Optimal. Leaf size=92 \[ \frac{\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac{2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac{2 \sin ^{n+4}(c+d x)}{a^2 d (n+4)}-\frac{\sin ^{n+5}(c+d x)}{a^2 d (n+5)} \]
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Rubi [A] time = 0.142898, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2836, 75} \[ \frac{\sin ^{n+1}(c+d x)}{a^2 d (n+1)}-\frac{2 \sin ^{n+2}(c+d x)}{a^2 d (n+2)}+\frac{2 \sin ^{n+4}(c+d x)}{a^2 d (n+4)}-\frac{\sin ^{n+5}(c+d x)}{a^2 d (n+5)} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 75
Rubi steps
\begin{align*} \int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 \left (\frac{x}{a}\right )^n (a+x) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^4 \left (\frac{x}{a}\right )^n-2 a^4 \left (\frac{x}{a}\right )^{1+n}+2 a^4 \left (\frac{x}{a}\right )^{3+n}-a^4 \left (\frac{x}{a}\right )^{4+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\sin ^{1+n}(c+d x)}{a^2 d (1+n)}-\frac{2 \sin ^{2+n}(c+d x)}{a^2 d (2+n)}+\frac{2 \sin ^{4+n}(c+d x)}{a^2 d (4+n)}-\frac{\sin ^{5+n}(c+d x)}{a^2 d (5+n)}\\ \end{align*}
Mathematica [A] time = 0.340678, size = 117, normalized size = 1.27 \[ \frac{\sin ^{n+1}(c+d x) \left (-\left (n^3+7 n^2+14 n+8\right ) \sin ^4(c+d x)+2 \left (n^3+8 n^2+17 n+10\right ) \sin ^3(c+d x)-2 \left (n^3+10 n^2+29 n+20\right ) \sin (c+d x)+n^3+11 n^2+38 n+40\right )}{a^2 d (n+1) (n+2) (n+4) (n+5)} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.848, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{7} \left ( \sin \left ( dx+c \right ) \right ) ^{n}}{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20429, size = 170, normalized size = 1.85 \begin{align*} -\frac{{\left ({\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \sin \left (d x + c\right )^{5} - 2 \,{\left (n^{3} + 8 \, n^{2} + 17 \, n + 10\right )} \sin \left (d x + c\right )^{4} + 2 \,{\left (n^{3} + 10 \, n^{2} + 29 \, n + 20\right )} \sin \left (d x + c\right )^{2} -{\left (n^{3} + 11 \, n^{2} + 38 \, n + 40\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{{\left (n^{4} + 12 \, n^{3} + 49 \, n^{2} + 78 \, n + 40\right )} a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22766, size = 414, normalized size = 4.5 \begin{align*} \frac{{\left (2 \,{\left (n^{3} + 8 \, n^{2} + 17 \, n + 10\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (n^{3} + 6 \, n^{2} + 5 \, n\right )} \cos \left (d x + c\right )^{2} - 4 \, n^{2} -{\left ({\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \cos \left (d x + c\right )^{2} - 4 \, n^{2} - 24 \, n - 32\right )} \sin \left (d x + c\right ) - 24 \, n - 20\right )} \sin \left (d x + c\right )^{n}}{a^{2} d n^{4} + 12 \, a^{2} d n^{3} + 49 \, a^{2} d n^{2} + 78 \, a^{2} d n + 40 \, a^{2} d} \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 [A] time = 1.58285, size = 127, normalized size = 1.38 \begin{align*} -\frac{\frac{\sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5}}{n + 5} - \frac{2 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4}}{n + 4} + \frac{2 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}}{n + 2} - \frac{\sin \left (d x + c\right )^{n + 1}}{n + 1}}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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