Integrand size = 21, antiderivative size = 123 \[ \int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx=\frac {7 (a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}+\frac {\operatorname {Hypergeometric2F1}(1,4+n,5+n,1+\sec (c+d x)) (a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}-\frac {5 (a+a \sec (c+d x))^{5+n}}{a^5 d (5+n)}+\frac {(a+a \sec (c+d x))^{6+n}}{a^6 d (6+n)} \]
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Time = 0.14 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3965, 90, 67} \[ \int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx=\frac {(a \sec (c+d x)+a)^{n+6}}{a^6 d (n+6)}-\frac {5 (a \sec (c+d x)+a)^{n+5}}{a^5 d (n+5)}+\frac {(a \sec (c+d x)+a)^{n+4} \operatorname {Hypergeometric2F1}(1,n+4,n+5,\sec (c+d x)+1)}{a^4 d (n+4)}+\frac {7 (a \sec (c+d x)+a)^{n+4}}{a^4 d (n+4)} \]
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Rule 67
Rule 90
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-a+a x)^3 (a+a x)^{3+n}}{x} \, dx,x,\sec (c+d x)\right )}{a^6 d} \\ & = \frac {\text {Subst}\left (\int \left (7 a^3 (a+a x)^{3+n}-\frac {a^3 (a+a x)^{3+n}}{x}-5 a^2 (a+a x)^{4+n}+a (a+a x)^{5+n}\right ) \, dx,x,\sec (c+d x)\right )}{a^6 d} \\ & = \frac {7 (a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}-\frac {5 (a+a \sec (c+d x))^{5+n}}{a^5 d (5+n)}+\frac {(a+a \sec (c+d x))^{6+n}}{a^6 d (6+n)}-\frac {\text {Subst}\left (\int \frac {(a+a x)^{3+n}}{x} \, dx,x,\sec (c+d x)\right )}{a^3 d} \\ & = \frac {7 (a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}+\frac {\operatorname {Hypergeometric2F1}(1,4+n,5+n,1+\sec (c+d x)) (a+a \sec (c+d x))^{4+n}}{a^4 d (4+n)}-\frac {5 (a+a \sec (c+d x))^{5+n}}{a^5 d (5+n)}+\frac {(a+a \sec (c+d x))^{6+n}}{a^6 d (6+n)} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.71 \[ \int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx=\frac {(1+\sec (c+d x))^4 (a (1+\sec (c+d x)))^n \left (\frac {7}{4+n}+\frac {\operatorname {Hypergeometric2F1}(1,4+n,5+n,1+\sec (c+d x))}{4+n}-\frac {5 (1+\sec (c+d x))}{5+n}+\frac {(1+\sec (c+d x))^2}{6+n}\right )}{d} \]
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\[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \tan \left (d x +c \right )^{7}d x\]
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\[ \int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{7} \,d x } \]
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\[ \int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \tan ^{7}{\left (c + d x \right )}\, dx \]
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\[ \int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{7} \,d x } \]
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\[ \int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{7} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^n \tan ^7(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^7\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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