Optimal. Leaf size=84 \[ -\frac{a^2 c 2^{m+\frac{3}{2}} \cos ^5(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left (\frac{5}{2},-m-\frac{1}{2};\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{5 f} \]
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Rubi [A] time = 0.155804, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2840, 2689, 70, 69} \[ -\frac{a^2 c 2^{m+\frac{3}{2}} \cos ^5(e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^{m-2} \, _2F_1\left (\frac{5}{2},-m-\frac{1}{2};\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{5 f} \]
Antiderivative was successfully verified.
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Rule 2840
Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x)) \, dx &=(a c) \int \cos ^4(e+f x) (a+a \sin (e+f x))^{-1+m} \, dx\\ &=\frac{\left (a^3 c \cos ^5(e+f x)\right ) \operatorname{Subst}\left (\int (a-a x)^{3/2} (a+a x)^{\frac{1}{2}+m} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}\\ &=\frac{\left (2^{\frac{1}{2}+m} a^3 c \cos ^5(e+f x) (a+a \sin (e+f x))^{-2+m} \left (\frac{a+a \sin (e+f x)}{a}\right )^{-\frac{1}{2}-m}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}+\frac{x}{2}\right )^{\frac{1}{2}+m} (a-a x)^{3/2} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x))^{5/2}}\\ &=-\frac{2^{\frac{3}{2}+m} a^2 c \cos ^5(e+f x) \, _2F_1\left (\frac{5}{2},-\frac{1}{2}-m;\frac{7}{2};\frac{1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac{1}{2}-m} (a+a \sin (e+f x))^{-2+m}}{5 f}\\ \end{align*}
Mathematica [F] time = 180.003, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [F] time = 1.707, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) \, 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{\left (c \sin \left (f x + e\right ) - c\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{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 (-{\left (c \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) - c \cos \left (f x + e\right )^{2}\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}, 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 -{\left (c \sin \left (f x + e\right ) - c\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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