Integrand size = 28, antiderivative size = 85 \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx=-\frac {6 i \sqrt [6]{2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{6},\frac {1}{6},\frac {1}{2} (1-i \tan (e+f x))\right ) \left (a^2+i a^2 \tan (e+f x)\right )}{5 f (d \sec (e+f x))^{5/3} \sqrt [6]{1+i \tan (e+f x)}} \]
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Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3586, 3604, 72, 71} \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx=-\frac {6 i \sqrt [6]{2} \left (a^2+i a^2 \tan (e+f x)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{6},\frac {1}{6},\frac {1}{2} (1-i \tan (e+f x))\right )}{5 f \sqrt [6]{1+i \tan (e+f x)} (d \sec (e+f x))^{5/3}} \]
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Rule 71
Rule 72
Rule 3586
Rule 3604
Rubi steps \begin{align*} \text {integral}& = \frac {\left ((a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6}\right ) \int \frac {(a+i a \tan (e+f x))^{7/6}}{(a-i a \tan (e+f x))^{5/6}} \, dx}{(d \sec (e+f x))^{5/3}} \\ & = \frac {\left (a^2 (a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))^{5/6}\right ) \text {Subst}\left (\int \frac {\sqrt [6]{a+i a x}}{(a-i a x)^{11/6}} \, dx,x,\tan (e+f x)\right )}{f (d \sec (e+f x))^{5/3}} \\ & = \frac {\left (\sqrt [6]{2} a^2 (a-i a \tan (e+f x))^{5/6} (a+i a \tan (e+f x))\right ) \text {Subst}\left (\int \frac {\sqrt [6]{\frac {1}{2}+\frac {i x}{2}}}{(a-i a x)^{11/6}} \, dx,x,\tan (e+f x)\right )}{f (d \sec (e+f x))^{5/3} \sqrt [6]{\frac {a+i a \tan (e+f x)}{a}}} \\ & = -\frac {6 i \sqrt [6]{2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{6},\frac {1}{6},\frac {1}{2} (1-i \tan (e+f x))\right ) \left (a^2+i a^2 \tan (e+f x)\right )}{5 f (d \sec (e+f x))^{5/3} \sqrt [6]{1+i \tan (e+f x)}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.24 \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx=\frac {3 a^2 \left (\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{2},\frac {1}{6},\sec ^2(e+f x)\right ) \tan (e+f x)+\operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},\sec ^2(e+f x)\right ) \tan (e+f x)-2 i \sqrt {-\tan ^2(e+f x)}\right )}{5 f (d \sec (e+f x))^{5/3} \sqrt {-\tan ^2(e+f x)}} \]
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\[\int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{2}}{\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}}}d x\]
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\[ \int \frac {(a+i a \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}} \,d x } \]
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\[ \int \frac {(a+i a \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx=- a^{2} \left (\int \left (- \frac {1}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}\right )\, dx + \int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}\, dx + \int \left (- \frac {2 i \tan {\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}\right )\, dx\right ) \]
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\[ \int \frac {(a+i a \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}} \,d x } \]
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\[ \int \frac {(a+i a \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2}}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(a+i a \tan (e+f x))^2}{(d \sec (e+f x))^{5/3}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}} \,d x \]
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