Integrand size = 14, antiderivative size = 62 \[ \int \frac {1}{\sqrt {b \tan ^n(e+f x)}} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (1,\frac {2-n}{4},\frac {6-n}{4},-\tan ^2(e+f x)\right ) \tan (e+f x)}{f (2-n) \sqrt {b \tan ^n(e+f x)}} \]
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Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3740, 3557, 371} \[ \int \frac {1}{\sqrt {b \tan ^n(e+f x)}} \, dx=\frac {2 \tan (e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {2-n}{4},\frac {6-n}{4},-\tan ^2(e+f x)\right )}{f (2-n) \sqrt {b \tan ^n(e+f x)}} \]
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Rule 371
Rule 3557
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \frac {\tan ^{\frac {n}{2}}(e+f x) \int \tan ^{-\frac {n}{2}}(e+f x) \, dx}{\sqrt {b \tan ^n(e+f x)}} \\ & = \frac {\tan ^{\frac {n}{2}}(e+f x) \text {Subst}\left (\int \frac {x^{-n/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f \sqrt {b \tan ^n(e+f x)}} \\ & = \frac {2 \operatorname {Hypergeometric2F1}\left (1,\frac {2-n}{4},\frac {6-n}{4},-\tan ^2(e+f x)\right ) \tan (e+f x)}{f (2-n) \sqrt {b \tan ^n(e+f x)}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {b \tan ^n(e+f x)}} \, dx=-\frac {2 \operatorname {Hypergeometric2F1}\left (1,\frac {2-n}{4},\frac {6-n}{4},-\tan ^2(e+f x)\right ) \tan (e+f x)}{f (-2+n) \sqrt {b \tan ^n(e+f x)}} \]
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\[\int \frac {1}{\sqrt {b \tan \left (f x +e \right )^{n}}}d x\]
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Exception generated. \[ \int \frac {1}{\sqrt {b \tan ^n(e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\sqrt {b \tan ^n(e+f x)}} \, dx=\int \frac {1}{\sqrt {b \tan ^{n}{\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {b \tan ^n(e+f x)}} \, dx=\int { \frac {1}{\sqrt {b \tan \left (f x + e\right )^{n}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {b \tan ^n(e+f x)}} \, dx=\int { \frac {1}{\sqrt {b \tan \left (f x + e\right )^{n}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {b \tan ^n(e+f x)}} \, dx=\int \frac {1}{\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^n}} \,d x \]
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