Integrand size = 12, antiderivative size = 103 \[ \int (a+b \cos (c+d x))^n \, dx=\frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right ) (a+b \cos (c+d x))^n \left (\frac {a+b \cos (c+d x)}{a+b}\right )^{-n} \sin (c+d x)}{d \sqrt {1+\cos (c+d x)}} \]
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Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2744, 144, 143} \[ \int (a+b \cos (c+d x))^n \, dx=\frac {\sqrt {2} \sin (c+d x) (a+b \cos (c+d x))^n \left (\frac {a+b \cos (c+d x)}{a+b}\right )^{-n} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right )}{d \sqrt {\cos (c+d x)+1}} \]
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Rule 143
Rule 144
Rule 2744
Rubi steps \begin{align*} \text {integral}& = -\frac {\sin (c+d x) \text {Subst}\left (\int \frac {(a+b x)^n}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{d \sqrt {1-\cos (c+d x)} \sqrt {1+\cos (c+d x)}} \\ & = -\frac {\left ((a+b \cos (c+d x))^n \left (-\frac {a+b \cos (c+d x)}{-a-b}\right )^{-n} \sin (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^n}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{d \sqrt {1-\cos (c+d x)} \sqrt {1+\cos (c+d x)}} \\ & = \frac {\sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right ) (a+b \cos (c+d x))^n \left (\frac {a+b \cos (c+d x)}{a+b}\right )^{-n} \sin (c+d x)}{d \sqrt {1+\cos (c+d x)}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.17 \[ \int (a+b \cos (c+d x))^n \, dx=-\frac {\operatorname {AppellF1}\left (1+n,\frac {1}{2},\frac {1}{2},2+n,\frac {a+b \cos (c+d x)}{a-b},\frac {a+b \cos (c+d x)}{a+b}\right ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {\frac {b (1+\cos (c+d x))}{-a+b}} (a+b \cos (c+d x))^{1+n} \csc (c+d x)}{b d (1+n)} \]
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\[\int \left (a +\cos \left (d x +c \right ) b \right )^{n}d x\]
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\[ \int (a+b \cos (c+d x))^n \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{n} \,d x } \]
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\[ \int (a+b \cos (c+d x))^n \, dx=\int \left (a + b \cos {\left (c + d x \right )}\right )^{n}\, dx \]
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\[ \int (a+b \cos (c+d x))^n \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{n} \,d x } \]
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\[ \int (a+b \cos (c+d x))^n \, dx=\int { {\left (b \cos \left (d x + c\right ) + a\right )}^{n} \,d x } \]
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Timed out. \[ \int (a+b \cos (c+d x))^n \, dx=\int {\left (a+b\,\cos \left (c+d\,x\right )\right )}^n \,d x \]
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