Integrand size = 23, antiderivative size = 59 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan (e+f x) \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b (c \sec (e+f x))^n}{a}\right ) \left (a+b (c \sec (e+f x))^n\right )^{1+p}}{a f n (1+p)} \]
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Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4224, 374, 12, 272, 67} \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan (e+f x) \, dx=-\frac {\left (a+b (c \sec (e+f x))^n\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b (c \sec (e+f x))^n}{a}+1\right )}{a f n (p+1)} \]
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Rule 12
Rule 67
Rule 272
Rule 374
Rule 4224
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b (c x)^n\right )^p}{x} \, dx,x,\sec (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {c \left (a+b x^n\right )^p}{x} \, dx,x,c \sec (e+f x)\right )}{c f} \\ & = \frac {\text {Subst}\left (\int \frac {\left (a+b x^n\right )^p}{x} \, dx,x,c \sec (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,(c \sec (e+f x))^n\right )}{f n} \\ & = -\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b (c \sec (e+f x))^n}{a}\right ) \left (a+b (c \sec (e+f x))^n\right )^{1+p}}{a f n (1+p)} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan (e+f x) \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b (c \sec (e+f x))^n}{a}\right ) \left (a+b (c \sec (e+f x))^n\right )^{1+p}}{a f n (1+p)} \]
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\[\int \left (a +b \left (c \sec \left (f x +e \right )\right )^{n}\right )^{p} \tan \left (f x +e \right )d x\]
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\[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan (e+f x) \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right ) \,d x } \]
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\[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan (e+f x) \, dx=\int \left (a + b \left (c \sec {\left (e + f x \right )}\right )^{n}\right )^{p} \tan {\left (e + f x \right )}\, dx \]
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\[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan (e+f x) \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right ) \,d x } \]
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\[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan (e+f x) \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right ) \,d x } \]
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Timed out. \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan (e+f x) \, dx=\int \mathrm {tan}\left (e+f\,x\right )\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]
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