Optimal. Leaf size=60 \[ -\frac{(c+d \sin (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d)} \]
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Rubi [A] time = 0.124484, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2833, 68} \[ -\frac{(c+d \sin (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d)} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 68
Rubi steps
\begin{align*} \int \frac{\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{d x}{a}\right )^n}{a+x} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=-\frac{\, _2F_1\left (1,1+n;2+n;\frac{c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a (c-d) f (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0694949, size = 60, normalized size = 1. \[ -\frac{(c+d \sin (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.207, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( fx+e \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}}{a+a\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 (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{a \sin \left (f x + e\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 (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{a \sin \left (f x + e\right ) + a}, 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 (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{a \sin \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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