Integrand size = 19, antiderivative size = 95 \[ \int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx=-\frac {13^{n/2} \cos ^{1+n}\left (c+d x-\arctan \left (\frac {3}{2}\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2\left (c+d x-\arctan \left (\frac {3}{2}\right )\right )\right ) \sin \left (c+d x-\arctan \left (\frac {3}{2}\right )\right )}{d (1+n) \sqrt {\sin ^2\left (c+d x-\arctan \left (\frac {3}{2}\right )\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3156, 2722} \[ \int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx=-\frac {13^{n/2} \sin \left (-\arctan \left (\frac {3}{2}\right )+c+d x\right ) \cos ^{n+1}\left (-\arctan \left (\frac {3}{2}\right )+c+d x\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2\left (c+d x-\arctan \left (\frac {3}{2}\right )\right )\right )}{d (n+1) \sqrt {\sin ^2\left (-\arctan \left (\frac {3}{2}\right )+c+d x\right )}} \]
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Rule 2722
Rule 3156
Rubi steps \begin{align*} \text {integral}& = 13^{n/2} \int \cos ^n\left (c+d x-\arctan \left (\frac {3}{2}\right )\right ) \, dx \\ & = -\frac {13^{n/2} \cos ^{1+n}\left (c+d x-\arctan \left (\frac {3}{2}\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},\cos ^2\left (c+d x-\arctan \left (\frac {3}{2}\right )\right )\right ) \sin \left (c+d x-\arctan \left (\frac {3}{2}\right )\right )}{d (1+n) \sqrt {\sin ^2\left (c+d x-\arctan \left (\frac {3}{2}\right )\right )}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3}{2},\cos ^2\left (c+d x+\arctan \left (\frac {2}{3}\right )\right )\right ) (2 \cos (c+d x)+3 \sin (c+d x))^n \sin ^2\left (c+d x+\arctan \left (\frac {2}{3}\right )\right )^{-\frac {1}{2}-\frac {n}{2}} \sin \left (2 \left (c+d x+\arctan \left (\frac {2}{3}\right )\right )\right )}{2 d} \]
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\[\int \left (2 \cos \left (d x +c \right )+3 \sin \left (d x +c \right )\right )^{n}d x\]
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\[ \int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx=\int { {\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{n} \,d x } \]
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\[ \int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx=\int \left (3 \sin {\left (c + d x \right )} + 2 \cos {\left (c + d x \right )}\right )^{n}\, dx \]
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\[ \int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx=\int { {\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{n} \,d x } \]
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\[ \int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx=\int { {\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{n} \,d x } \]
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Timed out. \[ \int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx=\int {\left (2\,\cos \left (c+d\,x\right )+3\,\sin \left (c+d\,x\right )\right )}^n \,d x \]
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