Integrand size = 19, antiderivative size = 69 \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=-\frac {(d \cos (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sin (a+b x)}{b d (1+n) \sqrt {\sin ^2(a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2656} \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=-\frac {\sin (a+b x) (d \cos (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(a+b x)\right )}{b d (n+1) \sqrt {\sin ^2(a+b x)}} \]
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Rule 2656
Rubi steps \begin{align*} \text {integral}& = -\frac {(d \cos (a+b x))^{1+n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sin (a+b x)}{b d (1+n) \sqrt {\sin ^2(a+b x)}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.99 \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=-\frac {(d \cos (a+b x))^n \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2(a+b x)\right ) \sin (2 (a+b x))}{2 b (1+n) \sqrt {\sin ^2(a+b x)}} \]
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\[\int \left (d \cos \left (b x +a \right )\right )^{n} \left (\sin ^{4}\left (b x +a \right )\right )d x\]
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\[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{n} \sin \left (b x + a\right )^{4} \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\text {Timed out} \]
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\[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{n} \sin \left (b x + a\right )^{4} \,d x } \]
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\[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\int { \left (d \cos \left (b x + a\right )\right )^{n} \sin \left (b x + a\right )^{4} \,d x } \]
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Timed out. \[ \int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^4\,{\left (d\,\cos \left (a+b\,x\right )\right )}^n \,d x \]
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