Integrand size = 23, antiderivative size = 105 \[ \int \frac {(a+a \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx=-\frac {\operatorname {AppellF1}\left (1-n,\frac {3}{4},\frac {3}{4}-n,2-n,\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{3/4} \cos (c+d x) (1+\cos (c+d x))^{\frac {3}{4}-n} (a+a \sec (c+d x))^n}{d (1-n) \sin ^{\frac {3}{2}}(c+d x)} \]
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Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3961, 2965, 140, 138} \[ \int \frac {(a+a \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx=-\frac {(1-\cos (c+d x))^{3/4} \cos (c+d x) (\cos (c+d x)+1)^{\frac {3}{4}-n} (a \sec (c+d x)+a)^n \operatorname {AppellF1}\left (1-n,\frac {3}{4},\frac {3}{4}-n,2-n,\cos (c+d x),-\cos (c+d x)\right )}{d (1-n) \sin ^{\frac {3}{2}}(c+d x)} \]
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Rule 138
Rule 140
Rule 2965
Rule 3961
Rubi steps \begin{align*} \text {integral}& = \left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{-n} (a+a \sec (c+d x))^n\right ) \int \frac {(-\cos (c+d x))^{-n} (-a-a \cos (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx \\ & = -\frac {\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{\frac {3}{4}-n} (-a+a \cos (c+d x))^{3/4} (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \frac {(-x)^{-n} (-a-a x)^{-\frac {3}{4}+n}}{(-a+a x)^{3/4}} \, dx,x,\cos (c+d x)\right )}{d \sin ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {\left ((-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {3}{4}-n} (-a+a \cos (c+d x))^{3/4} (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \frac {(-x)^{-n} (1+x)^{-\frac {3}{4}+n}}{(-a+a x)^{3/4}} \, dx,x,\cos (c+d x)\right )}{d \sin ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {\left ((1-\cos (c+d x))^{3/4} (-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {3}{4}-n} (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \frac {(-x)^{-n} (1+x)^{-\frac {3}{4}+n}}{(1-x)^{3/4}} \, dx,x,\cos (c+d x)\right )}{d \sin ^{\frac {3}{2}}(c+d x)} \\ & = -\frac {\operatorname {AppellF1}\left (1-n,\frac {3}{4},\frac {3}{4}-n,2-n,\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{3/4} \cos (c+d x) (1+\cos (c+d x))^{\frac {3}{4}-n} (a+a \sec (c+d x))^n}{d (1-n) \sin ^{\frac {3}{2}}(c+d x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(212\) vs. \(2(105)=210\).
Time = 3.53 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.02 \[ \int \frac {(a+a \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx=\frac {10 \operatorname {AppellF1}\left (\frac {1}{4},n,\frac {1}{2},\frac {5}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x)) (a (1+\sec (c+d x)))^n \sqrt {\sin (c+d x)}}{d \left (2 \left (\operatorname {AppellF1}\left (\frac {5}{4},n,\frac {3}{2},\frac {9}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 n \operatorname {AppellF1}\left (\frac {5}{4},1+n,\frac {1}{2},\frac {9}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) (-1+\cos (c+d x))+5 \operatorname {AppellF1}\left (\frac {1}{4},n,\frac {1}{2},\frac {5}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x))\right )} \]
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\[\int \frac {\left (a +a \sec \left (d x +c \right )\right )^{n}}{\sqrt {\sin \left (d x +c \right )}}d x\]
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\[ \int \frac {(a+a \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\sin \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+a \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx=\int \frac {\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n}}{\sqrt {\sin {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {(a+a \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\sin \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {(a+a \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\sin \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^n}{\sqrt {\sin (c+d x)}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{\sqrt {\sin \left (c+d\,x\right )}} \,d x \]
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