Optimal. Leaf size=95 \[ -\frac{2^{\frac{p}{2}-\frac{1}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left (\frac{3-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{a d e (p+1)} \]
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Rubi [A] time = 0.0990199, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2688, 69} \[ -\frac{2^{\frac{p}{2}-\frac{1}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left (\frac{3-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{a d e (p+1)} \]
Antiderivative was successfully verified.
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Rule 2688
Rule 69
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^p}{a+a \sin (c+d x)} \, dx &=\frac{\left ((e \cos (c+d x))^{1+p} (1-\sin (c+d x))^{\frac{1}{2} (-1-p)} (1+\sin (c+d x))^{\frac{1}{2} (-1-p)}\right ) \operatorname{Subst}\left (\int (1-x)^{\frac{1}{2} (-1+p)} (1+x)^{-1+\frac{1}{2} (-1+p)} \, dx,x,\sin (c+d x)\right )}{a d e}\\ &=-\frac{2^{-\frac{1}{2}+\frac{p}{2}} (e \cos (c+d x))^{1+p} \, _2F_1\left (\frac{3-p}{2},\frac{1+p}{2};\frac{3+p}{2};\frac{1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{\frac{1}{2} (-1-p)}}{a d e (1+p)}\\ \end{align*}
Mathematica [A] time = 0.165018, size = 94, normalized size = 0.99 \[ -\frac{2^{\frac{p-1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^p \, _2F_1\left (\frac{3-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{a d (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.133, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( e\cos \left ( dx+c \right ) \right ) ^{p}}{a+a\sin \left ( dx+c \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 (e \cos \left (d x + c\right )\right )^{p}}{a \sin \left (d x + c\right ) + a}\,{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 (e \cos \left (d x + c\right )\right )^{p}}{a \sin \left (d x + c\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\left (e \cos{\left (c + d x \right )}\right )^{p}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \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 (e \cos \left (d x + c\right )\right )^{p}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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