Integrand size = 23, antiderivative size = 245 \[ \int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx=\frac {3 a b^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {a^3 \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {3 a^2 b \cos ^2(e+f x)^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {2+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec (e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {b^3 \cos ^2(e+f x)^{\frac {4+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {4+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec ^3(e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]
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Time = 0.33 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3971, 3557, 371, 2697, 2687, 32} \[ \int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx=\frac {a^3 (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(e+f x)\right )}{d f (n+1)}+\frac {3 a^2 b \sec (e+f x) \cos ^2(e+f x)^{\frac {n+2}{2}} (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {n+2}{2},\frac {n+3}{2},\sin ^2(e+f x)\right )}{d f (n+1)}+\frac {3 a b^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac {b^3 \sec ^3(e+f x) \cos ^2(e+f x)^{\frac {n+4}{2}} (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {n+4}{2},\frac {n+3}{2},\sin ^2(e+f x)\right )}{d f (n+1)} \]
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Rule 32
Rule 371
Rule 2687
Rule 2697
Rule 3557
Rule 3971
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 (d \tan (e+f x))^n+3 a^2 b \sec (e+f x) (d \tan (e+f x))^n+3 a b^2 \sec ^2(e+f x) (d \tan (e+f x))^n+b^3 \sec ^3(e+f x) (d \tan (e+f x))^n\right ) \, dx \\ & = a^3 \int (d \tan (e+f x))^n \, dx+\left (3 a^2 b\right ) \int \sec (e+f x) (d \tan (e+f x))^n \, dx+\left (3 a b^2\right ) \int \sec ^2(e+f x) (d \tan (e+f x))^n \, dx+b^3 \int \sec ^3(e+f x) (d \tan (e+f x))^n \, dx \\ & = \frac {3 a^2 b \cos ^2(e+f x)^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {2+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec (e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {b^3 \cos ^2(e+f x)^{\frac {4+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {4+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec ^3(e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int (d x)^n \, dx,x,\tan (e+f x)\right )}{f}+\frac {\left (a^3 d\right ) \text {Subst}\left (\int \frac {x^n}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{f} \\ & = \frac {3 a b^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {a^3 \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {3 a^2 b \cos ^2(e+f x)^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {2+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec (e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {b^3 \cos ^2(e+f x)^{\frac {4+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {4+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec ^3(e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)} \\ \end{align*}
Time = 3.16 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.97 \[ \int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx=\frac {d (d \tan (e+f x))^{-1+n} \left (-\tan ^2(e+f x)\right )^{-n/2} \left (9 a^2 b (1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3}{2},\sec ^2(e+f x)\right ) \sec (e+f x) \sqrt {-\tan ^2(e+f x)}+b^3 (1+n) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1-n}{2},\frac {5}{2},\sec ^2(e+f x)\right ) \sec ^3(e+f x) \sqrt {-\tan ^2(e+f x)}-3 a^3 \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{\frac {2+n}{2}}+9 a b^2 \left (\sqrt {-\tan ^2(e+f x)}-\left (-\tan ^2(e+f x)\right )^{\frac {2+n}{2}}\right )\right )}{3 f (1+n)} \]
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\[\int \left (a +b \sec \left (f x +e \right )\right )^{3} \left (d \tan \left (f x +e \right )\right )^{n}d x\]
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\[ \int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{3} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx=\int \left (d \tan {\left (e + f x \right )}\right )^{n} \left (a + b \sec {\left (e + f x \right )}\right )^{3}\, dx \]
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\[ \int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{3} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \]
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\[ \int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{3} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \]
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Timed out. \[ \int (a+b \sec (e+f x))^3 (d \tan (e+f x))^n \, dx=\int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^3 \,d x \]
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