Integrand size = 23, antiderivative size = 251 \[ \int \frac {(c+d x)^m}{(a+i a \tan (e+f x))^3} \, dx=\frac {(c+d x)^{1+m}}{8 a^3 d (1+m)}+\frac {3 i 2^{-4-m} e^{-2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 i f (c+d x)}{d}\right )}{a^3 f}+\frac {3 i 2^{-5-2 m} e^{-4 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {4 i f (c+d x)}{d}\right )}{a^3 f}+\frac {i 2^{-4-m} 3^{-1-m} e^{-6 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {6 i f (c+d x)}{d}\right )}{a^3 f} \]
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Time = 0.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3810, 2212} \[ \int \frac {(c+d x)^m}{(a+i a \tan (e+f x))^3} \, dx=\frac {3 i 2^{-m-4} e^{-2 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 i f (c+d x)}{d}\right )}{a^3 f}+\frac {3 i 2^{-2 m-5} e^{-4 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {4 i f (c+d x)}{d}\right )}{a^3 f}+\frac {i 2^{-m-4} 3^{-m-1} e^{-6 i \left (e-\frac {c f}{d}\right )} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {6 i f (c+d x)}{d}\right )}{a^3 f}+\frac {(c+d x)^{m+1}}{8 a^3 d (m+1)} \]
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Rule 2212
Rule 3810
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(c+d x)^m}{8 a^3}+\frac {3 e^{-2 i e-2 i f x} (c+d x)^m}{8 a^3}+\frac {3 e^{-4 i e-4 i f x} (c+d x)^m}{8 a^3}+\frac {e^{-6 i e-6 i f x} (c+d x)^m}{8 a^3}\right ) \, dx \\ & = \frac {(c+d x)^{1+m}}{8 a^3 d (1+m)}+\frac {\int e^{-6 i e-6 i f x} (c+d x)^m \, dx}{8 a^3}+\frac {3 \int e^{-2 i e-2 i f x} (c+d x)^m \, dx}{8 a^3}+\frac {3 \int e^{-4 i e-4 i f x} (c+d x)^m \, dx}{8 a^3} \\ & = \frac {(c+d x)^{1+m}}{8 a^3 d (1+m)}+\frac {3 i 2^{-4-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^3 f}+\frac {3 i 2^{-5-2 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^3 f}+\frac {i 2^{-4-m} 3^{-1-m} e^{-6 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 {6 i f (c+d x)}{d}\right )}{a^3 f} \\ \end{align*}
Time = 13.03 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.07 \[ \int \frac {(c+d x)^m}{(a+i a \tan (e+f x))^3} \, dx=\frac {2^{-5-2 m} 3^{-1-m} e^{-3 i e} (c+d x)^m \left (\frac {i f (c+d x)}{d}\right )^{-m} \left (12^{1+m} e^{6 i e} f (c+d x) \left (\frac {i f (c+d x)}{d}\right )^m+i 2^{1+m} 3^{2+m} d e^{2 i \left (2 e+\frac {c f}{d}\right )} (1+m) \Gamma \left (1+m,\frac {2 i f (c+d x)}{d}\right )+i 3^{2+m} d e^{2 i e+\frac {4 i c f}{d}} (1+m) \Gamma \left (1+m,\frac {4 i f (c+d x)}{d}\right )+i 2^{1+m} d e^{\frac {6 i c f}{d}} (1+m) \Gamma \left (1+m,\frac {6 i f (c+d x)}{d}\right )\right ) \sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3}{d f (1+m) (a+i a \tan (e+f x))^3} \]
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\[\int \frac {\left (d x +c \right )^{m}}{\left (a +i a \tan \left (f x +e \right )\right )^{3}}d x\]
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none
Time = 0.09 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^m}{(a+i a \tan (e+f x))^3} \, dx=-\frac {18 \, {\left (-i \, d m - i \, d\right )} e^{\left (-\frac {d m \log \left (\frac {2 i \, f}{d}\right ) + 2 i \, d e - 2 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) + 9 \, {\left (-i \, d m - i \, d\right )} e^{\left (-\frac {d m \log \left (\frac {4 i \, f}{d}\right ) + 4 i \, d e - 4 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {4 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) + 2 \, {\left (-i \, d m - i \, d\right )} e^{\left (-\frac {d m \log \left (\frac {6 i \, f}{d}\right ) + 6 i \, d e - 6 i \, c f}{d}\right )} \Gamma \left (m + 1, -\frac {6 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) - 12 \, {\left (d f x + c f\right )} {\left (d x + c\right )}^{m}}{96 \, {\left (a^{3} d f m + a^{3} d f\right )}} \]
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\[ \int \frac {(c+d x)^m}{(a+i a \tan (e+f x))^3} \, dx=\frac {i \int \frac {\left (c + d x\right )^{m}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx}{a^{3}} \]
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\[ \int \frac {(c+d x)^m}{(a+i a \tan (e+f x))^3} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {(c+d x)^m}{(a+i a \tan (e+f x))^3} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^m}{(a+i a \tan (e+f x))^3} \, dx=\int \frac {{\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]
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