Integrand size = 19, antiderivative size = 131 \[ \int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx=-\frac {a \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) \sqrt {\sin ^2(c+d x)}}-\frac {a \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+m) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2827, 2722} \[ \int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx=-\frac {a \sin (c+d x) \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+1}{2},\frac {m+3}{2},\cos ^2(c+d x)\right )}{d (m+1) \sqrt {\sin ^2(c+d x)}}-\frac {a \sin (c+d x) \cos ^{m+2}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},\cos ^2(c+d x)\right )}{d (m+2) \sqrt {\sin ^2(c+d x)}} \]
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Rule 2722
Rule 2827
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^m(c+d x) \, dx+a \int \cos ^{1+m}(c+d x) \, dx \\ & = -\frac {a \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) \sqrt {\sin ^2(c+d x)}}-\frac {a \cos ^{2+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+m) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx=-\frac {a \cos ^{1+m}(c+d x) \csc (c+d x) \left ((2+m) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},\cos ^2(c+d x)\right )+(1+m) \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (1+m) (2+m)} \]
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\[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (a +\cos \left (d x +c \right ) a \right )d x\]
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\[ \int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{m} \,d x } \]
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\[ \int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx=a \left (\int \cos {\left (c + d x \right )} \cos ^{m}{\left (c + d x \right )}\, dx + \int \cos ^{m}{\left (c + d x \right )}\, dx\right ) \]
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\[ \int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{m} \,d x } \]
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\[ \int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx=\int { {\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{m} \,d x } \]
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Timed out. \[ \int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^m\,\left (a+a\,\cos \left (c+d\,x\right )\right ) \,d x \]
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