Optimal. Leaf size=171 \[ \frac {i 2^{-m-2} e^{2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 i f (c+d x)}{d}\right )}{a^2 f}-\frac {i 4^{-m-2} e^{4 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {4 i f (c+d x)}{d}\right )}{a^2 f}+\frac {(c+d x)^{m+1}}{4 a^2 d (m+1)} \]
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Rubi [A] time = 0.18, antiderivative size = 171, normalized size of antiderivative = 1.00, 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 2181
Rule 3729
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*}
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Mathematica [A] time = 2.83, size = 152, normalized size = 0.89 \[ \frac {(c+d x)^m \left (i 2^{2-m} e^{2 i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 i f (c+d x)}{d}\right )-i 4^{-m} e^{4 i \left (e-\frac {c f}{d}\right )} \left (-\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {4 i f (c+d x)}{d}\right )+\frac {4 f (c+d x)}{d (m+1)}\right )}{16 a^2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 140, normalized size = 0.82 \[ \frac {{\left (4 i \, d m + 4 i \, d\right )} e^{\left (-\frac {d m \log \left (-\frac {2 i \, f}{d}\right ) - 2 i \, d e + 2 i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {-2 i \, d f x - 2 i \, c f}{d}\right ) + {\left (-i \, d m - i \, d\right )} e^{\left (-\frac {d m \log \left (-\frac {4 i \, f}{d}\right ) - 4 i \, d e + 4 i \, c f}{d}\right )} \Gamma \left (m + 1, \frac {-4 i \, d f x - 4 i \, c f}{d}\right ) + 4 \, {\left (d f x + c f\right )} {\left (d x + c\right )}^{m}}{16 \, {\left (a^{2} d f m + a^{2} d f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d x + c\right )}^{m}}{{\left (i \, a \cot \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{m}}{\left (a +i a \cot \left (f x +e \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (d m + d\right )} \int {\left (d x + c\right )}^{m} \cos \left (4 \, f x + 4 \, e\right )\,{d x} - 2 \, {\left (d m + d\right )} \int {\left (d x + c\right )}^{m} \cos \left (2 \, f x + 2 \, e\right )\,{d x} + {\left (i \, d m + i \, d\right )} \int {\left (d x + c\right )}^{m} \sin \left (4 \, f x + 4 \, e\right )\,{d x} + {\left (-2 i \, d m - 2 i \, d\right )} \int {\left (d x + c\right )}^{m} \sin \left (2 \, f x + 2 \, e\right )\,{d x} + e^{\left (m \log \left (d x + c\right ) + \log \left (d x + c\right )\right )}}{4 \, {\left (a^{2} d m + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\left (c + d x\right )^{m}}{\cot ^{2}{\left (e + f x \right )} - 2 i \cot {\left (e + f x \right )} - 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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