Integrand size = 25, antiderivative size = 143 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^3(e+f x) \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b (c \sec (e+f x))^n}{a}\right ) \left (a+b (c \sec (e+f x))^n\right )^{1+p}}{a f n (1+p)}+\frac {\operatorname {Hypergeometric2F1}\left (\frac {2}{n},-p,\frac {2+n}{n},-\frac {b (c \sec (e+f x))^n}{a}\right ) \sec ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \left (1+\frac {b (c \sec (e+f x))^n}{a}\right )^{-p}}{2 f} \]
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Time = 0.38 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4224, 6874, 374, 12, 272, 67, 372, 371} \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^3(e+f x) \, dx=\frac {\sec ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \left (\frac {b (c \sec (e+f x))^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{n},-p,\frac {n+2}{n},-\frac {b (c \sec (e+f x))^n}{a}\right )}{2 f}+\frac {\left (a+b (c \sec (e+f x))^n\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b (c \sec (e+f x))^n}{a}+1\right )}{a f n (p+1)} \]
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Rule 12
Rule 67
Rule 272
Rule 371
Rule 372
Rule 374
Rule 4224
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (a+b (c x)^n\right )^p}{x} \, dx,x,\sec (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {\left (a+b (c x)^n\right )^p}{x}+x \left (a+b (c x)^n\right )^p\right ) \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\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 x \left (a+b (c x)^n\right )^p \, 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 {x \left (a+b x^n\right )^p}{c} \, 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 x \left (a+b x^n\right )^p \, dx,x,c \sec (e+f x)\right )}{c^2 f} \\ & = -\frac {\text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,(c \sec (e+f x))^n\right )}{f n}+\frac {\left (\left (a+b (c \sec (e+f x))^n\right )^p \left (1+\frac {b (c \sec (e+f x))^n}{a}\right )^{-p}\right ) \text {Subst}\left (\int x \left (1+\frac {b x^n}{a}\right )^p \, dx,x,c \sec (e+f x)\right )}{c^2 f} \\ & = \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)}+\frac {\operatorname {Hypergeometric2F1}\left (\frac {2}{n},-p,\frac {2+n}{n},-\frac {b (c \sec (e+f x))^n}{a}\right ) \sec ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \left (1+\frac {b (c \sec (e+f x))^n}{a}\right )^{-p}}{2 f} \\ \end{align*}
Time = 6.43 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.13 \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^3(e+f x) \, dx=\frac {\left (a+b (c \sec (e+f x))^n\right )^p \left (\frac {2 \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \left (c \sqrt {\sec ^2(e+f x)}\right )^n}{a}\right ) \left (a+b \left (c \sqrt {\sec ^2(e+f x)}\right )^n\right )}{a n (1+p)}+\operatorname {Hypergeometric2F1}\left (\frac {2}{n},-p,\frac {2+n}{n},-\frac {b \left (c \sqrt {\sec ^2(e+f x)}\right )^n}{a}\right ) \sec ^2(e+f x) \left (1+\frac {b \left (c \sqrt {\sec ^2(e+f x)}\right )^n}{a}\right )^{-p}\right )}{2 f} \]
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\[\int \left (a +b \left (c \sec \left (f x +e \right )\right )^{n}\right )^{p} \tan \left (f x +e \right )^{3}d x\]
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\[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^3(e+f x) \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{3} \,d x } \]
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\[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^3(e+f x) \, dx=\int \left (a + b \left (c \sec {\left (e + f x \right )}\right )^{n}\right )^{p} \tan ^{3}{\left (e + f x \right )}\, dx \]
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\[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^3(e+f x) \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{3} \,d x } \]
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\[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^3(e+f x) \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \tan \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \left (a+b (c \sec (e+f x))^n\right )^p \tan ^3(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^3\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]
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