Integrand size = 31, antiderivative size = 52 \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^4} \, dx=-\frac {i \operatorname {Hypergeometric2F1}\left (5,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{32 c^4 f m} \]
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Time = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 70} \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^4} \, dx=-\frac {i (a+i a \tan (e+f x))^m \operatorname {Hypergeometric2F1}\left (5,m,m+1,\frac {1}{2} (i \tan (e+f x)+1)\right )}{32 c^4 f m} \]
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Rule 70
Rule 3568
Rule 3603
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^8(e+f x) (a+i a \tan (e+f x))^{4+m} \, dx}{a^4 c^4} \\ & = -\frac {\left (i a^5\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{(a-x)^5} \, dx,x,i a \tan (e+f x)\right )}{c^4 f} \\ & = -\frac {i \operatorname {Hypergeometric2F1}\left (5,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{32 c^4 f m} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00 \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^4} \, dx=-\frac {i \operatorname {Hypergeometric2F1}\left (5,m,1+m,\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{32 c^4 f m} \]
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\[\int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{\left (c -i c \tan \left (f x +e \right )\right )^{4}}d x\]
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\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^4} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{4}} \,d x } \]
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\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^4} \, dx=\frac {\int \frac {\left (i a \tan {\left (e + f x \right )} + a\right )^{m}}{\tan ^{4}{\left (e + f x \right )} + 4 i \tan ^{3}{\left (e + f x \right )} - 6 \tan ^{2}{\left (e + f x \right )} - 4 i \tan {\left (e + f x \right )} + 1}\, dx}{c^{4}} \]
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Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^4} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^4} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^4} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^4} \,d x \]
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