Optimal. Leaf size=59 \[ \frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^3} \]
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Rubi [A] time = 0.103192, antiderivative size = 59, 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} \[ \frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^3} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 68
Rubi steps
\begin{align*} \int \frac{\cos (e+f x) (a+a \sin (e+f x))^m}{(c+d \sin (e+f x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^m}{\left (c+\frac{d x}{a}\right )^3} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\, _2F_1\left (3,1+m;2+m;-\frac{d (1+\sin (e+f x))}{c-d}\right ) (a+a \sin (e+f x))^{1+m}}{a (c-d)^3 f (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0904704, size = 59, normalized size = 1. \[ \frac{(a \sin (e+f x)+a)^{m+1} \, _2F_1\left (3,m+1;m+2;-\frac{d (\sin (e+f x)+1)}{c-d}\right )}{a f (m+1) (c-d)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.373, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( fx+e \right ) \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}}{ \left ( c+d\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 (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{{\left (d \sin \left (f x + e\right ) + c\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 (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{3 \, c d^{2} \cos \left (f x + e\right )^{2} - c^{3} - 3 \, c d^{2} +{\left (d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{2} d - d^{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 (a \sin \left (f x + e\right ) + a\right )}^{m} \cos \left (f x + e\right )}{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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