Integrand size = 17, antiderivative size = 72 \[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\frac {\cos (a+b x) \cos ^2(a+b x)^{n/2} \operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)} \]
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Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2697} \[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\frac {\cos (a+b x) \cos ^2(a+b x)^{n/2} (d \tan (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(a+b x)\right )}{b d (n+1)} \]
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Rule 2697
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (a+b x) \cos ^2(a+b x)^{n/2} \operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 2.02 (sec) , antiderivative size = 452, normalized size of antiderivative = 6.28 \[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=-\frac {2 \left (\operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-2 \operatorname {AppellF1}\left (\frac {1+n}{2},n,2,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \cos \left (\frac {1}{2} (a+b x)\right ) \cos (a+b x) \sin \left (\frac {1}{2} (a+b x)\right ) (d \tan (a+b x))^n}{b (1+n) \left (-\operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+\frac {\left (-\left (\left (\operatorname {AppellF1}\left (\frac {3+n}{2},n,2,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-4 \operatorname {AppellF1}\left (\frac {3+n}{2},n,3,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,1,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,2,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) (-1+\cos (a+b x))\right )+(3+n) \operatorname {AppellF1}\left (\frac {1+n}{2},n,2,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) (1+\cos (a+b x))\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right )}{3+n}\right )} \]
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\[\int \cos \left (b x +a \right ) \left (d \tan \left (b x +a \right )\right )^{n}d x\]
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\[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right ) \,d x } \]
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\[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int \left (d \tan {\left (a + b x \right )}\right )^{n} \cos {\left (a + b x \right )}\, dx \]
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\[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right ) \,d x } \]
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\[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int \cos \left (a+b\,x\right )\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n \,d x \]
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