Integrand size = 19, antiderivative size = 78 \[ \int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx=\frac {\cos ^3(a+b x) \cos ^2(a+b x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+n),\frac {1+n}{2},\frac {3+n}{2},\sin ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)} \]
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Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2697} \[ \int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx=\frac {\cos ^3(a+b x) \cos ^2(a+b x)^{\frac {n-2}{2}} (d \tan (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n-2}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(a+b x)\right )}{b d (n+1)} \]
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Rule 2697
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^3(a+b x) \cos ^2(a+b x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+n),\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 = 4.85 (sec) , antiderivative size = 1313, normalized size of antiderivative = 16.83 \[ \int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx=\frac {4 (3+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 )-6 \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 )+12 \operatorname {AppellF1}\left (\frac {1+n}{2},n,3,\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 )-8 \operatorname {AppellF1}\left (\frac {1+n}{2},n,4,\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 ^3\left (\frac {1}{2} (a+b x)\right ) \cos ^3(a+b x) \sin \left (\frac {1}{2} (a+b x)\right ) (d \tan (a+b x))^n}{b (1+n) \left ((3+n) \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 ) (1+\cos (a+b x))-2 \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 )-12 \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 )+36 \operatorname {AppellF1}\left (\frac {3+n}{2},n,4,\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 )-32 \operatorname {AppellF1}\left (\frac {3+n}{2},n,5,\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 )+6 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 )-12 n \operatorname {AppellF1}\left (\frac {3+n}{2},1+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 )+8 n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,4,\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 )+18 \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 ) \cos ^2\left (\frac {1}{2} (a+b x)\right )+6 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 ) \cos ^2\left (\frac {1}{2} (a+b x)\right )+8 (3+n) \operatorname {AppellF1}\left (\frac {1+n}{2},n,4,\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 ) \cos ^2\left (\frac {1}{2} (a+b x)\right )-\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 ) \cos (a+b x)+12 \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 ) \cos (a+b x)-36 \operatorname {AppellF1}\left (\frac {3+n}{2},n,4,\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 ) \cos (a+b x)+32 \operatorname {AppellF1}\left (\frac {3+n}{2},n,5,\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 ) \cos (a+b x)+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 ) \cos (a+b x)-6 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 ) \cos (a+b x)+12 n \operatorname {AppellF1}\left (\frac {3+n}{2},1+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 ) \cos (a+b x)-8 n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,4,\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 ) \cos (a+b x)-6 (3+n) \operatorname {AppellF1}\left (\frac {1+n}{2},n,3,\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 )\right )} \]
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\[\int \left (\cos ^{3}\left (b x +a \right )\right ) \left (d \tan \left (b x +a \right )\right )^{n}d x\]
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\[ \int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx=\text {Timed out} \]
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\[ \int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{3} \,d x } \]
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\[ \int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx=\int {\cos \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n \,d x \]
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