Optimal. Leaf size=88 \[ -\frac{4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d}+\frac{\sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac{4 \sin ^{n+1}(c+d x)}{d \left (a^4 \sin (c+d x)+a^4\right )} \]
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Rubi [A] time = 0.137456, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2836, 89, 80, 64} \[ -\frac{4 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d}+\frac{\sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac{4 \sin ^{n+1}(c+d x)}{d \left (a^4 \sin (c+d x)+a^4\right )} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 89
Rule 80
Rule 64
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 \left (\frac{x}{a}\right )^n}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{4 \sin ^{1+n}(c+d x)}{d \left (a^4+a^4 \sin (c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{(a (3+4 n)-x) \left (\frac{x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac{4 \sin ^{1+n}(c+d x)}{d \left (a^4+a^4 \sin (c+d x)\right )}-\frac{(4 (1+n)) \operatorname{Subst}\left (\int \frac{\left (\frac{x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac{\sin ^{1+n}(c+d x)}{a^4 d (1+n)}-\frac{4 \, _2F_1(1,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^4 d}+\frac{4 \sin ^{1+n}(c+d x)}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.105929, size = 72, normalized size = 0.82 \[ \frac{\sin ^{n+1}(c+d x) (-4 (n+1) (\sin (c+d x)+1) \, _2F_1(1,n+1;n+2;-\sin (c+d x))+\sin (c+d x)+4 n+5)}{a^4 d (n+1) (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.464, 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 ) ^{4}}}\, 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 )}^{4}}\,{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}}{a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \,{\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\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 )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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