Integrand size = 23, antiderivative size = 105 \[ \int (a+a \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=-\frac {\operatorname {AppellF1}\left (1-n,-\frac {1}{4},-\frac {1}{4}-n,2-n,\cos (c+d x),-\cos (c+d x)\right ) \cos (c+d x) (1+\cos (c+d x))^{-\frac {1}{4}-n} (a+a \sec (c+d x))^n \sqrt {\sin (c+d x)}}{d (1-n) \sqrt [4]{1-\cos (c+d x)}} \]
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Time = 0.38 (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} \[ \int (a+a \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=-\frac {\sqrt {\sin (c+d x)} \cos (c+d x) (\cos (c+d x)+1)^{-n-\frac {1}{4}} (a \sec (c+d x)+a)^n \operatorname {AppellF1}\left (1-n,-\frac {1}{4},-n-\frac {1}{4},2-n,\cos (c+d x),-\cos (c+d x)\right )}{d (1-n) \sqrt [4]{1-\cos (c+d x)}} \]
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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 (-\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 {1}{4}-n} (a+a \sec (c+d x))^n \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int (-x)^{-n} (-a-a x)^{\frac {1}{4}+n} \sqrt [4]{-a+a x} \, dx,x,\cos (c+d x)\right )}{d \sqrt [4]{-a+a \cos (c+d x)}} \\ & = -\frac {\left ((-\cos (c+d x))^n (1+\cos (c+d x))^{-\frac {1}{4}-n} (a+a \sec (c+d x))^n \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int (-x)^{-n} (1+x)^{\frac {1}{4}+n} \sqrt [4]{-a+a x} \, dx,x,\cos (c+d x)\right )}{d \sqrt [4]{-a+a \cos (c+d x)}} \\ & = -\frac {\left ((-\cos (c+d x))^n (1+\cos (c+d x))^{-\frac {1}{4}-n} (a+a \sec (c+d x))^n \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \sqrt [4]{1-x} (-x)^{-n} (1+x)^{\frac {1}{4}+n} \, dx,x,\cos (c+d x)\right )}{d \sqrt [4]{1-\cos (c+d x)}} \\ & = -\frac {\operatorname {AppellF1}\left (1-n,-\frac {1}{4},-\frac {1}{4}-n,2-n,\cos (c+d x),-\cos (c+d x)\right ) \cos (c+d x) (1+\cos (c+d x))^{-\frac {1}{4}-n} (a+a \sec (c+d x))^n \sqrt {\sin (c+d x)}}{d (1-n) \sqrt [4]{1-\cos (c+d x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(382\) vs. \(2(105)=210\).
Time = 16.60 (sec) , antiderivative size = 382, normalized size of antiderivative = 3.64 \[ \int (a+a \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\frac {10 \left (\operatorname {AppellF1}\left (\frac {1}{4},n,\frac {3}{2},\frac {5}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-\operatorname {AppellF1}\left (\frac {1}{4},n,\frac {5}{2},\frac {5}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) (1+\cos (c+d x)) (a (1+\sec (c+d x)))^n \sin ^{\frac {5}{2}}(c+d x)}{d \left (2 \left (3 \operatorname {AppellF1}\left (\frac {5}{4},n,\frac {5}{2},\frac {9}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-5 \operatorname {AppellF1}\left (\frac {5}{4},n,\frac {7}{2},\frac {9}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 n \operatorname {AppellF1}\left (\frac {5}{4},1+n,\frac {3}{2},\frac {9}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+2 n \operatorname {AppellF1}\left (\frac {5}{4},1+n,\frac {5}{2},\frac {9}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) (-1+\cos (c+d x))+5 \operatorname {AppellF1}\left (\frac {1}{4},n,\frac {3}{2},\frac {5}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x))-5 \operatorname {AppellF1}\left (\frac {1}{4},n,\frac {5}{2},\frac {5}{4},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+\cos (c+d x))\right )} \]
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\[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \sin \left (d x +c \right )^{\frac {3}{2}}d x\]
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\[ \int (a+a \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\text {Timed out} \]
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\[ \int (a+a \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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\[ \int (a+a \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^n \sin ^{\frac {3}{2}}(c+d x) \, dx=\int {\sin \left (c+d\,x\right )}^{3/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]
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