Optimal. Leaf size=269 \[ \frac{d (A-B) \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac{1}{2},2;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d)^2 (a-a \sin (e+f x)) \sqrt{a \sin (e+f x)+a}}-\frac{B \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac{1}{2},1;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d) (1-\sin (e+f x)) \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.469883, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {2987, 2788, 137, 136} \[ \frac{d (A-B) \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac{1}{2},2;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d)^2 (a-a \sin (e+f x)) \sqrt{a \sin (e+f x)+a}}-\frac{B \cos (e+f x) \sqrt{\frac{d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac{1}{2},1;n+2;\frac{c+d \sin (e+f x)}{c+d},\frac{c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d) (1-\sin (e+f x)) \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2987
Rule 2788
Rule 137
Rule 136
Rubi steps
\begin{align*} \int \frac{(A+B \sin (e+f x)) (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx &=(A-B) \int \frac{(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx+\frac{B \int \frac{(c+d \sin (e+f x))^n}{\sqrt{a+a \sin (e+f x)}} \, dx}{a}\\ &=\frac{\left (a^2 (A-B) \cos (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{a-a x} (a+a x)^2} \, dx,x,\sin (e+f x)\right )}{f \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}+\frac{(a B \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(c+d x)^n}{\sqrt{a-a x} (a+a x)} \, 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 (A-B) \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)^2 \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{\left (a B \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) \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{B F_1\left (1+n;\frac{1}{2},1;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) f (1+n) (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}}+\frac{(A-B) d F_1\left (1+n;\frac{1}{2},2;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)^2 f (1+n) (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [B] time = 10.1795, size = 603, normalized size = 2.24 \[ \frac{\sec (e+f x) (c+d \sin (e+f x))^n \left (a A (\sin (e+f x)+1) \left (a \sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1) \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\frac{d (\sin (e+f x)+1)}{d-c}\right )-\frac{4 \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} \left (\frac{c-d}{d \sin (e+f x)+d}+1\right )^{-n} \left (2 a (2 n+1) F_1\left (\frac{1}{2}-n;-\frac{1}{2},-n;\frac{3}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right )+a (2 n-1) (\sin (e+f x)+1) F_1\left (-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right )\right )}{4 n^2-1}\right )+a B (\sin (e+f x)+1) \left (a \sqrt{2-2 \sin (e+f x)} (\sin (e+f x)+1) \left (\frac{c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (1;\frac{1}{2},-n;2;\frac{1}{2} (\sin (e+f x)+1),\frac{d (\sin (e+f x)+1)}{d-c}\right )-\frac{4 \sqrt{\frac{\sin (e+f x)-1}{\sin (e+f x)+1}} \left (\frac{c-d}{d \sin (e+f x)+d}+1\right )^{-n} \left (a (2 n-1) (\sin (e+f x)+1) F_1\left (-n-\frac{1}{2};-\frac{1}{2},-n;\frac{1}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right )-2 a (2 n+1) F_1\left (\frac{1}{2}-n;-\frac{1}{2},-n;\frac{3}{2}-n;\frac{2}{\sin (e+f x)+1},\frac{d-c}{\sin (e+f x) d+d}\right )\right )}{4 n^2-1}\right )\right )}{8 a^3 f \sqrt{a (\sin (e+f x)+1)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.345, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+B\sin \left ( fx+e \right ) \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{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 (B \sin \left (f x + e\right ) + A\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{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{{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt{a \sin \left (f x + e\right ) + a}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, 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 (B \sin \left (f x + e\right ) + A\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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