Integrand size = 29, antiderivative size = 200 \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n) \sqrt {\cos ^2(c+d x)}}+\frac {2 a^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n) \sqrt {\cos ^2(c+d x)}}+\frac {a^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(c+d x)\right ) \sin ^{3+n}(c+d x)}{d (3+n) \sqrt {\cos ^2(c+d x)}} \]
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Time = 0.18 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2952, 2657} \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \cos (c+d x) \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(c+d x)\right )}{d (n+1) \sqrt {\cos ^2(c+d x)}}+\frac {2 a^2 \cos (c+d x) \sin ^{n+2}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(c+d x)\right )}{d (n+2) \sqrt {\cos ^2(c+d x)}}+\frac {a^2 \cos (c+d x) \sin ^{n+3}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n+3}{2},\frac {n+5}{2},\sin ^2(c+d x)\right )}{d (n+3) \sqrt {\cos ^2(c+d x)}} \]
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Rule 2657
Rule 2952
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cos ^6(c+d x) \sin ^n(c+d x)+2 a^2 \cos ^6(c+d x) \sin ^{1+n}(c+d x)+a^2 \cos ^6(c+d x) \sin ^{2+n}(c+d x)\right ) \, dx \\ & = a^2 \int \cos ^6(c+d x) \sin ^n(c+d x) \, dx+a^2 \int \cos ^6(c+d x) \sin ^{2+n}(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^6(c+d x) \sin ^{1+n}(c+d x) \, dx \\ & = \frac {a^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n) \sqrt {\cos ^2(c+d x)}}+\frac {2 a^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n) \sqrt {\cos ^2(c+d x)}}+\frac {a^2 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(c+d x)\right ) \sin ^{3+n}(c+d x)}{d (3+n) \sqrt {\cos ^2(c+d x)}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.82 \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \sqrt {\cos ^2(c+d x)} \sec (c+d x) \sin ^{1+n}(c+d x) \left (\left (6+5 n+n^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right )+(1+n) \sin (c+d x) \left (2 (3+n) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right )+(2+n) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(c+d x)\right ) \sin (c+d x)\right )\right )}{d (1+n) (2+n) (3+n)} \]
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\[\int \left (\cos ^{6}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{2}d x\]
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\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \]
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Timed out. \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \]
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\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \]
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\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{2} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \]
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Timed out. \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\sin \left (c+d\,x\right )}^n\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
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