Optimal. Leaf size=77 \[ -\frac{2^{m+\frac{3}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},-m-\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{c f} \]
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Rubi [A] time = 0.164882, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {2840, 2652, 2651} \[ -\frac{2^{m+\frac{3}{2}} \cos (e+f x) (\sin (e+f x)+1)^{-m-\frac{1}{2}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac{1}{2},-m-\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x))\right )}{c f} \]
Antiderivative was successfully verified.
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Rule 2840
Rule 2652
Rule 2651
Rubi steps
\begin{align*} \int \frac{\cos ^2(e+f x) (a+a \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx &=\frac{\int (a+a \sin (e+f x))^{1+m} \, dx}{a c}\\ &=\frac{\left ((1+\sin (e+f x))^{-m} (a+a \sin (e+f x))^m\right ) \int (1+\sin (e+f x))^{1+m} \, dx}{c}\\ &=-\frac{2^{\frac{3}{2}+m} \cos (e+f x) \, _2F_1\left (\frac{1}{2},-\frac{1}{2}-m;\frac{3}{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))^m}{c f}\\ \end{align*}
Mathematica [C] time = 19.9326, size = 6442, normalized size = 83.66 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.348, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}}{c-c\sin \left ( fx+e \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 \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{c \sin \left (f x + e\right ) - c}\,{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 (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{c \sin \left (f x + e\right ) - c}, 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 (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )^{2}}{c \sin \left (f x + e\right ) - c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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