Integrand size = 21, antiderivative size = 123 \[ \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx=\frac {(12-n) \cos ^4(c+d x) (a+a \sec (c+d x))^{3+n}}{20 a^3 d}-\frac {\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}+\frac {\left (32-13 n+n^2\right ) \operatorname {Hypergeometric2F1}(4,3+n,4+n,1+\sec (c+d x)) (a+a \sec (c+d x))^{3+n}}{20 a^3 d (3+n)} \]
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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 = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3958, 91, 79, 67} \[ \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx=\frac {\left (n^2-13 n+32\right ) (a \sec (c+d x)+a)^{n+3} \operatorname {Hypergeometric2F1}(4,n+3,n+4,\sec (c+d x)+1)}{20 a^3 d (n+3)}-\frac {\cos ^5(c+d x) (a \sec (c+d x)+a)^{n+3}}{5 a^3 d}+\frac {(12-n) \cos ^4(c+d x) (a \sec (c+d x)+a)^{n+3}}{20 a^3 d} \]
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Rule 67
Rule 79
Rule 91
Rule 3958
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(-a-a x)^2 (a-a x)^{2+n}}{x^6} \, dx,x,-\sec (c+d x)\right )}{a^4 d} \\ & = -\frac {\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}-\frac {\text {Subst}\left (\int \frac {(a-a x)^{2+n} \left (a^3 (12-n)+5 a^3 x\right )}{x^5} \, dx,x,-\sec (c+d x)\right )}{5 a^5 d} \\ & = \frac {(12-n) \cos ^4(c+d x) (a+a \sec (c+d x))^{3+n}}{20 a^3 d}-\frac {\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}-\frac {\left (32-13 n+n^2\right ) \text {Subst}\left (\int \frac {(a-a x)^{2+n}}{x^4} \, dx,x,-\sec (c+d x)\right )}{20 a^2 d} \\ & = \frac {(12-n) \cos ^4(c+d x) (a+a \sec (c+d x))^{3+n}}{20 a^3 d}-\frac {\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}+\frac {\left (32-13 n+n^2\right ) \operatorname {Hypergeometric2F1}(4,3+n,4+n,1+\sec (c+d x)) (a+a \sec (c+d x))^{3+n}}{20 a^3 d (3+n)} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68 \[ \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx=-\frac {\left ((3+n) \cos ^4(c+d x) (-12+n+4 \cos (c+d x))-\left (32-13 n+n^2\right ) \operatorname {Hypergeometric2F1}(4,3+n,4+n,1+\sec (c+d x))\right ) (1+\sec (c+d x))^3 (a (1+\sec (c+d x)))^n}{20 d (3+n)} \]
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\[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \sin \left (d x +c \right )^{5}d x\]
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\[ \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{5} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{5} \,d x } \]
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\[ \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{5} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx=\int {\sin \left (c+d\,x\right )}^5\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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