Integrand size = 33, antiderivative size = 67 \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\frac {i \operatorname {Hypergeometric2F1}\left (1,\frac {5}{2}+m,\frac {7}{2},\frac {1}{2} (1-i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2}}{5 f} \]
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Time = 0.14 (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} \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\frac {i 2^m (c-i c \tan (e+f x))^{5/2} (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2},1-m,\frac {7}{2},\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f} \]
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Rule 71
Rule 72
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int (a+i a x)^{-1+m} (c-i c x)^{3/2} \, 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 \left (\frac {1}{2}+\frac {i x}{2}\right )^{-1+m} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {i 2^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2},1-m,\frac {7}{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 (c-i c \tan (e+f x))^{5/2}}{5 f} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(67)=134\).
Time = 20.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.10 \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=-\frac {i 2^{\frac {3}{2}+m} c \left (e^{i f x}\right )^m \left (\frac {c}{1+e^{2 i (e+f x)}}\right )^{3/2} \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,1+m,-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m} \]
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Timed out.
\[\int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}d x\]
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\[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\int { {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\int { {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\int { {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^{5/2} \, dx=\int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
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