Integrand size = 21, antiderivative size = 83 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {2^{\frac {5}{2}+m} a^2 \cos ^5(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {3}{2}-m,\frac {7}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-m} (a+a \sin (c+d x))^{-2+m}}{5 d} \]
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Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2768, 72, 71} \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {a^2 2^{m+\frac {5}{2}} \cos ^5(c+d x) (\sin (c+d x)+1)^{-m-\frac {1}{2}} (a \sin (c+d x)+a)^{m-2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-m-\frac {3}{2},\frac {7}{2},\frac {1}{2} (1-\sin (c+d x))\right )}{5 d} \]
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Rule 71
Rule 72
Rule 2768
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \cos ^5(c+d x)\right ) \text {Subst}\left (\int (a-a x)^{3/2} (a+a x)^{\frac {3}{2}+m} \, dx,x,\sin (c+d x)\right )}{d (a-a \sin (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \\ & = \frac {\left (2^{\frac {3}{2}+m} a^3 \cos ^5(c+d x) (a+a \sin (c+d x))^{-2+m} \left (\frac {a+a \sin (c+d x)}{a}\right )^{-\frac {1}{2}-m}\right ) \text {Subst}\left (\int \left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {3}{2}+m} (a-a x)^{3/2} \, dx,x,\sin (c+d x)\right )}{d (a-a \sin (c+d x))^{5/2}} \\ & = -\frac {2^{\frac {5}{2}+m} a^2 \cos ^5(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {3}{2}-m,\frac {7}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {1}{2}-m} (a+a \sin (c+d x))^{-2+m}}{5 d} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {2^{\frac {5}{2}+m} \cos ^5(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-\frac {3}{2}-m,\frac {7}{2},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{-\frac {5}{2}-m} (a (1+\sin (c+d x)))^m}{5 d} \]
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\[\int \left (\cos ^{4}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}d x\]
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\[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{4} \,d x } \]
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\[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \cos ^{4}{\left (c + d x \right )}\, dx \]
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\[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{4} \,d x } \]
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\[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right )^{4} \,d x } \]
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Timed out. \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx=\int {\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m \,d x \]
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