Optimal. Leaf size=85 \[ \frac{4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)}-\frac{3 \sin ^{n+1}(c+d x)}{a^3 d (n+1)}+\frac{\sin ^{n+2}(c+d x)}{a^3 d (n+2)} \]
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Rubi [A] time = 0.144604, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 88, 64} \[ \frac{4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^3 d (n+1)}-\frac{3 \sin ^{n+1}(c+d x)}{a^3 d (n+1)}+\frac{\sin ^{n+2}(c+d x)}{a^3 d (n+2)} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 88
Rule 64
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 \left (\frac{x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-3 a \left (\frac{x}{a}\right )^n+a \left (\frac{x}{a}\right )^{1+n}+\frac{4 a^2 \left (\frac{x}{a}\right )^n}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac{3 \sin ^{1+n}(c+d x)}{a^3 d (1+n)}+\frac{\sin ^{2+n}(c+d x)}{a^3 d (2+n)}+\frac{4 \operatorname{Subst}\left (\int \frac{\left (\frac{x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac{3 \sin ^{1+n}(c+d x)}{a^3 d (1+n)}+\frac{4 \, _2F_1(1,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^3 d (1+n)}+\frac{\sin ^{2+n}(c+d x)}{a^3 d (2+n)}\\ \end{align*}
Mathematica [A] time = 0.0939923, size = 64, normalized size = 0.75 \[ \frac{\sin ^{n+1}(c+d x) (4 (n+2) \, _2F_1(1,n+1;n+2;-\sin (c+d x))+(n+1) \sin (c+d x)-3 (n+2))}{a^3 d (n+1) (n+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.217, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{5} \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 )^{5}}{{\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 )^{5}}{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 )^{5}}{{\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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