Optimal. Leaf size=104 \[ -\frac{\cos (e+f x) (c+d \sin (e+f x))^n \left (\frac{c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac{1}{2};-n,3;\frac{3}{2};\frac{d (1-\sin (e+f x))}{c+d},\frac{1}{2} (1-\sin (e+f x))\right )}{4 a^2 f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.177884, antiderivative size = 137, normalized size of antiderivative = 1.32, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2788, 137, 136} \[ -\frac{d^2 \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac{1}{2},3;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d)^3 \sqrt{a \sin (e+f x)+a} \left (a^2-a^2 \sin (e+f x)\right )} \]
Warning: Unable to verify antiderivative.
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Rule 2788
Rule 137
Rule 136
Rubi steps
\begin{align*} \int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{5/2}} \, dx &=\frac{\left (a^2 \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{a-a x} (a+a x)^3} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\\ &=\frac{\left (a^2 \cos (e+f x) \sqrt{\frac{d (a-a \sin (e+f x))}{a c+a d}}\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{(a+a x)^3 \sqrt{\frac{a d}{a c+a d}-\frac{a d x}{a c+a d}}} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}}\\ &=-\frac{d^2 F_1\left (1+n;\frac{1}{2},3;2+n;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right ) \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{1+n}}{(c-d)^3 f (1+n) \sqrt{a+a \sin (e+f x)} \left (a^2-a^2 \sin (e+f x)\right )}\\ \end{align*}
Mathematica [B] time = 9.46624, size = 414, normalized size = 3.98 \[ \frac{\sec (e+f x) (c+d \sin (e+f x))^n \left (a^3 \sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1)^3 \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\frac{d (\sin (e+f x)+1)}{d-c}\right )-\frac{4 a^2 (\sin (e+f x)+1) \sqrt{1-\frac{2}{\sin (e+f x)+1}} \left (\frac{c-d}{d \sin (e+f x)+d}+1\right )^{-n} \left (a \left (4 n^2-8 n+3\right ) (\sin (e+f x)+1)^2 F_1\left (-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right )+2 (2 n+1) \left (2 a (2 n-1) F_1\left (\frac{3}{2}-n;-\frac{1}{2},-n;\frac{5}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right )+a (2 n-3) (\sin (e+f x)+1) F_1\left (\frac{1}{2}-n;-\frac{1}{2},-n;\frac{3}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right )\right )\right )}{(2 n-3) (2 n-1) (2 n+1)}\right )}{16 a^4 f (a (\sin (e+f x)+1))^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.158, size = 0, normalized size = 0. \begin{align*} \int{ \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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