Optimal. Leaf size=63 \[ -\frac{d^2 (c+d \sin (e+f x))^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right )}{a^3 f (n+1) (c-d)^3} \]
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Rubi [A] time = 0.117261, antiderivative size = 63, 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} \[ -\frac{d^2 (c+d \sin (e+f x))^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{c+d \sin (e+f x)}{c-d}\right )}{a^3 f (n+1) (c-d)^3} \]
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))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{d x}{a}\right )^n}{(a+x)^3} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=-\frac{d^2 \, _2F_1\left (3,1+n;2+n;\frac{c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a^3 (c-d)^3 f (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0622529, size = 63, normalized size = 1. \[ \frac{d^2 (c+d \sin (e+f x))^{n+1} \, _2F_1\left (3,n+1;n+2;-\frac{c+d \sin (e+f x)}{d-c}\right )}{a^3 f (n+1) (d-c)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.088, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( fx+e \right ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{n}}{ \left ( a+a\sin \left ( fx+e \right ) \right ) ^{3}}}\, 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 )}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}\,{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 )}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, 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 )}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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