Optimal. Leaf size=97 \[ \frac{2^{1-\frac{m}{2}} (1-\sin (c+d x))^{\frac{m}{2}-1} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{2-m} \, _2F_1\left (\frac{m}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{d e (m+2)} \]
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Rubi [A] time = 0.106128, antiderivative size = 97, normalized size of antiderivative = 1., 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 = {2689, 70, 69} \[ \frac{2^{1-\frac{m}{2}} (1-\sin (c+d x))^{\frac{m}{2}-1} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{2-m} \, _2F_1\left (\frac{m}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{d e (m+2)} \]
Antiderivative was successfully verified.
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Rule 2689
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx &=\frac{\left (a^2 (e \cos (c+d x))^{2-m} (a-a \sin (c+d x))^{\frac{1}{2} (-2+m)} (a+a \sin (c+d x))^{\frac{1}{2} (-2+m)}\right ) \operatorname{Subst}\left (\int (a-a x)^{-m/2} (a+a x)^{m/2} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac{\left (2^{-m/2} a^2 (e \cos (c+d x))^{2-m} (a-a \sin (c+d x))^{\frac{1}{2} (-2+m)-\frac{m}{2}} \left (\frac{a-a \sin (c+d x)}{a}\right )^{m/2} (a+a \sin (c+d x))^{\frac{1}{2} (-2+m)}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{x}{2}\right )^{-m/2} (a+a x)^{m/2} \, dx,x,\sin (c+d x)\right )}{d e}\\ &=\frac{2^{1-\frac{m}{2}} (e \cos (c+d x))^{2-m} \, _2F_1\left (\frac{m}{2},\frac{2+m}{2};\frac{4+m}{2};\frac{1}{2} (1+\sin (c+d x))\right ) (1-\sin (c+d x))^{-1+\frac{m}{2}} (a+a \sin (c+d x))^m}{d e (2+m)}\\ \end{align*}
Mathematica [A] time = 0.254181, size = 97, normalized size = 1. \[ \frac{2^{\frac{m}{2}+1} (\sin (c+d x)+1)^{-\frac{m}{2}-1} (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{2-m} \, _2F_1\left (1-\frac{m}{2},-\frac{m}{2};2-\frac{m}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{d e (m-2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.15, size = 0, normalized size = 0. \begin{align*} \int \left ( e\cos \left ( dx+c \right ) \right ) ^{1-m} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{m}\, 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 (e \cos \left (d x + c\right )\right )^{-m + 1}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{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 (e \cos \left (d x + c\right )\right )^{-m + 1}{\left (a \sin \left (d x + c\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 (e \cos \left (d x + c\right )\right )^{-m + 1}{\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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