Integrand size = 23, antiderivative size = 78 \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx=-\frac {3 b}{5 f (d \sec (e+f x))^{5/3}}-\frac {3 a d \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(e+f x)\right ) \sin (e+f x)}{8 f (d \sec (e+f x))^{8/3} \sqrt {\sin ^2(e+f x)}} \]
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Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3567, 3857, 2722} \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx=-\frac {3 a d \sin (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(e+f x)\right )}{8 f \sqrt {\sin ^2(e+f x)} (d \sec (e+f x))^{8/3}}-\frac {3 b}{5 f (d \sec (e+f x))^{5/3}} \]
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Rule 2722
Rule 3567
Rule 3857
Rubi steps \begin{align*} \text {integral}& = -\frac {3 b}{5 f (d \sec (e+f x))^{5/3}}+a \int \frac {1}{(d \sec (e+f x))^{5/3}} \, dx \\ & = -\frac {3 b}{5 f (d \sec (e+f x))^{5/3}}+\left (a \sqrt [3]{\frac {\cos (e+f x)}{d}} \sqrt [3]{d \sec (e+f x)}\right ) \int \left (\frac {\cos (e+f x)}{d}\right )^{5/3} \, dx \\ & = -\frac {3 b}{5 f (d \sec (e+f x))^{5/3}}-\frac {3 a \cos ^3(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\cos ^2(e+f x)\right ) \sqrt [3]{d \sec (e+f x)} \sin (e+f x)}{8 d^2 f \sqrt {\sin ^2(e+f x)}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx=-\frac {3 \left (b+a \cot (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},\sec ^2(e+f x)\right ) \sqrt {-\tan ^2(e+f x)}\right )}{5 f (d \sec (e+f x))^{5/3}} \]
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\[\int \frac {a +b \tan \left (f x +e \right )}{\left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}}}d x\]
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\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}} \,d x } \]
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\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx=\int \frac {a + b \tan {\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}}}\, dx \]
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\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}} \,d x } \]
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\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{5/3}} \, dx=\int \frac {a+b\,\mathrm {tan}\left (e+f\,x\right )}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}} \,d x \]
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