Integrand size = 33, antiderivative size = 67 \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {i \operatorname {Hypergeometric2F1}\left (1,-\frac {5}{2}+m,-\frac {3}{2},\frac {1}{2} (1-i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{5 f (c-i c \tan (e+f x))^{5/2}} \]
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Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3604, 72, 71} \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {i 2^m (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1-m,-\frac {3}{2},\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f (c-i c \tan (e+f x))^{5/2}} \]
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Rule 71
Rule 72
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{-1+m}}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\left (2^{-1+m} c (a+i a \tan (e+f x))^m \left (\frac {a+i a \tan (e+f x)}{a}\right )^{-m}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {i x}{2}\right )^{-1+m}}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {i 2^m \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1-m,-\frac {3}{2},\frac {1}{2} (1-i \tan (e+f x))\right ) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m}{5 f (c-i c \tan (e+f x))^{5/2}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(67)=134\).
Time = 19.42 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.36 \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{5/2}} \, dx=-\frac {i 2^{-\frac {7}{2}+m} \left (e^{i f x}\right )^m \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \left (1+e^{2 i (e+f x)}\right )^4 \operatorname {Hypergeometric2F1}\left (1,\frac {7}{2},1+m,-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{c^3 f m} \]
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\[\int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{\left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{5/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{5/2}} \, dx=\int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m}}{\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{5/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+i a \tan (e+f x))^m}{(c-i c \tan (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
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