Integrand size = 23, antiderivative size = 105 \[ \int \frac {(a+a \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {\operatorname {AppellF1}\left (1-n,\frac {5}{4},\frac {5}{4}-n,2-n,\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{5/4} \cos (c+d x) (1+\cos (c+d x))^{\frac {5}{4}-n} (a+a \sec (c+d x))^n}{d (1-n) \sin ^{\frac {5}{2}}(c+d x)} \]
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Time = 0.31 (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} \[ \int \frac {(a+a \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {(1-\cos (c+d x))^{5/4} \cos (c+d x) (\cos (c+d x)+1)^{\frac {5}{4}-n} (a \sec (c+d x)+a)^n \operatorname {AppellF1}\left (1-n,\frac {5}{4},\frac {5}{4}-n,2-n,\cos (c+d x),-\cos (c+d x)\right )}{d (1-n) \sin ^{\frac {5}{2}}(c+d x)} \]
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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}{\sin ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {\left ((-\cos (c+d x))^n (-a-a \cos (c+d x))^{\frac {5}{4}-n} (-a+a \cos (c+d x))^{5/4} (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \frac {(-x)^{-n} (-a-a x)^{-\frac {5}{4}+n}}{(-a+a x)^{5/4}} \, dx,x,\cos (c+d x)\right )}{d \sin ^{\frac {5}{2}}(c+d x)} \\ & = -\left (-\frac {\left ((-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {1}{4}-n} (-a-a \cos (c+d x)) (-a+a \cos (c+d x))^{5/4} (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \frac {(-x)^{-n} (1+x)^{-\frac {5}{4}+n}}{(-a+a x)^{5/4}} \, dx,x,\cos (c+d x)\right )}{a d \sin ^{\frac {5}{2}}(c+d x)}\right ) \\ & = -\frac {\left (\sqrt [4]{1-\cos (c+d x)} (-\cos (c+d x))^n (1+\cos (c+d x))^{\frac {1}{4}-n} (-a-a \cos (c+d x)) (-a+a \cos (c+d x)) (a+a \sec (c+d x))^n\right ) \text {Subst}\left (\int \frac {(-x)^{-n} (1+x)^{-\frac {5}{4}+n}}{(1-x)^{5/4}} \, dx,x,\cos (c+d x)\right )}{a^2 d \sin ^{\frac {5}{2}}(c+d x)} \\ & = -\frac {\operatorname {AppellF1}\left (1-n,\frac {5}{4},\frac {5}{4}-n,2-n,\cos (c+d x),-\cos (c+d x)\right ) (1-\cos (c+d x))^{5/4} \cos (c+d x) (1+\cos (c+d x))^{\frac {5}{4}-n} (a+a \sec (c+d x))^n}{d (1-n) \sin ^{\frac {5}{2}}(c+d x)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(212\) vs. \(2(105)=210\).
Time = 4.38 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.02 \[ \int \frac {(a+a \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {6 \operatorname {AppellF1}\left (-\frac {1}{4},n,-\frac {1}{2},\frac {3}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x)) (a (1+\sec (c+d x)))^n}{d \left (-2 \left (\operatorname {AppellF1}\left (\frac {3}{4},n,\frac {1}{2},\frac {7}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+2 n \operatorname {AppellF1}\left (\frac {3}{4},1+n,-\frac {1}{2},\frac {7}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) (-1+\cos (c+d x))+3 \operatorname {AppellF1}\left (-\frac {1}{4},n,-\frac {1}{2},\frac {3}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x))\right ) \sqrt {\sin (c+d x)}} \]
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\[\int \frac {\left (a +a \sec \left (d x +c \right )\right )^{n}}{\sin \left (d x +c \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {(a+a \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sin \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(a+a \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n}}{\sin ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(a+a \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sin \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(a+a \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{n}}{\sin \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^n}{\sin ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^{3/2}} \,d x \]
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