Optimal. Leaf size=70 \[ \frac{i (e x)^{m+1}}{e (m+1)}-\frac{2 i (e x)^{m+1} \text{Hypergeometric2F1}\left (1,\frac{1}{2} (-m-1),\frac{1-m}{2},\frac{e^{2 i a}}{x^2}\right )}{e (m+1)} \]
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Rubi [F] time = 0.040328, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (e x)^m \cot (a+i \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (e x)^m \cot (a+i \log (x)) \, dx &=\int (e x)^m \cot (a+i \log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.241067, size = 103, normalized size = 1.47 \[ i x (e x)^m \left (\frac{x^2 (\cos (a)-i \sin (a))^2 \text{Hypergeometric2F1}\left (1,\frac{m+3}{2},\frac{m+5}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )}{m+3}+\frac{\text{Hypergeometric2F1}\left (1,\frac{m+1}{2},\frac{m+3}{2},x^2 (\cos (2 a)-i \sin (2 a))\right )}{m+1}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.158, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\cot \left ( a+i\ln \left ( x \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (e x\right )^{m}{\left (i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i\right )}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \cot{\left (a + i \log{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \cot \left (a + i \, \log \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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