Integrand size = 29, antiderivative size = 29 \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx=\text {Int}\left (\cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p,x\right ) \]
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Not integrable
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx=\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx \\ \end{align*}
Not integrable
Time = 5.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx=\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx \]
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Not integrable
Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00
\[\int \left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{p}d x\]
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Not integrable
Time = 0.60 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{p} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx=\text {Timed out} \]
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Not integrable
Time = 6.38 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{p} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4} \,d x } \]
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Not integrable
Time = 13.96 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{p} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4} \,d x } \]
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Not integrable
Time = 18.93 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^n\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^p \,d x \]
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