ \int 3^{2^x} 4^x \, dx=\frac {2^x 3^{2^x}}{\log (2) \log (3)}-\frac {3^{2^x}}{\log (2) \log ^2(3)} \]
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Rule 2207
Rule 2225
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int 3^x x \, dx,x,2^x\right )}{\log (2)} \\ & = \frac {2^x 3^{2^x}}{\log (2) \log (3)}-\frac {\text {Subst}\left (\int 3^x \, dx,x,2^x\right )}{\log (2) \log (3)} \\ & = -\frac {3^{2^x}}{\log (2) \log ^2(3)}+\frac {2^x 3^{2^x}}{\log (2) \log (3)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int 3^{2^x} 4^x \, dx=\frac {3^{2^x} \left (-1+2^x \log (3)\right )}{\log (2) \log ^2(3)} \]
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Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\frac {\left (2^{x} \ln \left (3\right )-1\right ) 3^{2^{x}}}{\ln \left (2\right ) \ln \left (3\right )^{2}}\) | \(23\) |
norman | \(\frac {{\mathrm e}^{x \ln \left (2\right )} {\mathrm e}^{{\mathrm e}^{x \ln \left (2\right )} \ln \left (3\right )}}{\ln \left (2\right ) \ln \left (3\right )}-\frac {{\mathrm e}^{{\mathrm e}^{x \ln \left (2\right )} \ln \left (3\right )}}{\ln \left (2\right ) \ln \left (3\right )^{2}}\) | \(44\) |
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none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int 3^{2^x} 4^x \, dx=\frac {{\left (2^{x} \log \left (3\right ) - 1\right )} 3^{\left (2^{x}\right )}}{\log \left (3\right )^{2} \log \left (2\right )} \]
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Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int 3^{2^x} 4^x \, dx=\frac {\left (e^{\frac {x \log {\left (4 \right )}}{2}} \log {\left (3 \right )} - 1\right ) e^{e^{\frac {x \log {\left (4 \right )}}{2}} \log {\left (3 \right )}}}{\log {\left (2 \right )} \log {\left (3 \right )}^{2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int 3^{2^x} 4^x \, dx=-\frac {4^{x} \Gamma \left (2, -4^{\frac {1}{2} \, x} \log \left (3\right )\right )}{4^{x} \log \left (3\right )^{2} \log \left (2\right )} \]
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none
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int 3^{2^x} 4^x \, dx=\frac {2^{x} e^{\left (2^{x} \log \left (3\right ) + 2 \, x \log \left (2\right )\right )} \log \left (3\right ) - e^{\left (2^{x} \log \left (3\right ) + 2 \, x \log \left (2\right )\right )}}{2^{2 \, x} \log \left (3\right )^{2} \log \left (2\right )} \]
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Time = 15.86 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int 3^{2^x} 4^x \, dx=\frac {3^{2^x}\,\left (2^x\,\ln \left (3\right )-1\right )}{\ln \left (2\right )\,{\ln \left (3\right )}^2} \]
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