Optimal. Leaf size=98 \[ \frac{(c+d x)^{m+1}}{2 a d (m+1)}+\frac{i 2^{-m-2} e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )}{a f} \]
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Rubi [A] time = 0.12381, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3727, 2181} \[ \frac{(c+d x)^{m+1}}{2 a d (m+1)}+\frac{i 2^{-m-2} e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )}{a f} \]
Antiderivative was successfully verified.
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Rule 3727
Rule 2181
Rubi steps
\begin{align*} \int \frac{(c+d x)^m}{a+i a \cot (e+f x)} \, dx &=\frac{(c+d x)^{1+m}}{2 a d (1+m)}+\frac{\int e^{2 i \left (e+\frac{\pi }{2}+f x\right )} (c+d x)^m \, dx}{2 a}\\ &=\frac{(c+d x)^{1+m}}{2 a d (1+m)}+\frac{i 2^{-2-m} e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{2 i f (c+d x)}{d}\right )}{a f}\\ \end{align*}
Mathematica [A] time = 1.21785, size = 190, normalized size = 1.94 \[ \frac{2^{-m-2} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^m \left (\frac{f^2 (c+d x)^2}{d^2}\right )^{-m} \left (\cos \left (e-\frac{c f}{d}\right )+i \sin \left (e-\frac{c f}{d}\right )\right ) \left (i d (m+1) \left (\cos \left (e-\frac{c f}{d}\right )+i \sin \left (e-\frac{c f}{d}\right )\right ) \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )+f 2^{m+1} (c+d x) \left (-\frac{i f (c+d x)}{d}\right )^m \left (\cos \left (e-\frac{c f}{d}\right )-i \sin \left (e-\frac{c f}{d}\right )\right )\right )}{a d f (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.267, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{m}}{a+ia\cot \left ( fx+e \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*} -\frac{{\left (d m + d\right )} \int{\left (d x + c\right )}^{m} \cos \left (2 \, f x + 2 \, e\right )\,{d x} +{\left (i \, d m + i \, d\right )} \int{\left (d x + c\right )}^{m} \sin \left (2 \, f x + 2 \, e\right )\,{d x} - e^{\left (m \log \left (d x + c\right ) + \log \left (d x + c\right )\right )}}{2 \,{\left (a d m + a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67098, size = 209, normalized size = 2.13 \begin{align*} \frac{{\left (i \, d m + i \, d\right )} e^{\left (-\frac{d m \log \left (-\frac{2 i \, f}{d}\right ) - 2 i \, d e + 2 i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{-2 i \, d f x - 2 i \, c f}{d}\right ) + 2 \,{\left (d f x + c f\right )}{\left (d x + c\right )}^{m}}{4 \,{\left (a d f m + a d f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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 \frac{{\left (d x + c\right )}^{m}}{i \, a \cot \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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