Optimal. Leaf size=38 \[ \frac{\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)} \]
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Rubi [A] time = 0.0741083, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2833, 64} \[ \frac{\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 64
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{x}{a}\right )^n}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\, _2F_1(3,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^3 d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0337427, size = 38, normalized size = 1. \[ \frac{\sin ^{n+1}(c+d x) \, _2F_1(3,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.917, size = 0, normalized size = 0. \begin{align*} \int{\frac{\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{n}}{ \left ( a+a\sin \left ( dx+c \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{\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )}{{\left (a \sin \left (d x + c\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{\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )}{3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\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{\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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