Optimal. Leaf size=216 \[ \frac{(A (1-2 m)-B (2 m+3)-C (2 m+7)) \cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left (1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right )}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}} \]
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Rubi [A] time = 0.677942, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.104, Rules used = {3035, 2973, 2745, 2667, 68} \[ \frac{(A (1-2 m)-B (2 m+3)-C (2 m+7)) \cos (e+f x) (a \sin (e+f x)+a)^m \, _2F_1\left (1,m+\frac{1}{2};m+\frac{3}{2};\frac{1}{2} (\sin (e+f x)+1)\right )}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(2 A m+A+2 B m+B+C (2 m+9)) \cos (e+f x) (a \sin (e+f x)+a)^m}{4 c f (2 m+1) \sqrt{c-c \sin (e+f x)}}+\frac{(A+B+C) \cos (e+f x) (a \sin (e+f x)+a)^{m+1}}{4 a f (c-c \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3035
Rule 2973
Rule 2745
Rule 2667
Rule 68
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^m \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{(c-c \sin (e+f x))^{3/2}} \, dx &=\frac{(A+B+C) \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac{\int \frac{(a+a \sin (e+f x))^m \left (-\frac{1}{2} a^2 (A (3-2 m)-(B+C) (5+2 m))+\frac{1}{2} a^2 (A+B+2 A m+2 B m+C (9+2 m)) \sin (e+f x)\right )}{\sqrt{c-c \sin (e+f x)}} \, dx}{4 a^2 c}\\ &=\frac{(A+B+C) \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{4 a f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B+2 A m+2 B m+C (9+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{4 c f (1+2 m) \sqrt{c-c \sin (e+f x)}}+\frac{(A (1-2 m)-B (3+2 m)-C (7+2 m)) \int \frac{(a+a \sin (e+f x))^m}{\sqrt{c-c \sin (e+f x)}} \, dx}{4 c}\\ &=\frac{(A+B+C) \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{4 a f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B+2 A m+2 B m+C (9+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{4 c f (1+2 m) \sqrt{c-c \sin (e+f x)}}+\frac{((A (1-2 m)-B (3+2 m)-C (7+2 m)) \cos (e+f x)) \int \sec (e+f x) (a+a \sin (e+f x))^{\frac{1}{2}+m} \, dx}{4 c \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{(A+B+C) \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{4 a f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B+2 A m+2 B m+C (9+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{4 c f (1+2 m) \sqrt{c-c \sin (e+f x)}}+\frac{(a (A (1-2 m)-B (3+2 m)-C (7+2 m)) \cos (e+f x)) \operatorname{Subst}\left (\int \frac{(a+x)^{-\frac{1}{2}+m}}{a-x} \, dx,x,a \sin (e+f x)\right )}{4 c f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{(A+B+C) \cos (e+f x) (a+a \sin (e+f x))^{1+m}}{4 a f (c-c \sin (e+f x))^{3/2}}+\frac{(A+B+2 A m+2 B m+C (9+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{4 c f (1+2 m) \sqrt{c-c \sin (e+f x)}}+\frac{(A (1-2 m)-B (3+2 m)-C (7+2 m)) \cos (e+f x) \, _2F_1\left (1,\frac{1}{2}+m;\frac{3}{2}+m;\frac{1}{2} (1+\sin (e+f x))\right ) (a+a \sin (e+f x))^m}{4 c f (1+2 m) \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 25.012, size = 14053, normalized size = 65.06 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.734, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) +C \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \left ( c-c\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 (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\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 (C \cos \left (f x + e\right )^{2} - B \sin \left (f x + e\right ) - A - C\right )} \sqrt{-c \sin \left (f x + e\right ) + c}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c^{2} \sin \left (f x + e\right ) - 2 \, c^{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 (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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