Integrand size = 13, antiderivative size = 60 \[ \int (e x)^m \tanh (a+2 \log (x)) \, dx=\frac {(e x)^{1+m}}{e (1+m)}-\frac {2 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},-e^{2 a} x^4\right )}{e (1+m)} \]
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Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5656, 470, 371} \[ \int (e x)^m \tanh (a+2 \log (x)) \, dx=\frac {(e x)^{m+1}}{e (m+1)}-\frac {2 (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{4},\frac {m+5}{4},-e^{2 a} x^4\right )}{e (m+1)} \]
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Rule 371
Rule 470
Rule 5656
Rubi steps \begin{align*} \text {integral}& = \int \frac {(e x)^m \left (-1+e^{2 a} x^4\right )}{1+e^{2 a} x^4} \, dx \\ & = \frac {(e x)^{1+m}}{e (1+m)}-2 \int \frac {(e x)^m}{1+e^{2 a} x^4} \, dx \\ & = \frac {(e x)^{1+m}}{e (1+m)}-\frac {2 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},-e^{2 a} x^4\right )}{e (1+m)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.78 \[ \int (e x)^m \tanh (a+2 \log (x)) \, dx=-\frac {x (e x)^m \left (-1+2 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{4},\frac {5+m}{4},-x^4 (\cosh (2 a)+\sinh (2 a))\right )\right )}{1+m} \]
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\[\int \left (e x \right )^{m} \tanh \left (a +2 \ln \left (x \right )\right )d x\]
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\[ \int (e x)^m \tanh (a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right ) \,d x } \]
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\[ \int (e x)^m \tanh (a+2 \log (x)) \, dx=\int \left (e x\right )^{m} \tanh {\left (a + 2 \log {\left (x \right )} \right )}\, dx \]
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\[ \int (e x)^m \tanh (a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right ) \,d x } \]
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\[ \int (e x)^m \tanh (a+2 \log (x)) \, dx=\int { \left (e x\right )^{m} \tanh \left (a + 2 \, \log \left (x\right )\right ) \,d x } \]
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Timed out. \[ \int (e x)^m \tanh (a+2 \log (x)) \, dx=\int \mathrm {tanh}\left (a+2\,\ln \left (x\right )\right )\,{\left (e\,x\right )}^m \,d x \]
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