Integrand size = 25, antiderivative size = 64 \[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\frac {2 \cos ^2(e+f x)^{7/12} \operatorname {Hypergeometric2F1}\left (\frac {7}{12},\frac {3}{4},\frac {7}{4},\sin ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{3 d f \sqrt [3]{b \sec (e+f x)}} \]
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Time = 0.05 (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} \[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\frac {2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{12},\frac {3}{4},\frac {7}{4},\sin ^2(e+f x)\right )}{3 d f \sqrt [3]{b \sec (e+f x)}} \]
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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 {3}{4},\frac {7}{4},\sin ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{3 d f \sqrt [3]{b \sec (e+f x)}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=-\frac {3 d \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{4},\frac {5}{6},\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}}{f \sqrt [3]{b \sec (e+f x)} \sqrt {d \tan (e+f x)}} \]
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\[\int \frac {\sqrt {d \tan \left (f x +e \right )}}{\left (b \sec \left (f x +e \right )\right )^{\frac {1}{3}}}d x\]
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\[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {d \tan \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int \frac {\sqrt {d \tan {\left (e + f x \right )}}}{\sqrt [3]{b \sec {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {d \tan \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {d \tan \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int \frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \]
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