Integrand size = 25, antiderivative size = 64 \[ \int \frac {(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx=\frac {2 \cos ^2(e+f x)^{7/12} \operatorname {Hypergeometric2F1}\left (\frac {7}{12},\frac {5}{4},\frac {9}{4},\sin ^2(e+f x)\right ) (d \tan (e+f x))^{5/2}}{5 d f (b \sec (e+f x))^{4/3}} \]
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Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2697} \[ \int \frac {(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx=\frac {2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{12},\frac {5}{4},\frac {9}{4},\sin ^2(e+f x)\right )}{5 d f (b \sec (e+f x))^{4/3}} \]
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Rule 2697
Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos ^2(e+f x)^{7/12} \operatorname {Hypergeometric2F1}\left (\frac {7}{12},\frac {5}{4},\frac {9}{4},\sin ^2(e+f x)\right ) (d \tan (e+f x))^{5/2}}{5 d f (b \sec (e+f x))^{4/3}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.11 \[ \int \frac {(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx=\frac {3 \cot ^3(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {1}{3},\sec ^2(e+f x)\right ) (d \tan (e+f x))^{3/2} \left (-\tan ^2(e+f x)\right )^{3/4}}{4 f (b \sec (e+f x))^{4/3}} \]
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\[\int \frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{\left (b \sec \left (f x +e \right )\right )^{\frac {4}{3}}}d x\]
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\[ \int \frac {(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx=\int { \frac {\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx=\int \frac {\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx=\int { \frac {\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx=\int { \frac {\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(d \tan (e+f x))^{3/2}}{(b \sec (e+f x))^{4/3}} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{4/3}} \,d x \]
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