Optimal. Leaf size=171 \[ \frac{i 2^{-m-2} e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )}{a^2 f}-\frac{i 4^{-m-2} e^{4 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{4 i f (c+d x)}{d}\right )}{a^2 f}+\frac{(c+d x)^{m+1}}{4 a^2 d (m+1)} \]
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Rubi [A] time = 0.184724, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3729, 2181} \[ \frac{i 2^{-m-2} e^{2 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )}{a^2 f}-\frac{i 4^{-m-2} e^{4 i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{4 i f (c+d x)}{d}\right )}{a^2 f}+\frac{(c+d x)^{m+1}}{4 a^2 d (m+1)} \]
Antiderivative was successfully verified.
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Rule 3729
Rule 2181
Rubi steps
\begin{align*} \int \frac{(c+d x)^m}{(a+i a \cot (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^m}{4 a^2}-\frac{e^{2 i e+2 i f x} (c+d x)^m}{2 a^2}+\frac{e^{4 i e+4 i f x} (c+d x)^m}{4 a^2}\right ) \, dx\\ &=\frac{(c+d x)^{1+m}}{4 a^2 d (1+m)}+\frac{\int e^{4 i e+4 i f x} (c+d x)^m \, dx}{4 a^2}-\frac{\int e^{2 i e+2 i f x} (c+d x)^m \, dx}{2 a^2}\\ &=\frac{(c+d x)^{1+m}}{4 a^2 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^2 f}-\frac{i 4^{-2-m} e^{4 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{4 i f (c+d x)}{d}\right )}{a^2 f}\\ \end{align*}
Mathematica [A] time = 2.55082, size = 152, normalized size = 0.89 \[ \frac{(c+d x)^m \left (i 2^{2-m} e^{2 i \left (e-\frac{c f}{d}\right )} \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 i f (c+d x)}{d}\right )-i 4^{-m} e^{4 i \left (e-\frac{c f}{d}\right )} \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{4 i f (c+d x)}{d}\right )+\frac{4 f (c+d x)}{d (m+1)}\right )}{16 a^2 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{m}}{ \left ( a+ia\cot \left ( fx+e \right ) \right ) ^{2}}}\, 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 (4 \, f x + 4 \, e\right )\,{d x} - 2 \,{\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 (4 \, f x + 4 \, e\right )\,{d x} +{\left (-2 i \, d m - 2 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 )}}{4 \,{\left (a^{2} d m + a^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72403, size = 360, normalized size = 2.11 \begin{align*} \frac{{\left (4 i \, d m + 4 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 ) +{\left (-i \, d m - i \, d\right )} e^{\left (-\frac{d m \log \left (-\frac{4 i \, f}{d}\right ) - 4 i \, d e + 4 i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{-4 i \, d f x - 4 i \, c f}{d}\right ) + 4 \,{\left (d f x + c f\right )}{\left (d x + c\right )}^{m}}{16 \,{\left (a^{2} d f m + a^{2} 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}}{{\left (i \, a \cot \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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